part i. order, posets, lattices and residuated lattices in...
TRANSCRIPT
![Page 1: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/1.jpg)
Introduction Introduction Introduction
PART I.Order, posets, lattices and residuated lattices in
logic
October 22, 2007
[Latest updated version]
PART I. Order, posets, lattices and residuated lattices in logic
![Page 2: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/2.jpg)
Introduction Introduction Introduction
Applied Logics
Classical Boolean logic is the logic of mathematics, whosehistorical challenger has been Intuitionistic Logic.
Recently, there has appeared applied logical systems such as• Girard’s Linear Logic,• Zadeh’s Fuzzy Logic,• Hajek’s GUHA Logic.In 1920’s Lukasiewicz introduced Many-valued Logic.All such logics are special cases of Hohle’s Monoidal Logic; we arenow going to study this logic from an algebraic point of view; eachabove mentioned logic has an algebraic counterpart which is aresiduated lattice.We are going to show that first order Monoidal Logic and all itspresented axiomatic extensions are complete logics.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 3: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/3.jpg)
Introduction Introduction Introduction
Applied Logics
Classical Boolean logic is the logic of mathematics, whosehistorical challenger has been Intuitionistic Logic.Recently, there has appeared applied logical systems such as• Girard’s Linear Logic,
• Zadeh’s Fuzzy Logic,• Hajek’s GUHA Logic.In 1920’s Lukasiewicz introduced Many-valued Logic.All such logics are special cases of Hohle’s Monoidal Logic; we arenow going to study this logic from an algebraic point of view; eachabove mentioned logic has an algebraic counterpart which is aresiduated lattice.We are going to show that first order Monoidal Logic and all itspresented axiomatic extensions are complete logics.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 4: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/4.jpg)
Introduction Introduction Introduction
Applied Logics
Classical Boolean logic is the logic of mathematics, whosehistorical challenger has been Intuitionistic Logic.Recently, there has appeared applied logical systems such as• Girard’s Linear Logic,• Zadeh’s Fuzzy Logic,
• Hajek’s GUHA Logic.In 1920’s Lukasiewicz introduced Many-valued Logic.All such logics are special cases of Hohle’s Monoidal Logic; we arenow going to study this logic from an algebraic point of view; eachabove mentioned logic has an algebraic counterpart which is aresiduated lattice.We are going to show that first order Monoidal Logic and all itspresented axiomatic extensions are complete logics.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 5: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/5.jpg)
Introduction Introduction Introduction
Applied Logics
Classical Boolean logic is the logic of mathematics, whosehistorical challenger has been Intuitionistic Logic.Recently, there has appeared applied logical systems such as• Girard’s Linear Logic,• Zadeh’s Fuzzy Logic,• Hajek’s GUHA Logic.
In 1920’s Lukasiewicz introduced Many-valued Logic.All such logics are special cases of Hohle’s Monoidal Logic; we arenow going to study this logic from an algebraic point of view; eachabove mentioned logic has an algebraic counterpart which is aresiduated lattice.We are going to show that first order Monoidal Logic and all itspresented axiomatic extensions are complete logics.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 6: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/6.jpg)
Introduction Introduction Introduction
Applied Logics
Classical Boolean logic is the logic of mathematics, whosehistorical challenger has been Intuitionistic Logic.Recently, there has appeared applied logical systems such as• Girard’s Linear Logic,• Zadeh’s Fuzzy Logic,• Hajek’s GUHA Logic.In 1920’s Lukasiewicz introduced Many-valued Logic.
All such logics are special cases of Hohle’s Monoidal Logic; we arenow going to study this logic from an algebraic point of view; eachabove mentioned logic has an algebraic counterpart which is aresiduated lattice.We are going to show that first order Monoidal Logic and all itspresented axiomatic extensions are complete logics.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 7: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/7.jpg)
Introduction Introduction Introduction
Applied Logics
Classical Boolean logic is the logic of mathematics, whosehistorical challenger has been Intuitionistic Logic.Recently, there has appeared applied logical systems such as• Girard’s Linear Logic,• Zadeh’s Fuzzy Logic,• Hajek’s GUHA Logic.In 1920’s Lukasiewicz introduced Many-valued Logic.All such logics are special cases of Hohle’s Monoidal Logic; we arenow going to study this logic from an algebraic point of view; eachabove mentioned logic has an algebraic counterpart which is aresiduated lattice.
We are going to show that first order Monoidal Logic and all itspresented axiomatic extensions are complete logics.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 8: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/8.jpg)
Introduction Introduction Introduction
Applied Logics
Classical Boolean logic is the logic of mathematics, whosehistorical challenger has been Intuitionistic Logic.Recently, there has appeared applied logical systems such as• Girard’s Linear Logic,• Zadeh’s Fuzzy Logic,• Hajek’s GUHA Logic.In 1920’s Lukasiewicz introduced Many-valued Logic.All such logics are special cases of Hohle’s Monoidal Logic; we arenow going to study this logic from an algebraic point of view; eachabove mentioned logic has an algebraic counterpart which is aresiduated lattice.We are going to show that first order Monoidal Logic and all itspresented axiomatic extensions are complete logics.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 9: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/9.jpg)
Introduction Introduction Introduction
Order is a fundamental concept in logic. Indeed, already in twovalued logic we assume that the truth value true is in some sense’more’ or ’bigger’ than the truth value false. When talking aboutlogics were truth is by degrees we implicitely assume that thesedegrees are put to some order, the top and bottom elements beingtrue and false, respectively.
Moreover, it is natural to assume that,given two degrees of truth, if they are not comparable, they atleast have the greatest lower bound and the least upper bound. Inthis way we have entered the realm of lattices. Anotherfundamental assumpion is that logical connectives and and impliesare bound by the following: a sentence α and β has at most thesame degree of truth that a sentence γ if, and only if α has atmost the same degree of truth that a sentence β implies γ. Thenwe talk about residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 10: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/10.jpg)
Introduction Introduction Introduction
Order is a fundamental concept in logic. Indeed, already in twovalued logic we assume that the truth value true is in some sense’more’ or ’bigger’ than the truth value false. When talking aboutlogics were truth is by degrees we implicitely assume that thesedegrees are put to some order, the top and bottom elements beingtrue and false, respectively. Moreover, it is natural to assume that,given two degrees of truth, if they are not comparable, they atleast have the greatest lower bound and the least upper bound. Inthis way we have entered the realm of lattices.
Anotherfundamental assumpion is that logical connectives and and impliesare bound by the following: a sentence α and β has at most thesame degree of truth that a sentence γ if, and only if α has atmost the same degree of truth that a sentence β implies γ. Thenwe talk about residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 11: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/11.jpg)
Introduction Introduction Introduction
Order is a fundamental concept in logic. Indeed, already in twovalued logic we assume that the truth value true is in some sense’more’ or ’bigger’ than the truth value false. When talking aboutlogics were truth is by degrees we implicitely assume that thesedegrees are put to some order, the top and bottom elements beingtrue and false, respectively. Moreover, it is natural to assume that,given two degrees of truth, if they are not comparable, they atleast have the greatest lower bound and the least upper bound. Inthis way we have entered the realm of lattices. Anotherfundamental assumpion is that logical connectives and and impliesare bound by the following: a sentence α and β has at most thesame degree of truth that a sentence γ if, and only if α has atmost the same degree of truth that a sentence β implies γ. Thenwe talk about residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 12: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/12.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 13: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/13.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 14: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/14.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 15: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/15.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 16: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/16.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.
A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 17: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/17.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤.
A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 18: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/18.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 19: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/19.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 20: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/20.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 21: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/21.jpg)
Introduction Introduction Introduction
Definition
Assume A is a non–void set and R a binary relation on A. R is apre–order (or quasi–order) on A if
R is reflexive if ∀x ∈ A, xRx and
R is transitive if xRy , yRz imlies xRz where x , y , z ∈ A.
A pre–order R is a partial order on A if
R is anti-symmetric: if xRy , yRx then x = y where x , y ∈ A.A partial order is denoted by ≤. A partial order is a totalorder on A if ∀x , y ∈ A, x ≤ y or y ≤ x .
A pre–order R is an equivalence on A if
R is symmetric: if xRy then yRx where x , y ∈ A.An equivalence relation on A is denoted by ∼.
Denote |x | = {y ∈ A|x ∼ y}, x ∈ A, the equivalence class definedby x , A/ ∼= {|x |; x ∈ A}, set of all equivalence classes.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 22: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/22.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,x ∼ y iff |x | = |y | iff x ∈ |y |,if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A. If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f . Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 23: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/23.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,x ∼ y iff |x | = |y | iff x ∈ |y |,if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A. If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f . Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 24: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/24.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,
x ∼ y iff |x | = |y | iff x ∈ |y |,if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A. If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f . Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 25: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/25.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,x ∼ y iff |x | = |y | iff x ∈ |y |,
if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A. If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f . Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 26: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/26.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,x ∼ y iff |x | = |y | iff x ∈ |y |,if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A. If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f . Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 27: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/27.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,x ∼ y iff |x | = |y | iff x ∈ |y |,if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A. If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f . Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 28: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/28.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,x ∼ y iff |x | = |y | iff x ∈ |y |,if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A.
If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f . Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 29: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/29.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,x ∼ y iff |x | = |y | iff x ∈ |y |,if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A. If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f .
Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 30: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/30.jpg)
Introduction Introduction Introduction
A poset A is called a chain if the partial order ≤ is a total order.
Lemma (1)
Let ∼ is an equivalence relation on A. Then for all x , y ∈ A
x ∈ |x |,x ∼ y iff |x | = |y | iff x ∈ |y |,if x ∼ y does not hold, then |x | ∩ |y | = ∅.
Proof. Exercise.
Let ∼ is an equivalence relation on A and f : Am 7→ A an m–aryoperation on A. If conditions a1 ∼ b1, · · · , am ∼ bm implyf (a1, · · · , am) ∼ f (b1, · · · , bm) then ∼ is a congruence withrespect to f . Then define g(|a1|, · · · , |am|) = |f (a1), · · · , f (am)|and we obtain m–ary operation on A/ ∼.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 31: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/31.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A. Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E. Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}. A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y . An upper boundand a lower bound of any subset X ⊆ A are defined similarly. Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 32: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/32.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A.
Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E. Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}. A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y . An upper boundand a lower bound of any subset X ⊆ A are defined similarly. Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 33: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/33.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A. Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E.
Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}. A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y . An upper boundand a lower bound of any subset X ⊆ A are defined similarly. Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 34: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/34.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A. Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E. Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}. A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y . An upper boundand a lower bound of any subset X ⊆ A are defined similarly. Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 35: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/35.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A. Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E. Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.
Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}. A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y . An upper boundand a lower bound of any subset X ⊆ A are defined similarly. Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 36: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/36.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A. Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E. Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}.
A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y . An upper boundand a lower bound of any subset X ⊆ A are defined similarly. Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 37: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/37.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A. Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E. Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}. A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y .
An upper boundand a lower bound of any subset X ⊆ A are defined similarly. Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 38: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/38.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A. Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E. Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}. A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y . An upper boundand a lower bound of any subset X ⊆ A are defined similarly.
Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 39: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/39.jpg)
Introduction Introduction Introduction
Lemma (2)
Let R be a pre–order on A. Define a binary operation E on A by
xEy iff xRy and yRx.
Then E is an equivalence relation on A. Moreover, for eachx , y ∈ A define on A/E a binary relation S by
|x |S |y | iff xRy.
Then S is an order relation on A/E. Proof. Exercise.
If there is a partial order ≤ on a set A, then A is called a poset.Assume A is a poset and x , y ∈ A. Any element z ∈ A such thatx , y ≤ z is called an upper bound of {x , y}. A lower bound of{x , y} is an element w ∈ A such that w ≤ x , y . An upper boundand a lower bound of any subset X ⊆ A are defined similarly. Theleast upper bound of {x , y} is an upper bound z such that z ≤ z0
for any other upper bound of {x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 40: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/40.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly.
We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 41: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/41.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively.
If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 42: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/42.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice.
A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 43: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/43.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A.
It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 44: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/44.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 45: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/45.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 46: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/46.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 47: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/47.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 48: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/48.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 49: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/49.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 50: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/50.jpg)
Introduction Introduction Introduction
The greatest lower bound of {x , y} is defined similarly. We denoteby x ∧ y and x ∨ y the greatest lower bound and the least upperbound of {x , y}, respectively. If x ∧ y and x ∨ y exist in A for allelements x , y ∈ A, then A is a lattice. A lattice A is (countable)complete if
∨{x |x ∈ X} and
∧{x |x ∈ X} exist in A for any
(countable) subset X ⊆ A. It is an exercise to prove the following
Lemma (3)
Let A be a poset. Then (whenever the equations exist in A),
x ∧ x = x, x ∨ x = x (idempotency)
x ∧ y = y ∧ x, x ∨ y = y ∨ x (commutativity)
(x ∧ y) ∧ z = x ∧ (y ∧ z), (x ∨ y) ∨ z = x ∨ (y ∨ z) (assoc.)
x ∧ (x ∨ y) = x ∨ (x ∧ y) = x (absoption)
x ≤ y iff x ∧ y = x iff x ∨ y = y (consistency)
∧ and ∨ are isotone.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 51: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/51.jpg)
Introduction Introduction Introduction
Lemma (4)
Let L be a lattice. Then for all elements x , y , z ∈ L,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (1)
if, and only if x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (2)
Proof. We show that (1) implies (2). The converse is similar.
x ∨ (y ∧ z) = x ∨ (z ∧ y) (commut.)
= [(x ∨ (z ∧ x)] ∨ (z ∧ y) (absorb.)
= (x ∨ [(z ∧ x) ∨ (z ∧ y)] (assoc.)
= (x ∨ [(z ∧ (x ∨ y)] (by (1))
= [x ∧ (x ∨ y)] ∨ [z ∧ (x ∨ y)] (absorb.)
= [(x ∨ y) ∧ x ] ∨ [(x ∨ y) ∧ z ] (commut.)
= (x ∨ y) ∧ (x ∨ z) (by (1)).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 52: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/52.jpg)
Introduction Introduction Introduction
Lemma (4)
Let L be a lattice. Then for all elements x , y , z ∈ L,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (1)
if, and only if x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (2)
Proof. We show that (1) implies (2). The converse is similar.
x ∨ (y ∧ z) = x ∨ (z ∧ y) (commut.)
= [(x ∨ (z ∧ x)] ∨ (z ∧ y) (absorb.)
= (x ∨ [(z ∧ x) ∨ (z ∧ y)] (assoc.)
= (x ∨ [(z ∧ (x ∨ y)] (by (1))
= [x ∧ (x ∨ y)] ∨ [z ∧ (x ∨ y)] (absorb.)
= [(x ∨ y) ∧ x ] ∨ [(x ∨ y) ∧ z ] (commut.)
= (x ∨ y) ∧ (x ∨ z) (by (1)).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 53: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/53.jpg)
Introduction Introduction Introduction
Lemma (4)
Let L be a lattice. Then for all elements x , y , z ∈ L,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (1)
if, and only if x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (2)
Proof. We show that (1) implies (2). The converse is similar.
x ∨ (y ∧ z) = x ∨ (z ∧ y) (commut.)
= [(x ∨ (z ∧ x)] ∨ (z ∧ y) (absorb.)
= (x ∨ [(z ∧ x) ∨ (z ∧ y)] (assoc.)
= (x ∨ [(z ∧ (x ∨ y)] (by (1))
= [x ∧ (x ∨ y)] ∨ [z ∧ (x ∨ y)] (absorb.)
= [(x ∨ y) ∧ x ] ∨ [(x ∨ y) ∧ z ] (commut.)
= (x ∨ y) ∧ (x ∨ z) (by (1)).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 54: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/54.jpg)
Introduction Introduction Introduction
Lemma (4)
Let L be a lattice. Then for all elements x , y , z ∈ L,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (1)
if, and only if x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (2)
Proof. We show that (1) implies (2). The converse is similar.
x ∨ (y ∧ z) = x ∨ (z ∧ y) (commut.)
= [(x ∨ (z ∧ x)] ∨ (z ∧ y) (absorb.)
= (x ∨ [(z ∧ x) ∨ (z ∧ y)] (assoc.)
= (x ∨ [(z ∧ (x ∨ y)] (by (1))
= [x ∧ (x ∨ y)] ∨ [z ∧ (x ∨ y)] (absorb.)
= [(x ∨ y) ∧ x ] ∨ [(x ∨ y) ∧ z ] (commut.)
= (x ∨ y) ∧ (x ∨ z) (by (1)).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 55: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/55.jpg)
Introduction Introduction Introduction
Lemma (4)
Let L be a lattice. Then for all elements x , y , z ∈ L,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (1)
if, and only if x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (2)
Proof. We show that (1) implies (2). The converse is similar.
x ∨ (y ∧ z) = x ∨ (z ∧ y) (commut.)
= [(x ∨ (z ∧ x)] ∨ (z ∧ y) (absorb.)
= (x ∨ [(z ∧ x) ∨ (z ∧ y)] (assoc.)
= (x ∨ [(z ∧ (x ∨ y)] (by (1))
= [x ∧ (x ∨ y)] ∨ [z ∧ (x ∨ y)] (absorb.)
= [(x ∨ y) ∧ x ] ∨ [(x ∨ y) ∧ z ] (commut.)
= (x ∨ y) ∧ (x ∨ z) (by (1)).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 56: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/56.jpg)
Introduction Introduction Introduction
Lemma (4)
Let L be a lattice. Then for all elements x , y , z ∈ L,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (1)
if, and only if x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (2)
Proof. We show that (1) implies (2). The converse is similar.
x ∨ (y ∧ z) = x ∨ (z ∧ y) (commut.)
= [(x ∨ (z ∧ x)] ∨ (z ∧ y) (absorb.)
= (x ∨ [(z ∧ x) ∨ (z ∧ y)] (assoc.)
= (x ∨ [(z ∧ (x ∨ y)] (by (1))
= [x ∧ (x ∨ y)] ∨ [z ∧ (x ∨ y)] (absorb.)
= [(x ∨ y) ∧ x ] ∨ [(x ∨ y) ∧ z ] (commut.)
= (x ∨ y) ∧ (x ∨ z) (by (1)).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 57: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/57.jpg)
Introduction Introduction Introduction
Lemma (4)
Let L be a lattice. Then for all elements x , y , z ∈ L,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (1)
if, and only if x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (2)
Proof. We show that (1) implies (2). The converse is similar.
x ∨ (y ∧ z) = x ∨ (z ∧ y) (commut.)
= [(x ∨ (z ∧ x)] ∨ (z ∧ y) (absorb.)
= (x ∨ [(z ∧ x) ∨ (z ∧ y)] (assoc.)
= (x ∨ [(z ∧ (x ∨ y)] (by (1))
= [x ∧ (x ∨ y)] ∨ [z ∧ (x ∨ y)] (absorb.)
= [(x ∨ y) ∧ x ] ∨ [(x ∨ y) ∧ z ] (commut.)
= (x ∨ y) ∧ (x ∨ z) (by (1)).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 58: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/58.jpg)
Introduction Introduction Introduction
A lattice L such that equation (1) holds (thus (2) holds, too) iscalled a distributive lattice.
Definition (Boolean algebra)
Assume in a distributive lattice L, for all elements x ∈ L, there isan element x∗ ∈ L (called lattice complement of x) such that, forall y ∈ L holds
(x ∧ x∗) ∨ y = y and (x ∨ x∗) ∧ y = y .
Then the lattice L = 〈L,≤,∨,∧,∗ 〉 is a Boolean algebra.
Exercise. Prove that in a Boolean algebra (a) the latticecomplement x∗ of x ∈ L is unique, (b) for all x , y ∈ L,x ∧ x∗ = y ∧ y∗ and x ∨ x∗ = y ∨ y∗, (c) x ∧ x∗ is the leastelement of L (a therefore noted by 0), (d) x ∨ x∗ is the greatestelement of L (a therefore noted by 1).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 59: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/59.jpg)
Introduction Introduction Introduction
A lattice L such that equation (1) holds (thus (2) holds, too) iscalled a distributive lattice.
Definition (Boolean algebra)
Assume in a distributive lattice L, for all elements x ∈ L, there isan element x∗ ∈ L (called lattice complement of x) such that, forall y ∈ L holds
(x ∧ x∗) ∨ y = y and (x ∨ x∗) ∧ y = y .
Then the lattice L = 〈L,≤,∨,∧,∗ 〉 is a Boolean algebra.
Exercise. Prove that in a Boolean algebra (a) the latticecomplement x∗ of x ∈ L is unique, (b) for all x , y ∈ L,x ∧ x∗ = y ∧ y∗ and x ∨ x∗ = y ∨ y∗, (c) x ∧ x∗ is the leastelement of L (a therefore noted by 0), (d) x ∨ x∗ is the greatestelement of L (a therefore noted by 1).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 60: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/60.jpg)
Introduction Introduction Introduction
A lattice L such that equation (1) holds (thus (2) holds, too) iscalled a distributive lattice.
Definition (Boolean algebra)
Assume in a distributive lattice L, for all elements x ∈ L, there isan element x∗ ∈ L (called lattice complement of x) such that, forall y ∈ L holds
(x ∧ x∗) ∨ y = y and (x ∨ x∗) ∧ y = y .
Then the lattice L = 〈L,≤,∨,∧,∗ 〉 is a Boolean algebra.
Exercise. Prove that in a Boolean algebra (a) the latticecomplement x∗ of x ∈ L is unique,
(b) for all x , y ∈ L,x ∧ x∗ = y ∧ y∗ and x ∨ x∗ = y ∨ y∗, (c) x ∧ x∗ is the leastelement of L (a therefore noted by 0), (d) x ∨ x∗ is the greatestelement of L (a therefore noted by 1).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 61: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/61.jpg)
Introduction Introduction Introduction
A lattice L such that equation (1) holds (thus (2) holds, too) iscalled a distributive lattice.
Definition (Boolean algebra)
Assume in a distributive lattice L, for all elements x ∈ L, there isan element x∗ ∈ L (called lattice complement of x) such that, forall y ∈ L holds
(x ∧ x∗) ∨ y = y and (x ∨ x∗) ∧ y = y .
Then the lattice L = 〈L,≤,∨,∧,∗ 〉 is a Boolean algebra.
Exercise. Prove that in a Boolean algebra (a) the latticecomplement x∗ of x ∈ L is unique, (b) for all x , y ∈ L,x ∧ x∗ = y ∧ y∗ and x ∨ x∗ = y ∨ y∗,
(c) x ∧ x∗ is the leastelement of L (a therefore noted by 0), (d) x ∨ x∗ is the greatestelement of L (a therefore noted by 1).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 62: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/62.jpg)
Introduction Introduction Introduction
A lattice L such that equation (1) holds (thus (2) holds, too) iscalled a distributive lattice.
Definition (Boolean algebra)
Assume in a distributive lattice L, for all elements x ∈ L, there isan element x∗ ∈ L (called lattice complement of x) such that, forall y ∈ L holds
(x ∧ x∗) ∨ y = y and (x ∨ x∗) ∧ y = y .
Then the lattice L = 〈L,≤,∨,∧,∗ 〉 is a Boolean algebra.
Exercise. Prove that in a Boolean algebra (a) the latticecomplement x∗ of x ∈ L is unique, (b) for all x , y ∈ L,x ∧ x∗ = y ∧ y∗ and x ∨ x∗ = y ∨ y∗, (c) x ∧ x∗ is the leastelement of L (a therefore noted by 0),
(d) x ∨ x∗ is the greatestelement of L (a therefore noted by 1).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 63: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/63.jpg)
Introduction Introduction Introduction
A lattice L such that equation (1) holds (thus (2) holds, too) iscalled a distributive lattice.
Definition (Boolean algebra)
Assume in a distributive lattice L, for all elements x ∈ L, there isan element x∗ ∈ L (called lattice complement of x) such that, forall y ∈ L holds
(x ∧ x∗) ∨ y = y and (x ∨ x∗) ∧ y = y .
Then the lattice L = 〈L,≤,∨,∧,∗ 〉 is a Boolean algebra.
Exercise. Prove that in a Boolean algebra (a) the latticecomplement x∗ of x ∈ L is unique, (b) for all x , y ∈ L,x ∧ x∗ = y ∧ y∗ and x ∨ x∗ = y ∨ y∗, (c) x ∧ x∗ is the leastelement of L (a therefore noted by 0), (d) x ∨ x∗ is the greatestelement of L (a therefore noted by 1).
PART I. Order, posets, lattices and residuated lattices in logic
![Page 64: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/64.jpg)
Introduction Introduction Introduction
Not all lattices are distributive and a distributive lattice need not tobe a Boolean algebra.
A lattice L is called completely distributive if
x ∧∨i∈Γ
yi =∨i∈Γ
(x ∧ yi ), (3)
x ∨∧i∈Γ
yi =∧i∈Γ
(x ∨ yi ) (4)
hold for all elements x ∈ L and all subsets {yi |i ∈ Γ} ⊆ L. Ofcourse, only complete lattices can (but need not) be completelydistributive. Unlike equations (1) and (2), the equations (3) and(4) are not equivalent conditions. It is easy to find examples oflattices that do not contain the least element 0 nor the largestelement 1. In logic applications is, however, natural to assume thatwe have the elements 0 and 1 available. We are now ready todefine residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 65: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/65.jpg)
Introduction Introduction Introduction
Not all lattices are distributive and a distributive lattice need not tobe a Boolean algebra. A lattice L is called completely distributive if
x ∧∨i∈Γ
yi =∨i∈Γ
(x ∧ yi ), (3)
x ∨∧i∈Γ
yi =∧i∈Γ
(x ∨ yi ) (4)
hold for all elements x ∈ L and all subsets {yi |i ∈ Γ} ⊆ L.
Ofcourse, only complete lattices can (but need not) be completelydistributive. Unlike equations (1) and (2), the equations (3) and(4) are not equivalent conditions. It is easy to find examples oflattices that do not contain the least element 0 nor the largestelement 1. In logic applications is, however, natural to assume thatwe have the elements 0 and 1 available. We are now ready todefine residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 66: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/66.jpg)
Introduction Introduction Introduction
Not all lattices are distributive and a distributive lattice need not tobe a Boolean algebra. A lattice L is called completely distributive if
x ∧∨i∈Γ
yi =∨i∈Γ
(x ∧ yi ), (3)
x ∨∧i∈Γ
yi =∧i∈Γ
(x ∨ yi ) (4)
hold for all elements x ∈ L and all subsets {yi |i ∈ Γ} ⊆ L. Ofcourse, only complete lattices can (but need not) be completelydistributive.
Unlike equations (1) and (2), the equations (3) and(4) are not equivalent conditions. It is easy to find examples oflattices that do not contain the least element 0 nor the largestelement 1. In logic applications is, however, natural to assume thatwe have the elements 0 and 1 available. We are now ready todefine residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 67: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/67.jpg)
Introduction Introduction Introduction
Not all lattices are distributive and a distributive lattice need not tobe a Boolean algebra. A lattice L is called completely distributive if
x ∧∨i∈Γ
yi =∨i∈Γ
(x ∧ yi ), (3)
x ∨∧i∈Γ
yi =∧i∈Γ
(x ∨ yi ) (4)
hold for all elements x ∈ L and all subsets {yi |i ∈ Γ} ⊆ L. Ofcourse, only complete lattices can (but need not) be completelydistributive. Unlike equations (1) and (2), the equations (3) and(4) are not equivalent conditions.
It is easy to find examples oflattices that do not contain the least element 0 nor the largestelement 1. In logic applications is, however, natural to assume thatwe have the elements 0 and 1 available. We are now ready todefine residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 68: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/68.jpg)
Introduction Introduction Introduction
Not all lattices are distributive and a distributive lattice need not tobe a Boolean algebra. A lattice L is called completely distributive if
x ∧∨i∈Γ
yi =∨i∈Γ
(x ∧ yi ), (3)
x ∨∧i∈Γ
yi =∧i∈Γ
(x ∨ yi ) (4)
hold for all elements x ∈ L and all subsets {yi |i ∈ Γ} ⊆ L. Ofcourse, only complete lattices can (but need not) be completelydistributive. Unlike equations (1) and (2), the equations (3) and(4) are not equivalent conditions. It is easy to find examples oflattices that do not contain the least element 0 nor the largestelement 1.
In logic applications is, however, natural to assume thatwe have the elements 0 and 1 available. We are now ready todefine residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 69: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/69.jpg)
Introduction Introduction Introduction
Not all lattices are distributive and a distributive lattice need not tobe a Boolean algebra. A lattice L is called completely distributive if
x ∧∨i∈Γ
yi =∨i∈Γ
(x ∧ yi ), (3)
x ∨∧i∈Γ
yi =∧i∈Γ
(x ∨ yi ) (4)
hold for all elements x ∈ L and all subsets {yi |i ∈ Γ} ⊆ L. Ofcourse, only complete lattices can (but need not) be completelydistributive. Unlike equations (1) and (2), the equations (3) and(4) are not equivalent conditions. It is easy to find examples oflattices that do not contain the least element 0 nor the largestelement 1. In logic applications is, however, natural to assume thatwe have the elements 0 and 1 available.
We are now ready todefine residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 70: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/70.jpg)
Introduction Introduction Introduction
Not all lattices are distributive and a distributive lattice need not tobe a Boolean algebra. A lattice L is called completely distributive if
x ∧∨i∈Γ
yi =∨i∈Γ
(x ∧ yi ), (3)
x ∨∧i∈Γ
yi =∧i∈Γ
(x ∨ yi ) (4)
hold for all elements x ∈ L and all subsets {yi |i ∈ Γ} ⊆ L. Ofcourse, only complete lattices can (but need not) be completelydistributive. Unlike equations (1) and (2), the equations (3) and(4) are not equivalent conditions. It is easy to find examples oflattices that do not contain the least element 0 nor the largestelement 1. In logic applications is, however, natural to assume thatwe have the elements 0 and 1 available. We are now ready todefine residuated lattices.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 71: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/71.jpg)
Introduction Introduction Introduction
Definition (Residuated lattice)
Assume L = 〈L,≤,∧,∨, 0, 1〉 is a lattice with the least and thegreatest elements 0, 1, respectively.
Assume further that L has amonoidal structure, i.e. there is a binary operation � calledproduct on L such that � is associative, commutative, isotone andx � 1 = x holds for all x ∈ L. Moreover, assume there is anotherbinary operation → called residuum of � on L such that, for allx , y , z ∈ L holds a Galois connection
x � y ≤ z if, and only if x ≤ y → z . (5)
Then the structure L = 〈L,≤,∧,∨,�,→, 0, 1〉 is a residuatedlattice and the couple 〈�,→〉 is an adjoint couple.
Before giving examples of residuated lattices we present twoTheorems
PART I. Order, posets, lattices and residuated lattices in logic
![Page 72: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/72.jpg)
Introduction Introduction Introduction
Definition (Residuated lattice)
Assume L = 〈L,≤,∧,∨, 0, 1〉 is a lattice with the least and thegreatest elements 0, 1, respectively. Assume further that L has amonoidal structure, i.e. there is a binary operation � calledproduct on L such that � is associative, commutative, isotone andx � 1 = x holds for all x ∈ L.
Moreover, assume there is anotherbinary operation → called residuum of � on L such that, for allx , y , z ∈ L holds a Galois connection
x � y ≤ z if, and only if x ≤ y → z . (5)
Then the structure L = 〈L,≤,∧,∨,�,→, 0, 1〉 is a residuatedlattice and the couple 〈�,→〉 is an adjoint couple.
Before giving examples of residuated lattices we present twoTheorems
PART I. Order, posets, lattices and residuated lattices in logic
![Page 73: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/73.jpg)
Introduction Introduction Introduction
Definition (Residuated lattice)
Assume L = 〈L,≤,∧,∨, 0, 1〉 is a lattice with the least and thegreatest elements 0, 1, respectively. Assume further that L has amonoidal structure, i.e. there is a binary operation � calledproduct on L such that � is associative, commutative, isotone andx � 1 = x holds for all x ∈ L. Moreover, assume there is anotherbinary operation → called residuum of � on L such that, for allx , y , z ∈ L holds a Galois connection
x � y ≤ z if, and only if x ≤ y → z . (5)
Then the structure L = 〈L,≤,∧,∨,�,→, 0, 1〉 is a residuatedlattice and the couple 〈�,→〉 is an adjoint couple.
Before giving examples of residuated lattices we present twoTheorems
PART I. Order, posets, lattices and residuated lattices in logic
![Page 74: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/74.jpg)
Introduction Introduction Introduction
Definition (Residuated lattice)
Assume L = 〈L,≤,∧,∨, 0, 1〉 is a lattice with the least and thegreatest elements 0, 1, respectively. Assume further that L has amonoidal structure, i.e. there is a binary operation � calledproduct on L such that � is associative, commutative, isotone andx � 1 = x holds for all x ∈ L. Moreover, assume there is anotherbinary operation → called residuum of � on L such that, for allx , y , z ∈ L holds a Galois connection
x � y ≤ z if, and only if x ≤ y → z . (5)
Then the structure L = 〈L,≤,∧,∨,�,→, 0, 1〉 is a residuatedlattice and the couple 〈�,→〉 is an adjoint couple.
Before giving examples of residuated lattices we present twoTheorems
PART I. Order, posets, lattices and residuated lattices in logic
![Page 75: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/75.jpg)
Introduction Introduction Introduction
Definition (Residuated lattice)
Assume L = 〈L,≤,∧,∨, 0, 1〉 is a lattice with the least and thegreatest elements 0, 1, respectively. Assume further that L has amonoidal structure, i.e. there is a binary operation � calledproduct on L such that � is associative, commutative, isotone andx � 1 = x holds for all x ∈ L. Moreover, assume there is anotherbinary operation → called residuum of � on L such that, for allx , y , z ∈ L holds a Galois connection
x � y ≤ z if, and only if x ≤ y → z . (5)
Then the structure L = 〈L,≤,∧,∨,�,→, 0, 1〉 is a residuatedlattice and the couple 〈�,→〉 is an adjoint couple.
Before giving examples of residuated lattices we present twoTheorems
PART I. Order, posets, lattices and residuated lattices in logic
![Page 76: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/76.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.Since � is isotone, x � yi ≤ x �
∨i∈Γ yi for all i ∈ Γ, therefore∨
i∈Γ(x � yi ) ≤ x �∨
i∈Γ yi . Conversely, as x � yi ≤∨
i∈Γ(x � yi )holds for all i ∈ Γ, we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ, and therefore also∨
i∈Γ yi ≤ x → [∨
i∈Γ(x � yi )]. Again by (5) we reasonx �
∨i∈Γ yi ≤
∨i∈Γ(x � yi ), and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 77: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/77.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.
Since � is isotone, x � yi ≤ x �∨
i∈Γ yi for all i ∈ Γ, therefore∨i∈Γ(x � yi ) ≤ x �
∨i∈Γ yi . Conversely, as x � yi ≤
∨i∈Γ(x � yi )
holds for all i ∈ Γ, we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ, and therefore also∨
i∈Γ yi ≤ x → [∨
i∈Γ(x � yi )]. Again by (5) we reasonx �
∨i∈Γ yi ≤
∨i∈Γ(x � yi ), and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 78: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/78.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.Since � is isotone, x � yi ≤ x �
∨i∈Γ yi for all i ∈ Γ,
therefore∨i∈Γ(x � yi ) ≤ x �
∨i∈Γ yi . Conversely, as x � yi ≤
∨i∈Γ(x � yi )
holds for all i ∈ Γ, we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ, and therefore also∨
i∈Γ yi ≤ x → [∨
i∈Γ(x � yi )]. Again by (5) we reasonx �
∨i∈Γ yi ≤
∨i∈Γ(x � yi ), and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 79: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/79.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.Since � is isotone, x � yi ≤ x �
∨i∈Γ yi for all i ∈ Γ, therefore∨
i∈Γ(x � yi ) ≤ x �∨
i∈Γ yi .
Conversely, as x � yi ≤∨
i∈Γ(x � yi )holds for all i ∈ Γ, we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ, and therefore also∨
i∈Γ yi ≤ x → [∨
i∈Γ(x � yi )]. Again by (5) we reasonx �
∨i∈Γ yi ≤
∨i∈Γ(x � yi ), and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 80: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/80.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.Since � is isotone, x � yi ≤ x �
∨i∈Γ yi for all i ∈ Γ, therefore∨
i∈Γ(x � yi ) ≤ x �∨
i∈Γ yi . Conversely, as x � yi ≤∨
i∈Γ(x � yi )holds for all i ∈ Γ,
we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ, and therefore also∨
i∈Γ yi ≤ x → [∨
i∈Γ(x � yi )]. Again by (5) we reasonx �
∨i∈Γ yi ≤
∨i∈Γ(x � yi ), and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 81: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/81.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.Since � is isotone, x � yi ≤ x �
∨i∈Γ yi for all i ∈ Γ, therefore∨
i∈Γ(x � yi ) ≤ x �∨
i∈Γ yi . Conversely, as x � yi ≤∨
i∈Γ(x � yi )holds for all i ∈ Γ, we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ,
and therefore also∨i∈Γ yi ≤ x → [
∨i∈Γ(x � yi )]. Again by (5) we reason
x �∨
i∈Γ yi ≤∨
i∈Γ(x � yi ), and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 82: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/82.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.Since � is isotone, x � yi ≤ x �
∨i∈Γ yi for all i ∈ Γ, therefore∨
i∈Γ(x � yi ) ≤ x �∨
i∈Γ yi . Conversely, as x � yi ≤∨
i∈Γ(x � yi )holds for all i ∈ Γ, we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ, and therefore also∨
i∈Γ yi ≤ x → [∨
i∈Γ(x � yi )].
Again by (5) we reasonx �
∨i∈Γ yi ≤
∨i∈Γ(x � yi ), and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 83: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/83.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.Since � is isotone, x � yi ≤ x �
∨i∈Γ yi for all i ∈ Γ, therefore∨
i∈Γ(x � yi ) ≤ x �∨
i∈Γ yi . Conversely, as x � yi ≤∨
i∈Γ(x � yi )holds for all i ∈ Γ, we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ, and therefore also∨
i∈Γ yi ≤ x → [∨
i∈Γ(x � yi )]. Again by (5) we reasonx �
∨i∈Γ yi ≤
∨i∈Γ(x � yi ),
and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 84: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/84.jpg)
Introduction Introduction Introduction
Theorem (1)
Assume 〈�,→〉 is an adjoint couple of a residuated lattice L. Then
x �∨i∈Γ
yi =∨i∈Γ
(x � yi ) (6)
whenever these joins exist in L.
Proof. Let x ∈ L, {yi |i ∈ Γ} ⊆ L and both sides of (6) exist in L.Since � is isotone, x � yi ≤ x �
∨i∈Γ yi for all i ∈ Γ, therefore∨
i∈Γ(x � yi ) ≤ x �∨
i∈Γ yi . Conversely, as x � yi ≤∨
i∈Γ(x � yi )holds for all i ∈ Γ, we obtain by the Galois connection (5) thatyi ≤ x → [
∨i∈Γ(x � yi )] holds for all i ∈ Γ, and therefore also∨
i∈Γ yi ≤ x → [∨
i∈Γ(x � yi )]. Again by (5) we reasonx �
∨i∈Γ yi ≤
∨i∈Γ(x � yi ), and therefore the claim follows.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 85: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/85.jpg)
Introduction Introduction Introduction
Theorem (2)
Assume condition (6) always holds in a lattice L with a monoidalstructure. Then L can be (uniquely) residuated via
x → y =∨{z ∈ L| x � z ≤ y}. (7)
Proof. If a� b ≤ c then a ≤∨{z ∈ L| b � z ≤ c} = b → c and
vice versa. Thus, the Galois connection (5) holds.
By condition (7), residuum → of a product � is unique, we maytalk about the residuum of �.
The above Theorems imply that if (6) holds, then we can definethe corresponding residuum by (7) and vice versa.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 86: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/86.jpg)
Introduction Introduction Introduction
Theorem (2)
Assume condition (6) always holds in a lattice L with a monoidalstructure. Then L can be (uniquely) residuated via
x → y =∨{z ∈ L| x � z ≤ y}. (7)
Proof. If a� b ≤ c then a ≤∨{z ∈ L| b � z ≤ c} = b → c and
vice versa.
Thus, the Galois connection (5) holds.
By condition (7), residuum → of a product � is unique, we maytalk about the residuum of �.
The above Theorems imply that if (6) holds, then we can definethe corresponding residuum by (7) and vice versa.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 87: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/87.jpg)
Introduction Introduction Introduction
Theorem (2)
Assume condition (6) always holds in a lattice L with a monoidalstructure. Then L can be (uniquely) residuated via
x → y =∨{z ∈ L| x � z ≤ y}. (7)
Proof. If a� b ≤ c then a ≤∨{z ∈ L| b � z ≤ c} = b → c and
vice versa. Thus, the Galois connection (5) holds.
By condition (7), residuum → of a product � is unique, we maytalk about the residuum of �.
The above Theorems imply that if (6) holds, then we can definethe corresponding residuum by (7) and vice versa.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 88: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/88.jpg)
Introduction Introduction Introduction
Theorem (2)
Assume condition (6) always holds in a lattice L with a monoidalstructure. Then L can be (uniquely) residuated via
x → y =∨{z ∈ L| x � z ≤ y}. (7)
Proof. If a� b ≤ c then a ≤∨{z ∈ L| b � z ≤ c} = b → c and
vice versa. Thus, the Galois connection (5) holds.
By condition (7), residuum → of a product � is unique, we maytalk about the residuum of �.
The above Theorems imply that if (6) holds, then we can definethe corresponding residuum by (7) and vice versa.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 89: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/89.jpg)
Introduction Introduction Introduction
Theorem (2)
Assume condition (6) always holds in a lattice L with a monoidalstructure. Then L can be (uniquely) residuated via
x → y =∨{z ∈ L| x � z ≤ y}. (7)
Proof. If a� b ≤ c then a ≤∨{z ∈ L| b � z ≤ c} = b → c and
vice versa. Thus, the Galois connection (5) holds.
By condition (7), residuum → of a product � is unique, we maytalk about the residuum of �.
The above Theorems imply that if (6) holds, then we can definethe corresponding residuum by (7) and vice versa.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 90: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/90.jpg)
Introduction Introduction Introduction
As a consequence we have, for example, that any completelydistributive lattice is residuated.
Indeed, they have bottom and topelements 0, 1, the meet operation ∧ is associative, commutative,isotone and for all x ∈ L, x ∧ 1 = x . Moreover, completelydistributiveness is exatly equation (6). Such algebras are Heytingalgebras, the algebras corresponding to Intuitionistic Logic.Another familiar class of residuated lattices are Boolean algebras.There again � = ∧ and the residuum is given via x → y = x∗ ∨ y -to see all the details is an exercise. Boolean algebras are thealgebras of Classical Logic. The real unit interval [0, 1] is a latticewith bottom and top elements, the order is the order ≤ of reals,x ∧ y = min{x , y} and x ∨ y = max{x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 91: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/91.jpg)
Introduction Introduction Introduction
As a consequence we have, for example, that any completelydistributive lattice is residuated. Indeed, they have bottom and topelements 0, 1, the meet operation ∧ is associative, commutative,isotone and for all x ∈ L, x ∧ 1 = x . Moreover, completelydistributiveness is exatly equation (6).
Such algebras are Heytingalgebras, the algebras corresponding to Intuitionistic Logic.Another familiar class of residuated lattices are Boolean algebras.There again � = ∧ and the residuum is given via x → y = x∗ ∨ y -to see all the details is an exercise. Boolean algebras are thealgebras of Classical Logic. The real unit interval [0, 1] is a latticewith bottom and top elements, the order is the order ≤ of reals,x ∧ y = min{x , y} and x ∨ y = max{x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 92: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/92.jpg)
Introduction Introduction Introduction
As a consequence we have, for example, that any completelydistributive lattice is residuated. Indeed, they have bottom and topelements 0, 1, the meet operation ∧ is associative, commutative,isotone and for all x ∈ L, x ∧ 1 = x . Moreover, completelydistributiveness is exatly equation (6). Such algebras are Heytingalgebras, the algebras corresponding to Intuitionistic Logic.
Another familiar class of residuated lattices are Boolean algebras.There again � = ∧ and the residuum is given via x → y = x∗ ∨ y -to see all the details is an exercise. Boolean algebras are thealgebras of Classical Logic. The real unit interval [0, 1] is a latticewith bottom and top elements, the order is the order ≤ of reals,x ∧ y = min{x , y} and x ∨ y = max{x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 93: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/93.jpg)
Introduction Introduction Introduction
As a consequence we have, for example, that any completelydistributive lattice is residuated. Indeed, they have bottom and topelements 0, 1, the meet operation ∧ is associative, commutative,isotone and for all x ∈ L, x ∧ 1 = x . Moreover, completelydistributiveness is exatly equation (6). Such algebras are Heytingalgebras, the algebras corresponding to Intuitionistic Logic.Another familiar class of residuated lattices are Boolean algebras.There again � = ∧ and the residuum is given via x → y = x∗ ∨ y
-to see all the details is an exercise. Boolean algebras are thealgebras of Classical Logic. The real unit interval [0, 1] is a latticewith bottom and top elements, the order is the order ≤ of reals,x ∧ y = min{x , y} and x ∨ y = max{x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 94: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/94.jpg)
Introduction Introduction Introduction
As a consequence we have, for example, that any completelydistributive lattice is residuated. Indeed, they have bottom and topelements 0, 1, the meet operation ∧ is associative, commutative,isotone and for all x ∈ L, x ∧ 1 = x . Moreover, completelydistributiveness is exatly equation (6). Such algebras are Heytingalgebras, the algebras corresponding to Intuitionistic Logic.Another familiar class of residuated lattices are Boolean algebras.There again � = ∧ and the residuum is given via x → y = x∗ ∨ y -to see all the details is an exercise.
Boolean algebras are thealgebras of Classical Logic. The real unit interval [0, 1] is a latticewith bottom and top elements, the order is the order ≤ of reals,x ∧ y = min{x , y} and x ∨ y = max{x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 95: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/95.jpg)
Introduction Introduction Introduction
As a consequence we have, for example, that any completelydistributive lattice is residuated. Indeed, they have bottom and topelements 0, 1, the meet operation ∧ is associative, commutative,isotone and for all x ∈ L, x ∧ 1 = x . Moreover, completelydistributiveness is exatly equation (6). Such algebras are Heytingalgebras, the algebras corresponding to Intuitionistic Logic.Another familiar class of residuated lattices are Boolean algebras.There again � = ∧ and the residuum is given via x → y = x∗ ∨ y -to see all the details is an exercise. Boolean algebras are thealgebras of Classical Logic.
The real unit interval [0, 1] is a latticewith bottom and top elements, the order is the order ≤ of reals,x ∧ y = min{x , y} and x ∨ y = max{x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 96: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/96.jpg)
Introduction Introduction Introduction
As a consequence we have, for example, that any completelydistributive lattice is residuated. Indeed, they have bottom and topelements 0, 1, the meet operation ∧ is associative, commutative,isotone and for all x ∈ L, x ∧ 1 = x . Moreover, completelydistributiveness is exatly equation (6). Such algebras are Heytingalgebras, the algebras corresponding to Intuitionistic Logic.Another familiar class of residuated lattices are Boolean algebras.There again � = ∧ and the residuum is given via x → y = x∗ ∨ y -to see all the details is an exercise. Boolean algebras are thealgebras of Classical Logic. The real unit interval [0, 1] is a latticewith bottom and top elements, the order is the order ≤ of reals,x ∧ y = min{x , y} and x ∨ y = max{x , y}.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 97: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/97.jpg)
Introduction Introduction Introduction
Examples of residuated structures on [0, 1]
Godel algebra (BL):
x � y = min{x , y}, x → y =
{1 if x ≤ yy otherwise
Product algebra (BL):
x � y = xy , x → y =
{1 if x ≤ y
y/x otherwiseLukasiewicz algebra (MV):x � y = max{0, x + y − 1}, x → y = min{1, 1− x + y}.A structure Lc (MTL) where 0 < c < 1:
x � y =
{0 if x + y ≤ cmin{x , y} elsewhere
,
x → y =
{1 if x ≤ y ,max{c − x , y} elsewhere.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 98: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/98.jpg)
Introduction Introduction Introduction
Examples of residuated structures on [0, 1]Godel algebra (BL):
x � y = min{x , y}, x → y =
{1 if x ≤ yy otherwise
Product algebra (BL):
x � y = xy , x → y =
{1 if x ≤ y
y/x otherwiseLukasiewicz algebra (MV):x � y = max{0, x + y − 1}, x → y = min{1, 1− x + y}.A structure Lc (MTL) where 0 < c < 1:
x � y =
{0 if x + y ≤ cmin{x , y} elsewhere
,
x → y =
{1 if x ≤ y ,max{c − x , y} elsewhere.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 99: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/99.jpg)
Introduction Introduction Introduction
Examples of residuated structures on [0, 1]Godel algebra (BL):
x � y = min{x , y}, x → y =
{1 if x ≤ yy otherwise
Product algebra (BL):
x � y = xy , x → y =
{1 if x ≤ y
y/x otherwise
Lukasiewicz algebra (MV):x � y = max{0, x + y − 1}, x → y = min{1, 1− x + y}.A structure Lc (MTL) where 0 < c < 1:
x � y =
{0 if x + y ≤ cmin{x , y} elsewhere
,
x → y =
{1 if x ≤ y ,max{c − x , y} elsewhere.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 100: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/100.jpg)
Introduction Introduction Introduction
Examples of residuated structures on [0, 1]Godel algebra (BL):
x � y = min{x , y}, x → y =
{1 if x ≤ yy otherwise
Product algebra (BL):
x � y = xy , x → y =
{1 if x ≤ y
y/x otherwiseLukasiewicz algebra (MV):x � y = max{0, x + y − 1}, x → y = min{1, 1− x + y}.
A structure Lc (MTL) where 0 < c < 1:
x � y =
{0 if x + y ≤ cmin{x , y} elsewhere
,
x → y =
{1 if x ≤ y ,max{c − x , y} elsewhere.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 101: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/101.jpg)
Introduction Introduction Introduction
Examples of residuated structures on [0, 1]Godel algebra (BL):
x � y = min{x , y}, x → y =
{1 if x ≤ yy otherwise
Product algebra (BL):
x � y = xy , x → y =
{1 if x ≤ y
y/x otherwiseLukasiewicz algebra (MV):x � y = max{0, x + y − 1}, x → y = min{1, 1− x + y}.A structure Lc (MTL) where 0 < c < 1:
x � y =
{0 if x + y ≤ cmin{x , y} elsewhere
,
x → y =
{1 if x ≤ y ,max{c − x , y} elsewhere.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 102: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/102.jpg)
Introduction Introduction Introduction
A structure LsD (SD):
x � y =
{0 if x , y ∈ [0, 1
2 ]min{x , y} elsewhere
,
x → y =
1 if x ≤ y ,12 if y < x ≤ 1
2 ,y if y < x , 1
2 < x .
Theorem (Basic properties of residuated lattices)
For all elements x , y , x1, x2, y1, y2 ∈ L the following hold
x = 1 → x , (8)
1 = x → x , (9)
x � y ≤ x , y , (10)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 103: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/103.jpg)
Introduction Introduction Introduction
A structure LsD (SD):
x � y =
{0 if x , y ∈ [0, 1
2 ]min{x , y} elsewhere
,
x → y =
1 if x ≤ y ,12 if y < x ≤ 1
2 ,y if y < x , 1
2 < x .
Theorem (Basic properties of residuated lattices)
For all elements x , y , x1, x2, y1, y2 ∈ L the following hold
x = 1 → x , (8)
1 = x → x , (9)
x � y ≤ x , y , (10)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 104: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/104.jpg)
Introduction Introduction Introduction
A structure LsD (SD):
x � y =
{0 if x , y ∈ [0, 1
2 ]min{x , y} elsewhere
,
x → y =
1 if x ≤ y ,12 if y < x ≤ 1
2 ,y if y < x , 1
2 < x .
Theorem (Basic properties of residuated lattices)
For all elements x , y , x1, x2, y1, y2 ∈ L the following hold
x = 1 → x , (8)
1 = x → x , (9)
x � y ≤ x , y , (10)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 105: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/105.jpg)
Introduction Introduction Introduction
A structure LsD (SD):
x � y =
{0 if x , y ∈ [0, 1
2 ]min{x , y} elsewhere
,
x → y =
1 if x ≤ y ,12 if y < x ≤ 1
2 ,y if y < x , 1
2 < x .
Theorem (Basic properties of residuated lattices)
For all elements x , y , x1, x2, y1, y2 ∈ L the following hold
x = 1 → x , (8)
1 = x → x , (9)
x � y ≤ x , y , (10)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 106: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/106.jpg)
Introduction Introduction Introduction
A structure LsD (SD):
x � y =
{0 if x , y ∈ [0, 1
2 ]min{x , y} elsewhere
,
x → y =
1 if x ≤ y ,12 if y < x ≤ 1
2 ,y if y < x , 1
2 < x .
Theorem (Basic properties of residuated lattices)
For all elements x , y , x1, x2, y1, y2 ∈ L the following hold
x = 1 → x , (8)
1 = x → x , (9)
x � y ≤ x , y , (10)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 107: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/107.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 108: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/108.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 109: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/109.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 110: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/110.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 111: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/111.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 112: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/112.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 113: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/113.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 114: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/114.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 115: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/115.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 1)
x � y ≤ x ∧ y , (11)
y ≤ x → y , (12)
x � y ≤ x → y , (13)
x ≤ y iff 1 = x → y , (14)
1 = x → y = y → x iff x = y , (15)
x → 1 = 1, (16)
0 → x = 1, (17)
x → (y → x) = 1, (18)
(x → y) → [(y → z) → (x → z)] = 1, (19)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 116: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/116.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 117: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/117.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 118: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/118.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 119: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/119.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 120: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/120.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21).
By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 121: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/121.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y)
≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 122: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/122.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y
≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 123: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/123.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z ,
and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 124: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/124.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z .
Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 125: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/125.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff
[(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 126: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/126.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff
[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 127: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/127.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff
[(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 128: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/128.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.
Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 129: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/129.jpg)
Introduction Introduction Introduction
Theorem (Basic properties of residuated lattices, continuation 2)
(x → y) → [(z → x) → (z → y)] = 1, (20)
(x � y) → z = x → (y → z), (21)
x → (y → z) = y → (x → z), (22)
(x1 → y1) → {(y2 → x2) → [(y1 → y2) → (x1 → x2)]} = 1. (23)
Proof. We establish only (21). By Galois connection (5) we have[x → (y → z)]� (x � y) ≤ (y → z)� y ≤ z , and again by (5),[x → (y → z) ≤ (x � y) → z . Conversely,(x � y) → z ≤ x → (y → z) iff [(x � y) → z ]� x ≤ y → z iff[(x � y) → z ]� (x � y) ≤ z iff [(x � y) → z ] ≤ (x � y) → z , true.Thus, (21) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 130: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/130.jpg)
Introduction Introduction Introduction
When we later talk about logics, the residuum operation will playan important role when considering implication. For the needs ofnegation we introduce a complement (do not mix up with latticecomplement!) of an element x ∈ L by stipulating x∗ = x → 0.
Theorem (Properties of complement)
Let L be a residuated lattice and x , y ∈ L. Then
x � x∗ = 0, (24)
x ≤ x∗∗, (25)
1∗ = 0, (26)
0∗ = 1, (27)
x → y ≤ y∗ → x∗, (28)
x∗ = x∗∗∗. (29)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 131: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/131.jpg)
Introduction Introduction Introduction
When we later talk about logics, the residuum operation will playan important role when considering implication. For the needs ofnegation we introduce a complement (do not mix up with latticecomplement!) of an element x ∈ L by stipulating x∗ = x → 0.
Theorem (Properties of complement)
Let L be a residuated lattice and x , y ∈ L. Then
x � x∗ = 0, (24)
x ≤ x∗∗, (25)
1∗ = 0, (26)
0∗ = 1, (27)
x → y ≤ y∗ → x∗, (28)
x∗ = x∗∗∗. (29)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 132: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/132.jpg)
Introduction Introduction Introduction
When we later talk about logics, the residuum operation will playan important role when considering implication. For the needs ofnegation we introduce a complement (do not mix up with latticecomplement!) of an element x ∈ L by stipulating x∗ = x → 0.
Theorem (Properties of complement)
Let L be a residuated lattice and x , y ∈ L. Then
x � x∗ = 0, (24)
x ≤ x∗∗, (25)
1∗ = 0, (26)
0∗ = 1, (27)
x → y ≤ y∗ → x∗, (28)
x∗ = x∗∗∗. (29)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 133: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/133.jpg)
Introduction Introduction Introduction
When we later talk about logics, the residuum operation will playan important role when considering implication. For the needs ofnegation we introduce a complement (do not mix up with latticecomplement!) of an element x ∈ L by stipulating x∗ = x → 0.
Theorem (Properties of complement)
Let L be a residuated lattice and x , y ∈ L. Then
x � x∗ = 0, (24)
x ≤ x∗∗, (25)
1∗ = 0, (26)
0∗ = 1, (27)
x → y ≤ y∗ → x∗, (28)
x∗ = x∗∗∗. (29)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 134: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/134.jpg)
Introduction Introduction Introduction
When we later talk about logics, the residuum operation will playan important role when considering implication. For the needs ofnegation we introduce a complement (do not mix up with latticecomplement!) of an element x ∈ L by stipulating x∗ = x → 0.
Theorem (Properties of complement)
Let L be a residuated lattice and x , y ∈ L. Then
x � x∗ = 0, (24)
x ≤ x∗∗, (25)
1∗ = 0, (26)
0∗ = 1, (27)
x → y ≤ y∗ → x∗, (28)
x∗ = x∗∗∗. (29)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 135: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/135.jpg)
Introduction Introduction Introduction
When we later talk about logics, the residuum operation will playan important role when considering implication. For the needs ofnegation we introduce a complement (do not mix up with latticecomplement!) of an element x ∈ L by stipulating x∗ = x → 0.
Theorem (Properties of complement)
Let L be a residuated lattice and x , y ∈ L. Then
x � x∗ = 0, (24)
x ≤ x∗∗, (25)
1∗ = 0, (26)
0∗ = 1, (27)
x → y ≤ y∗ → x∗, (28)
x∗ = x∗∗∗. (29)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 136: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/136.jpg)
Introduction Introduction Introduction
When we later talk about logics, the residuum operation will playan important role when considering implication. For the needs ofnegation we introduce a complement (do not mix up with latticecomplement!) of an element x ∈ L by stipulating x∗ = x → 0.
Theorem (Properties of complement)
Let L be a residuated lattice and x , y ∈ L. Then
x � x∗ = 0, (24)
x ≤ x∗∗, (25)
1∗ = 0, (26)
0∗ = 1, (27)
x → y ≤ y∗ → x∗, (28)
x∗ = x∗∗∗. (29)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 137: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/137.jpg)
Introduction Introduction Introduction
The proof is left as an exercise.
We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.Bi–residuum will turn useful when we talk about fuzzy equivalencerelations. The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 138: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/138.jpg)
Introduction Introduction Introduction
The proof is left as an exercise. We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.
Bi–residuum will turn useful when we talk about fuzzy equivalencerelations. The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 139: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/139.jpg)
Introduction Introduction Introduction
The proof is left as an exercise. We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.Bi–residuum will turn useful when we talk about fuzzy equivalencerelations.
The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 140: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/140.jpg)
Introduction Introduction Introduction
The proof is left as an exercise. We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.Bi–residuum will turn useful when we talk about fuzzy equivalencerelations. The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 141: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/141.jpg)
Introduction Introduction Introduction
The proof is left as an exercise. We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.Bi–residuum will turn useful when we talk about fuzzy equivalencerelations. The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 142: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/142.jpg)
Introduction Introduction Introduction
The proof is left as an exercise. We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.Bi–residuum will turn useful when we talk about fuzzy equivalencerelations. The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 143: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/143.jpg)
Introduction Introduction Introduction
The proof is left as an exercise. We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.Bi–residuum will turn useful when we talk about fuzzy equivalencerelations. The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 144: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/144.jpg)
Introduction Introduction Introduction
The proof is left as an exercise. We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.Bi–residuum will turn useful when we talk about fuzzy equivalencerelations. The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 145: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/145.jpg)
Introduction Introduction Introduction
The proof is left as an exercise. We introduce a derived operationbi–residuum by setting x ↔ y = (x → y) ∧ (y → x) on L.Bi–residuum will turn useful when we talk about fuzzy equivalencerelations. The proof of the following theorem is an exercise
Theorem (Properties of bi–residuum)
Let L be a residuated lattice. Then for all elements in L hold
x ↔ 1 = x , (30)
x = y iff x ↔ y = 1, (31)
x ↔ y = y ↔ x , (32)
(x ↔ y)� (y ↔ z) ≤ (x ↔ z) (33)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∧ x2) ↔ (y1 ∧ y2), (34)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 146: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/146.jpg)
Introduction Introduction Introduction
Theorem (Properties of bi–residuum, continuation)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∨ x2) ↔ (y1 ∨ y2), (35)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 � x2) ↔ (y1 � y2), (36)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 → x2) ↔ (y1 → y2). (37)
Theorem (Properties of a complete residuated lattice L)
Assume x ∈ L, {yi |i ∈ Γ} ⊆ L. Then
x →∧i∈Γ
yi =∧i∈Γ
(x → yi ), (38)∨i∈Γ
yi → x =∧i∈Γ
(yi → x) (39)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 147: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/147.jpg)
Introduction Introduction Introduction
Theorem (Properties of bi–residuum, continuation)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∨ x2) ↔ (y1 ∨ y2), (35)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 � x2) ↔ (y1 � y2), (36)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 → x2) ↔ (y1 → y2). (37)
Theorem (Properties of a complete residuated lattice L)
Assume x ∈ L, {yi |i ∈ Γ} ⊆ L. Then
x →∧i∈Γ
yi =∧i∈Γ
(x → yi ), (38)∨i∈Γ
yi → x =∧i∈Γ
(yi → x) (39)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 148: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/148.jpg)
Introduction Introduction Introduction
Theorem (Properties of bi–residuum, continuation)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∨ x2) ↔ (y1 ∨ y2), (35)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 � x2) ↔ (y1 � y2), (36)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 → x2) ↔ (y1 → y2). (37)
Theorem (Properties of a complete residuated lattice L)
Assume x ∈ L, {yi |i ∈ Γ} ⊆ L. Then
x →∧i∈Γ
yi =∧i∈Γ
(x → yi ), (38)∨i∈Γ
yi → x =∧i∈Γ
(yi → x) (39)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 149: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/149.jpg)
Introduction Introduction Introduction
Theorem (Properties of bi–residuum, continuation)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∨ x2) ↔ (y1 ∨ y2), (35)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 � x2) ↔ (y1 � y2), (36)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 → x2) ↔ (y1 → y2). (37)
Theorem (Properties of a complete residuated lattice L)
Assume x ∈ L, {yi |i ∈ Γ} ⊆ L. Then
x →∧i∈Γ
yi =∧i∈Γ
(x → yi ), (38)
∨i∈Γ
yi → x =∧i∈Γ
(yi → x) (39)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 150: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/150.jpg)
Introduction Introduction Introduction
Theorem (Properties of bi–residuum, continuation)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∨ x2) ↔ (y1 ∨ y2), (35)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 � x2) ↔ (y1 � y2), (36)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 → x2) ↔ (y1 → y2). (37)
Theorem (Properties of a complete residuated lattice L)
Assume x ∈ L, {yi |i ∈ Γ} ⊆ L. Then
x →∧i∈Γ
yi =∧i∈Γ
(x → yi ), (38)∨i∈Γ
yi → x =∧i∈Γ
(yi → x)
(39)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 151: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/151.jpg)
Introduction Introduction Introduction
Theorem (Properties of bi–residuum, continuation)
(x1 ↔ y1) ∧ (x2 ↔ y2) ≤ (x1 ∨ x2) ↔ (y1 ∨ y2), (35)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 � x2) ↔ (y1 � y2), (36)
(x1 ↔ y1)� (x2 ↔ y2) ≤ (x1 → x2) ↔ (y1 → y2). (37)
Theorem (Properties of a complete residuated lattice L)
Assume x ∈ L, {yi |i ∈ Γ} ⊆ L. Then
x →∧i∈Γ
yi =∧i∈Γ
(x → yi ), (38)∨i∈Γ
yi → x =∧i∈Γ
(yi → x) (39)
PART I. Order, posets, lattices and residuated lattices in logic
![Page 152: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/152.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)
∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable) and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 153: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/153.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi ,
(41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable) and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 154: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/154.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable) and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 155: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/155.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise),
for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable) and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 156: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/156.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that
(yi → x) ≤∧
i∈Γ yi → x holds for each indexi ∈ Γ (as → is antitone in the first variable) and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 157: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/157.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable)
and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 158: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/158.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable) and therefore (40)holds.
Particular instances of (39) and (40) are the following(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 159: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/159.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable) and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 160: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/160.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable) and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 161: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/161.jpg)
Introduction Introduction Introduction
Theorem (Properties of a complete ... continuation)
∨i∈Γ
(yi → x) ≤∧i∈Γ
yi → x , (40)∨i∈Γ
(x → yi ) ≤ x →∨i∈Γ
yi , (41)
The definitions of∨
and∧
as well as the Galois connection (5)are essential in the prooves (an exercise), for example to prove(40), we realize that (yi → x) ≤
∧i∈Γ yi → x holds for each index
i ∈ Γ (as → is antitone in the first variable) and therefore (40)holds. Particular instances of (39) and (40) are the following
(∨
i∈Γ yi )∗ =
∧i∈Γ y∗i ,
∨i∈Γ y∗i ≤ (
∧i∈Γ yi )
∗.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 162: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/162.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology.
Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.∗ a Heyting algebra if the product operation � coincides with theoperation ∧. – Involutive Heyting algebras are Boolean algebras.∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L. –Another name is MTL–algebra.∗ a BL–algebra if it is prelinear and divisible.∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 163: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/163.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology. Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.
∗ a Heyting algebra if the product operation � coincides with theoperation ∧. – Involutive Heyting algebras are Boolean algebras.∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L. –Another name is MTL–algebra.∗ a BL–algebra if it is prelinear and divisible.∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 164: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/164.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology. Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.∗ a Heyting algebra if the product operation � coincides with theoperation ∧.
– Involutive Heyting algebras are Boolean algebras.∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L. –Another name is MTL–algebra.∗ a BL–algebra if it is prelinear and divisible.∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 165: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/165.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology. Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.∗ a Heyting algebra if the product operation � coincides with theoperation ∧. – Involutive Heyting algebras are Boolean algebras.
∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L. –Another name is MTL–algebra.∗ a BL–algebra if it is prelinear and divisible.∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 166: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/166.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology. Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.∗ a Heyting algebra if the product operation � coincides with theoperation ∧. – Involutive Heyting algebras are Boolean algebras.∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.
∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L. –Another name is MTL–algebra.∗ a BL–algebra if it is prelinear and divisible.∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 167: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/167.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology. Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.∗ a Heyting algebra if the product operation � coincides with theoperation ∧. – Involutive Heyting algebras are Boolean algebras.∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L.
–Another name is MTL–algebra.∗ a BL–algebra if it is prelinear and divisible.∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 168: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/168.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology. Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.∗ a Heyting algebra if the product operation � coincides with theoperation ∧. – Involutive Heyting algebras are Boolean algebras.∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L. –Another name is MTL–algebra.
∗ a BL–algebra if it is prelinear and divisible.∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 169: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/169.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology. Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.∗ a Heyting algebra if the product operation � coincides with theoperation ∧. – Involutive Heyting algebras are Boolean algebras.∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L. –Another name is MTL–algebra.∗ a BL–algebra if it is prelinear and divisible.
∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 170: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/170.jpg)
Introduction Introduction Introduction
We finish this introduction by recalling some terminology. Aresiduated lattice L is∗ a (commutative) Girard monoid if the complement is involutive,i.e., x = x∗∗ for all x ∈ L. Complete Girard monoids – that is,Girard monoids whose lattice structure contains all meets and joins– are called Girard quantales.∗ a Heyting algebra if the product operation � coincides with theoperation ∧. – Involutive Heyting algebras are Boolean algebras.∗ called divisible if, for all x , y ∈ L, x ≤ y , there is z ∈ L such thatx = z � y . This condition is equivalent to x ∧ y = x � (x → y) forall x , y ∈ L.∗ called prelinear if (x → y)∨ (y → x) = 1 holds for all x , y ∈ L. –Another name is MTL–algebra.∗ a BL–algebra if it is prelinear and divisible.∗ an MV –algebra if it is a divisible Girard monoid or, equivalenty, aBL–algebra with an involutive complement.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 171: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/171.jpg)
Introduction Introduction Introduction
∗ called semi–divisible if, for all x , y ∈ L,
(x∗ → y∗) → y∗ = (y∗ → x∗) → x∗. (42)
Equation (42) is equivalent to
(x∗ ∧ y∗)∗ = [x∗ � (x∗ → y∗)]∗ (43)
and, moreover, a subset MV (L) = {x∗| x ∈ L} (non–void as0, 1 ∈ MV (L) and called the MV–center of L) generates anMV –algebra, where the operations �MV and ∨MV are defined via
x∗ �MV y∗ = (x∗ � y∗)∗∗, x∗ ∨MV y∗ = (x∗ ∨ y∗)∗∗. (44)
The order ≤ and the operations →,∗ ,∧ are those on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 172: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/172.jpg)
Introduction Introduction Introduction
∗ called semi–divisible if, for all x , y ∈ L,
(x∗ → y∗) → y∗ = (y∗ → x∗) → x∗. (42)
Equation (42) is equivalent to
(x∗ ∧ y∗)∗ = [x∗ � (x∗ → y∗)]∗ (43)
and, moreover, a subset MV (L) = {x∗| x ∈ L} (non–void as0, 1 ∈ MV (L) and called the MV–center of L) generates anMV –algebra, where the operations �MV and ∨MV are defined via
x∗ �MV y∗ = (x∗ � y∗)∗∗, x∗ ∨MV y∗ = (x∗ ∨ y∗)∗∗. (44)
The order ≤ and the operations →,∗ ,∧ are those on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 173: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/173.jpg)
Introduction Introduction Introduction
∗ called semi–divisible if, for all x , y ∈ L,
(x∗ → y∗) → y∗ = (y∗ → x∗) → x∗. (42)
Equation (42) is equivalent to
(x∗ ∧ y∗)∗ = [x∗ � (x∗ → y∗)]∗ (43)
and, moreover, a subset MV (L) = {x∗| x ∈ L} (non–void as0, 1 ∈ MV (L) and called the MV–center of L) generates anMV –algebra, where the operations �MV and ∨MV are defined via
x∗ �MV y∗ = (x∗ � y∗)∗∗, x∗ ∨MV y∗ = (x∗ ∨ y∗)∗∗. (44)
The order ≤ and the operations →,∗ ,∧ are those on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 174: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/174.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 175: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/175.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif
(i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 176: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/176.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F .
Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 177: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/177.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if
(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 178: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/178.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 179: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/179.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 180: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/180.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A.
Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 181: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/181.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A.
Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 182: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/182.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L.
If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 183: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/183.jpg)
Introduction Introduction Introduction
Filters and deductive systems on residuated lattices
Definition
Let L be a residuated lattice. A non–void set F ⊆ L is a filter on Lif (i) x , y ∈ F implies x � y ∈ F and (ii) x ∈ F , x ≤ y impliesy ∈ F . Further a non–void set D ⊆ L is a deductive system on L if(a) 1 ∈ D and (b) x , x → y ∈ D implies y ∈ D.
Lemma
A ⊆ L is a filter on L iff A is a deductive system on L.
Proof. Let A be a filter on L. Then by (ii) 1 ∈ A. Letx , x → y ∈ A. Then by (i), (x → y)� x ∈ A. But(x → y)� x ≤ y so that by (ii) y ∈ A. Conversely, let A be a dson L. If x ∈ A, x ≤ y , then x → y = 1 ∈ A which by (b) impliesy ∈ A, consequently (ii) holds.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 184: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/184.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that
y → (x � y) ∈ A, moreover, sincey ∈ A, also x � y ∈ A by (b). Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X . It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L. A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 185: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/185.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that y → (x � y) ∈ A,
moreover, sincey ∈ A, also x � y ∈ A by (b). Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X . It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L. A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 186: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/186.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that y → (x � y) ∈ A, moreover, sincey ∈ A, also x � y ∈ A by (b).
Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X . It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L. A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 187: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/187.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that y → (x � y) ∈ A, moreover, sincey ∈ A, also x � y ∈ A by (b). Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X . It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L. A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 188: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/188.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that y → (x � y) ∈ A, moreover, sincey ∈ A, also x � y ∈ A by (b). Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X .
It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L. A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 189: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/189.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that y → (x � y) ∈ A, moreover, sincey ∈ A, also x � y ∈ A by (b). Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X . It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L. A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 190: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/190.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that y → (x � y) ∈ A, moreover, sincey ∈ A, also x � y ∈ A by (b). Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X . It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L. A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 191: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/191.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that y → (x � y) ∈ A, moreover, sincey ∈ A, also x � y ∈ A by (b). Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X . It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L.
A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 192: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/192.jpg)
Introduction Introduction Introduction
To see that (i) holds too, let x , y ∈ A. Since (ii) holds andx ≤ y → (x � y), it follows that y → (x � y) ∈ A, moreover, sincey ∈ A, also x � y ∈ A by (b). Hence (i) is valid. The proof iscomplete.
If X ⊆ L, a filter generated by X , denoted G (X ), is the smallestfilter containing X . It is an exercise to prove that G (∅) = {1} andif X 6= ∅ then
G (X ) = {y ∈ L| x1 � · · · � xn ≤ y for some x1, · · · , xn ∈ X}.
Further, if F is a filter on L and x ∈ L then
G (F , x) = {y ∈ L| f � xn ≤ y for some f ∈ F , n ∈ N}.
A filter F is proper if F 6= L. A proper filter is maximal if it is notstrictly contained in any proper filter on L.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 193: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/193.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F . ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .Conversely, suppose that the condition holds and that F is proper.Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N . Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal. The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 194: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/194.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F .
ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .Conversely, suppose that the condition holds and that F is proper.Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N . Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal. The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 195: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/195.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F . ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .
Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .Conversely, suppose that the condition holds and that F is proper.Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N . Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal. The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 196: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/196.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F . ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .
Conversely, suppose that the condition holds and that F is proper.Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N . Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal. The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 197: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/197.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F . ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .Conversely, suppose that the condition holds and that F is proper.
Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N . Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal. The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 198: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/198.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F . ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .Conversely, suppose that the condition holds and that F is proper.Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N .
Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal. The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 199: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/199.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F . ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .Conversely, suppose that the condition holds and that F is proper.Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N . Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).
Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal. The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 200: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/200.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F . ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .Conversely, suppose that the condition holds and that F is proper.Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N . Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal.
The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 201: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/201.jpg)
Introduction Introduction Introduction
Lemma
A proper filter F ⊆ L is maximal if and only if for all x ∈ L \ F ,there is an n ∈ N such that (xn)∗ ∈ F .
Proof. Let F be a maximal filter and x ∈ L \ F . ThenG (F , x) = L, so that 0 = f � xn for some f ∈ F and some n ∈ N .Then, by residuation, f ≤ (xn)∗ which implies (xn)∗ ∈ F .Conversely, suppose that the condition holds and that F is proper.Let x /∈ F . Then x ∈ G (F , x), so xm ∈ G (F , x) for all m ∈ N . Onthe other hand there is an n ∈ N such that (xn)∗ ∈ F ⊆ G (F , x).Then, however, 0 = xn � (xn)∗ ∈ G (F , x), so that G (F , x) = Lwhich implies that F is maximal. The proof is complete.
PART I. Order, posets, lattices and residuated lattices in logic
![Page 202: PART I. Order, posets, lattices and residuated lattices in logiceturunen/AppliedLogics007/Order.pdf · 2007-10-22 · Introduction Introduction Introduction PART I. Order, posets,](https://reader034.vdocuments.net/reader034/viewer/2022050511/5f9bf456f2c20c3488547e06/html5/thumbnails/202.jpg)
Introduction Introduction Introduction
ExercisesExercise 1. Assume A is the set of all human beings. Define a binary relation Ron A by xRy if person x understands person’s y language. Is R a quasi–orderon A?
Exercise 2. Prove Lemma 1.
Exercise 3. Prove Lemma 2.
Exercise 4. Prove Lemma 3.
Exercise 5. Prove that in a Boolean algebra (a) the lattice complement x∗ ofx ∈ L is unique, (b) for all x , y ∈ L, x ∧ x∗ = y ∧ y∗ and x ∨ x∗ = y ∨ y∗, (c)x ∧ x∗ is the least element of L, (d) x ∨ x∗ is the greatest element of L.
Exercise 6. Prove that the above mentioned structures (Godel, Product,Lukasiewicz, Lc and LsD) are residuated lattices.
Exercise 7. Prove equations (8) – (23).
Exercise 8. Prove equations (24) – (29).
Exercise 9. Prove equations (30) – (33).
Exercise 10. Prove equations (38), (39) and (41).
PART I. Order, posets, lattices and residuated lattices in logic