IEGULDIJUMS TAVA NAKOTNE

Projekts Nr. 2009/0216/1DP/1.1.1.2.0/09/APIA/VIAA/044

ORTHOPOSETS: A CONSTRUCTION

OF QUANTIFIERS

Janis CırulisUniversity of Latvia

email: jc@lanet.lv

AAA 81

Salzburg, February 4–6, 2011

Thesis: A quantifier on an ordered algebra A is a closure re-

traction (i.e., a closure operator whose range is a subalgebra of A).

(Need not be a homomorphism!)

Aim: We are interested in systems of quantifiers – indexed

families (∇t: t ∈ T ) of quantifiers on A, where¦ T is a (meet) semilattice,¦ ∇s∇t = ∇s∧t,¦ every element of A belongs to the range of some ∇t.

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OVERWIEV

1 Quantifiers on a Boolean algebra: basic facts

2 What do hold true in orthoposets?

3 Maximal orthogonal subsets in an orthoposet

4 Quantifiers induced by maximal orthogonal subsets

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1. QUANTIFIERS ON A BOOLEAN ALGEBRA

1.1 Standard quantifier axioms (A.Tarski & F.B.Tompson

1952, P.Halmos 1955)

A quantifier on a Boolean algebra B is a unary operation ∇ such

that¦ ∇0 = 0,¦ a ≤ ∇a,¦ ∇(a ∧∇b) = ∇a ∧∇b.

Proposition. Every quantifier is an additive (even completely

additive) closure operator.

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1.2 Quantifier axioms: another (equivalent) version (Ch. Davis

1954)

A quantifier on a Boolean algebra B is a unary operation ∇ such

that¦ a ≤ ∇a,¦ if a ≤ b, then ∇a ≤ ∇b,¦ ∇(∼∇a) = ∼∇a.

Origin: modal S5 operators.

Another name: a symmetric closure operator.

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1.3 Quantifiers are closure retractions

Proposition (P. Halmos 1955).

An operation ∇ is a quantifier on a Boolean algebra B iff it is a

closure operator whose range is a subalgebra of B.

A subset M of B is the range of a closure operator C iff, for every p ∈ B,

C(p) = min{x ∈ M : p ≤ x}.

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1.4 Quantifiers and partitions

Proposition (Ch. Davis 1950).

Let X be any set. An operation ∇ on P(X) is a quantifier iff

there is a partition U of X such that

∇(A) =⋃(Y ∈ U : Y ∩A 6= ∅)

for every A ∈ P(X).

(if this is the case, then ∇(A) =⋃

(∇({x}): x ∈ A))

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2. QUANTIFIERS ON ORTHOPOSETS

2.1 Preliminaries: orthoposets

An orthoposet (or orthocomplemented poset) is a system

(P,≤,∼ ,1), where¦ (P,≤ 1) is a poset with the greatest element,¦ ∼ is a unary operation on P such that

¦ p ≤ q implies that ∼ q ≤ ∼ p,¦ ∼∼ p = p,¦ 1 = p ∨ ∼ p.

Let 0 := ∼1; then 0 is the least element of P and¦ 0 = p ∧ ∼ p.

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Elements p and q of P are orthogonal (in symbols, p ⊥ q) if

p ≤ ∼ q.

Properties of the orthogonality relation:

¦ if p ⊥ q, then q ⊥ p,¦ p ⊥ p iff p = 0,¦ if p ≤ q and q ⊥ r, then p ⊥ q,¦ p ≤ q iff p ⊥ r for all r with q ⊥ r,¦ if P0 ⊆ P and

∨P0 exists, then

r ⊥ ∨P0 iff r ⊥ p for every p ∈ P0.

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P – an orthoposet.

A suborthoposet of P is any subset P0 of P containing 1 and

closed under ∼ .

We may view P as a partial ortholattice (P,∨,∧,∼ ,1).

A suborthoposet of P is called a partial subortholattice if it is

closed also under existing joins and, hence, meets.

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2.2 Quantifiers

Proposition (M.F. Janowitz, 1963).

On an ortomodular lattice L,

(a) every standard quantifier is a symmetric closure operator,

(b) every center-valued symmetric closure operator is a standard

quantifier,

(c) there are symmetric closure operators that are not standard

quantifiers.

However,¦ a closure retract on an orthomodular lattice need not be a

standard quantifier,¦ already in an ortholattice, a standard quantifier need not be

a symmetric closure operator.

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By a quantifier on an orthoposet P we shall mean a symmetric

closure operator

¦ a ≤ ∇a, ¦ if a ≤ b, then ∇a ≤ ∇b, ¦ ∇(∼∇a) = ∼∇a.

On ortholattices: I. Chajda & H. Langer 2009.

Lemma (after Ch.Davis, 1954).

Every quantifier has the following properties:¦ ∇1 = 1, ∇0 = 0,¦ ∇∇p = ∇p,¦ p ≤ ∇p iff ∃p ≤ ∇q,¦ the range of ∇ is closed under existing meets and joins,¦ if p ∨ q exists, then ∇(p ∨ q) = ∇(p) ∨∇(q).

Corollary. An operation on P is a quantifier iff it is a closure

retraction. The range of a quantifier is even a partial subortho-

lattice of P .11

3 MAXIMAL ORTHOGONAL SETS

3.1 Some definitions

P – an orthoposet.

A subset P0 ⊆ P is¦ orthogonal if it is empty or either its elements differ from 0

and are pairwise orthogonal,¦ orthocomplete, if every its subset has a l.u.b.

Every orthogonal subset is included in a maximal one.

Let V stand for the set of all maximal orthogonal subsets of P .

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For V ∈ V, let

V ∗:= {p ∈ P : for every v ∈ V , either v ≤ p or v ⊥ p}.

Lemma.

(a) An element belongs to V ∗ iff it is the join of some (uniquely

defined) subset of V ,

(namely, p =∨

(v ∈ V : v ≤ p))

(b) the relation ≤ on V defined by

U ≤ V iff U ⊆ V ∗ (iff U∗ ⊆ V ∗)is an ordering with the least element ⊥ := {1},(c) V has the greatest element iff P = V ∗ for some V ∈ V.

If V is orthocomplete, then V ∗ ' P(V ).

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3.2 Quantifiers induced by subsets from V

Theorem 1.

Suppose that P is an orthoposet and V is a maximal orthogonal

subset in it. Then

(a) the mapping

∇V : p 7→ ∨(v ∈ V : v 6⊥ p)

is a quantifier with range V ∗,(b) a mapping ∇: P → V ∗ is a quantifier iff ∇ = ∇U for some

U ≤ V

Quantifiers of this type will be called V-quantifiers.

Warning: the needed suprema in (a) may not exist for some p

(i.e., the set V induces a totally defined quantifier only if it is

”sufficiently complete”).

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Once more the theorem:

Suppose that V ∈ V. Then

(a) the mapping ∇V defined by

∇V (p) :=∨(v ∈ V : v 6⊥ p)

is a quantifier with range V ∗,(b) a mapping ∇: P → V ∗ is a quantifier iff ∇ = ∇U for some

U ≤ V .

Recall:

Proposition.

Let X be any set. An operation ∇ on P(X) is a quantifier iff there is a

partition U of X such that

∇(A) =⋃

(Y ∈ U : Y ∩A 6= ∅)

for every A ∈ P(X).

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Examples.

(a) The simple quantifier ∇⊥:

∇⊥(p) =

{1 if p = 1,0 otherwise.

(b) The discrete quantifier Υ defined by

Υ(p) = p for all p

is a V-quantifier iff V has the greatest element.

(c) If V := {p,∼ p}, then V ∗ = {0, p,∼ p,1} and

∇V (q) =

0 if q = 0,p if 0 6= q ≤ p,

∼ p if 0 6= q ⊥ p,1 otherwise.

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4 THE FAMILY OF V-QUANTIFIERS

Recall:

We are interested in systems of quantifiers – indexed families (∇t: t ∈ T ) of

quantifiers on A, where¦ T is a (meet) semilattice,¦ ∇s∇t = ∇s∧t,¦ every element of A belongs to the range of some ∇t.

A := P, T := V.

Questions: Is it the case that

(a) the poset V is a semilattice?(b) ∇U∇V = ∇U∧V ?(c) every element of P is in the range of some V-quantifier?

(a) – yes, if P is orthocomplete,(b) – not for all U and V ,(c) – yes (evidently).

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4.1 Structure of V

Recall:

the relation ≤ on V defined by

U ≤ V iff U ⊆ V ∗ (iff U∗ ⊆ V ∗)

is an ordering with the least element ⊥ := {1}.

Theorem 2. If P is orthocomplete, then the poset V is bounded

complete

i.e., every subset of V having an upper bound has the l.u.b.;

equivalently, every non-empty subset of V has the g.l.b..

In particular, V is then a meet semilattice (V,∧,⊥).

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Let U, V, W ∈ V and U, V ≤ W .

For u ∈ U , v ∈ V , let Wu,v := {w ∈ W : w ≤ u, v}.Then U ∨ V = {∨ Wu,v: u ∈ U, v ∈ V and Wu,v 6= ∅}.

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4.2 Permutation

Theorem 3. Suppose that quantifiers ∇U un ∇V are defined.

Then the following conditions are equivalent:¦ U ≤ V ,¦ ∇U∇V = ∇U ,¦ ∇V∇U = ∇U¦ ran∇U ⊆ ran∇V ,¦ ∇U ≥ ∇V (i.e., ∇U(p) ≥ ∇V (p) for every p ∈ P ).

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Let us call two sets U, V ∈ V correlated if, for all u ∈ U, v ∈ V ,

u ⊥ v iff u and v have no upper bound in U ∧ V .

It is the case, for instance, when U ≤ V or when u ⊥ v never

hold.

Theorem 4. Suppose that U, V ∈ V and that both quantifiers

∇U and ∇V exist. Then the following are equivalent:¦ ∇U∧V exists, and ∇U∇V = ∇U∧V ,¦ U and V are correlated.

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Once more:

Let us call two sets U, V ∈ V correlated if, for all u ∈ U, v ∈ V ,

u ⊥ v iff u and v have no upper bound in U ∧ V .

Example. Suppose that U = {p,∼ p}, V = {q,∼ q} and p < q.

Then ∼ q < ∼ p, and¦ U ∧ V = ⊥,¦ U and V are not correlated, and¦ ∇U∇V (p) = ∇U(q) = 1, but ∇V∇U(p) = ∇V (p) = q.

Example. Suppose that U = {p,∼ p}, V = {q,∼ q}, p 6⊥ q and

p 6≤ q. Then¦ U ∧ V = ⊥,¦ U and V are correlated, and¦ ∇U∇V = ∇U∧V = ∇V∇U .

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