umberto rivieccio - peoplepeople.maths.ox.ac.uk/hap/rivieccioslides.pdf · 2012-06-18 ·...

Implicative twist-structures

Umberto Rivieccio

University of Birmingham

W1 on Duality Theory in Algebra,Logic and Computer Science

14 June, Oxford

U. Rivieccio (UOB) Implicative twist-structures Duality W1, 14 June, Oxford 1 / 18

Outline

1 The Arieli-Avron bilattice logic

2 Its algebraic semantics: implicative bilattices

3 The implication-negation fragment of the logic

4 Its algebraic counterpart: implicative twist-structures

Outline

Introduction

The Arieli-Avron logic of bilattices LBIn 1996 O. Arieli and A. Avron introduced the logic LB of “logicalbilattices” with implication(s), a logic in the languagex^,_,b,‘,Ą, y (two conjunctions, two disjunctions, a negationand an implication), which is:

a conservative expansion of the Belnap-Dunn four-valued logic(in the language x^,_, y)

a conservative expansion of the positive fragment of classicallogic (in the language x^,_,Ąy and also xb,‘,Ąy)

Introduction

The Arieli-Avron logic of bilattices LB

axiomatized by modus ponens, the axioms of positive classicallogic for both x^,_,Ąy and xb,‘,Ąy and the following ones:

p ^q pp ^ qq ” p p _ qq

p _q pp _ qq ” p p ^ qq

p bq pp b qq ” p p b qq

p ‘q pp ‘ qq ” p p ‘ qq

p Ąq pp Ą qq ” pp ^ qq

p q p ” p

where p ” q abbreviates p Ą q and q Ą p

Introduction

The Arieli-Avron logic of bilattices LB

axiomatized by modus ponens, the axioms of positive classicallogic for both x^,_,Ąy and xb,‘,Ąy and the following ones:

p ^q pp ^ qq ” p p _ qq

p _q pp _ qq ” p p ^ qq

p bq pp b qq ” p p b qq

p ‘q pp ‘ qq ” p p ‘ qq

p Ąq pp Ą qq ” pp ^ qq

p q p ” p

where p ” q abbreviates p Ą q and q Ą p

Introduction

The Arieli-Avron logic of bilattices LBSome facts:

LB enjoys the classical DDT with respect to Ą

LB is algebraizable in the sense of Blok-Pigozzi, with definingequation tϕ « ϕ Ą ϕu and equivalence formulastϕ Ą ψ, ψ Ą ϕ, ϕ Ą ψ, ψ Ą ϕu

the equivalent algebraic semantics of LB is a variety ofdistributive bilattices with an additional implication operationintroduced in the dissertation (Rivieccio, 2010), namedimplicative bilattices.

Introduction

Algebraic semantics of LBThe variety of implicative bilattices is:

a discriminator variety(congruence-distributive, congruence-permutable, has EDPC)

generated by its four-element member FOURĄ

hence, dualizable via natural duality(or using Priestley duality for distributive bilattices)

equivalent as a category to generalized Boolean algebras(i.e., 0-free subreducts of Boolean algebras).

Introduction

A remark

From an algebraic logic point of view, the implication-negationfragment is the ‘core’ of of LB, as these connectives establish thelink between the logic and its algebraic semantics.

A problem

How to axiomatize and study the implication-negation fragment ofLB (and its algebraic counterpart)?

A solution

Use some kind of twist-structure construction.

Introduction

A remark

A problem

A solution

Introduction

A remark

A problem

A solution

Twist-structures

The twist-structure construction is a way to represent severalalgebras related to non-classical logics (Nelson lattices,bilattices, involutive residuated lattices, BK-lattices) as aspecial power of (a twist-structure over) a better knownalgebraic structure (lattice, (generalized) Heyting/Booleanalgebra, modal algebra).

It has also been used to introduce new algebraic structures,e.g., to construct a “residuated bilattice” as a special power ofan arbitrary residuated lattice.

An application: the categorial equivalence between implicativebilattices and generalized Boolean algebras can be establishedby proving that any implicative bilattice is isomorphic to atwist-structure over a generalized Boolean algebra.

Twist-structures

The full twist-structure construction

Let L “ xL,[,\,Ñy be a lattice with an implication Ñ (e.g.,Heyting or Boolean implication). The full twist-structure over L isthe algebra L’ “ xLˆ L,^,_,b,‘,Ą, y with operations definedas follows:

^ “ [ˆ\

_ “ \ˆ[

b “ [ˆ[

‘ “ \ˆ\

and, for all xa1, a2y , xb1, b2y P Lˆ L,

xa1, a2y Ą xb1, b2y “ xa1 Ñ b1, a1 [ b2y

xa1, a2y “ xa2, a1y

Twist-structures

^ “ [ˆ\

_ “ \ˆ[

b “ [ˆ[

‘ “ \ˆ\

Twist-structures

^ “ [ˆ\

_ “ \ˆ[

b “ [ˆ[

‘ “ \ˆ\

Twist-structures

Known results

Any interlaced bilattice (with implication, conflation) can berepresented as a full twist-structure, in the appropriatelanguage, over a lattice (with implication, involution).This extends in all cases to a categorial equivalence between acategory of (enriched) bilattices and a category of (enriched)lattices.

Nelson lattices and some related algebras can be embeddedinto full twist-structures (not an equivalence).

These representation results can be proved in several ways: weare here interested in one that only use the implication and thenegation.

Twist-structures

Known results

Twist-structures

Known results

Twist-structures

How to exploit the twist-structure construction

1 Introduce a variety of algebras in the language xĄ, y that wecall I-algebras.

2 Prove that any algebra in this variety is embeddable into a fulltwist-structure.

3 Invoke the representation of implicative bilattices as fulltwist-structures to establish that I-algebras are precisely thetĄ, u-subreducts of implicative bilattices.

4 Obtain an axiomatization of the tĄ, u-fragment of LB bydefining a logic whose equivalent algebraic semantics is theclass of I-algebras.

Twist-structures

I-algebrasAn I-algebra is an algebra A “ xA,Ą, y satisfying the followingequations:

(I1) px Ą xq Ą y « y

(I2) x Ą py Ą zq « px Ą yq Ą px Ą zq « y Ą px Ą zq

(I3) ppx Ą yq Ą xq Ą x « x Ą x

(I4) x Ą py Ą zq « px ˚ yq Ą z

(I5) x « x

(I6) px Ø yq Ą x « px Ø yq Ą y

where x ˚ y :“ px Ą yqx Ø y :“ ppx Ą yq ˚ p y Ą xqq ˚ ppy Ą xq ˚ p x Ą yqq

I-algebras

...and their logic

I-algebras are the equivalent algebraic semantics of the logic definedby the following axioms (with modus ponens as the only inferencerule):

p Ą pq Ą pq (Ą 1)

pp Ą pq Ą rqq Ą ppp Ą qq Ą pp Ą rqq (Ą 2)

ppp Ą qq Ą pq Ą p (Ą 3)

pp ˚ qq Ą p pp ˚ qq Ą q (˚ 1)

p Ą pq Ą pp ˚ qqq (˚ 2)

p ” p ( )

Main result

Any I-algebra is embeddable into a full twist-structure.

In fact, any I-algebra isomorphic to an implicative twist-structureover a generalized Boolean algebra L, defined as follows.

An implicative twist-structure over L is an arbitrary subalgebra A,w.r.t. to the language tĄ, u, of the full twist-structure L’ s.t.π1pAq “ L, where π1pAq “ ta1 P L : xa1, a2y P Au.

Main result

Sketch of the proof

Given an I-algebra A “ xA,Ą, y

define a relation ∼ Ď Aˆ A as: a ∼ b if and only ifa Ą b “ pa Ą bq Ą pa Ą bq and b Ą a “ pb Ą aq Ą pb Ą aq

∼ is compatible with t˚,Ąu but not with

the quotient algebra xA{∼, ˚,Ąy is the conjunction-implicationsubreduct of a Boolean algebra

xA{∼, ˚,\,Ąy is a generalized Boolean algebra, where

ras \ rbs :“ rpa Ą bq Ą bs

Sketch of the proof

Hence,

xA{∼, ˚,\,Ąy’ is a full twist-structure(i.e., an implicative bilattice)

the map h : AÑ A{∼ ˆ A{∼ given by hpaq “ xras, r asyis a tĄ, u-embedding.

Moreover,

xA{∼, ˚,\,Ąy’ (viewed as a bilattice) is generated by hpAq

if f : AÑ L’ is a tĄ, u-homomorphism from A to a fulltwist-structure L’, then there is a uniquetĄ, u-homomorphism f 1 : A{∼ ˆ A{∼ Ñ L’ such thatf 1 ¨ h “ f .

Sketch of the proof

Hence,

Moreover,

Sketch of the proof

Hence,

Moreover,

Sketch of the proof

Hence,

Moreover,

Corollaries

The previous result can be used to obtain the following:

there is an adjunction between the category of I-algebras andthe category of implicative bilattices

the variety of I-algebras is generated by its four-elementmember, which is the tĄ, u-reduct of the four-elementimplicative bilattice FOURĄ

the congruences of any I-algebra are isomorphic to those of itsassociated implicative bilattice (which are also isomorphic tothose of the underlying generalized Boolean algebra)

we can determine the subdirectly irreducible I-algebras andaxiomatize the subvarieties of I-algebras (there are exactlyfour proper and non-trivial ones).

Corollaries

Future work

Some topics to be further investigated:

give a characterization of the subsets of full twist-structures(implicative bilattices) that are carrier sets of implicativetwist-structures (I-algebras)

describe the lattice of sub-quasi-varieties of I-algebras

study implicative twist-structures from a topological point ofview (via natural duality?).

Future work

umberto rivieccio - peoplepeople.maths.ox.ac.uk/hap/rivieccioslides.pdf · 2012-06-18 ·...

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