quantum logics as relational...

Quantum logics as relational monoids

Gejza JencaAnna Jencová

June 24, 2015

I Every effect algebra is a monoid in Rel.I Some well-known definitions in the theory of effect algebras,

appear to come from 2-categorial structure of RelMon.I Maybe I will sketch some related result concerning test

spaces.I Almost everything here has almost nonexistent proof.

I Every effect algebra is a monoid in Rel.

I Some well-known definitions in the theory of effect algebras,appear to come from 2-categorial structure of RelMon.

I Maybe I will sketch some related result concerning testspaces.

I Almost everything here has almost nonexistent proof.

appear to come from 2-categorial structure of RelMon.

I Maybe I will sketch some related result concerning testspaces.

spaces.

spaces.I Almost everything here has almost nonexistent proof.

The category of sets and relations

...denoted by Rel.

I Objects: sets.I Morphisms: binary relations; f : A → B in Rel is a set of pairs

f ⊆ A × B.I Identities: idA : A → A is the identity relation.I Composition: if f : A → B and g : B → C, then (a, c) ∈ g ◦ f iff

there exists b ∈ B such that (a, b) ∈ f and (b , c) ∈ g.

...denoted by Rel.I Objects: sets.

I Morphisms: binary relations; f : A → B in Rel is a set of pairsf ⊆ A × B.

I Identities: idA : A → A is the identity relation.I Composition: if f : A → B and g : B → C, then (a, c) ∈ g ◦ f iff

...denoted by Rel.I Objects: sets.I Morphisms: binary relations; f : A → B in Rel is a set of pairs

f ⊆ A × B.

I Identities: idA : A → A is the identity relation.I Composition: if f : A → B and g : B → C, then (a, c) ∈ g ◦ f iff

...denoted by Rel.I Objects: sets.I Morphisms: binary relations; f : A → B in Rel is a set of pairs

f ⊆ A × B.I Identities: idA : A → A is the identity relation.I Composition: if f : A → B and g : B → C, then (a, c) ∈ g ◦ f iff

Rel is a monoidal category

I The usual direct product of sets × : Rel × Rel→ Rel is abifunctor...

I ...so (Rel,×, 1) is a monoidal category...I ...because (Set,×, 1) is one.I However, × is not the product in Rel, because...I ...disjoint union t is product and, at the same time, coproduct

in Rel.

I ...so (Rel,×, 1) is a monoidal category...

I ...because (Set,×, 1) is one.I However, × is not the product in Rel, because...I ...disjoint union t is product and, at the same time, coproduct

in Rel.

I ...so (Rel,×, 1) is a monoidal category...I ...because (Set,×, 1) is one.

I However, × is not the product in Rel, because...I ...disjoint union t is product and, at the same time, coproduct

in Rel.

I ...so (Rel,×, 1) is a monoidal category...I ...because (Set,×, 1) is one.I However, × is not the product in Rel, because...

I ...disjoint union t is product and, at the same time, coproductin Rel.

I ...so (Rel,×, 1) is a monoidal category...I ...because (Set,×, 1) is one.I However, × is not the product in Rel, because...I ...disjoint union t is product and, at the same time, coproduct

in Rel.

Monoids in a monoidal category

Recall, that a monoid is a monoidal category (C,⊗, 1) is a triple(M,m, e), where M is an object of C, m : M ⊗M → M ande : 1→ M are arrows such that the diagrams

1 ⊗Me⊗idM //

M ⊗M

m��

M ⊗ 1idM⊗eoo

(M ⊗M) ⊗M ' //

m⊗idM��

M ⊗ (M ⊗M)

idM⊗m��

M ⊗M

Examples of monoids

I Monoids in (Set,×, 1) are ordinary monoids.

I Monoids in the monoidal category of Abelian groups(Ab,⊗,Z) are rings.

I Monoids in the monoidal category of complete joinsemilattices (Sup,⊗, 2) are quantales.

I Monoids in the monoidal category of ordinary monoids(Mon,×, 1) are commutative monoids.

Examples of monoids

I Monoids in (Set,×, 1) are ordinary monoids.I Monoids in the monoidal category of Abelian groups(Ab,⊗,Z) are rings.

Examples of monoids

Monoids in Rel

So a monoid in the monoidal category (Rel,×, 1) consists ofI a set M,I a relation e : 1→ M andI a relation ∗ : M ×M → M.

such that some diagrams commute.We call these objects relational monoids

When dealing with monoids in Rel, one should bear in mind thatboth m : A × A → A and e : 1→ A are relations, and notmappings.

That means, among other things, thatI e is (essentially) a subset of the underlying set, rather than an

element;I it is misleading to write a ∗ b = c to express the fact that(a, b) is in the relation ∗ with c;

I we write (a, b)∗7−→ c instead.

When dealing with monoids in Rel, one should bear in mind thatboth m : A × A → A and e : 1→ A are relations, and notmappings.That means, among other things, thatI e is (essentially) a subset of the underlying set, rather than an

element;I it is misleading to write a ∗ b = c to express the fact that(a, b) is in the relation ∗ with c;

I we write (a, b)∗7−→ c instead.

Classes of monoids in Rel

I Ordinary monoids are monoids in Rel.

I Hypergroups/hypermonoids are monoids in Rel.I Partial monoids (including effect algebras and some of their

generalizations) are monoids in Rel.I Small categories are monoids in Rel:

I elements are arrows,I the e : 1→ M is the selection of identity arrows.

I Ordinary monoids are monoids in Rel.I Hypergroups/hypermonoids are monoids in Rel.

I Partial monoids (including effect algebras and some of theirgeneralizations) are monoids in Rel.

I Small categories are monoids in Rel:I elements are arrows,I the e : 1→ M is the selection of identity arrows.

I Ordinary monoids are monoids in Rel.I Hypergroups/hypermonoids are monoids in Rel.I Partial monoids (including effect algebras and some of their

generalizations) are monoids in Rel.

I Small categories are monoids in Rel:I elements are arrows,I the e : 1→ M is the selection of identity arrows.

I elements are arrows,

I the e : 1→ M is the selection of identity arrows.

Morphisms of monoids in Rel

The class of monoids in a monoidal category comes equipped witha standard notion of morphism:

M ×M h×h //

��

M′ ×M′

��M h // M′

homewer, this notion does not work in the examples we areinterested in.

Rel as a 2-category

I A relation f ⊆ A × B is a set of pairs, soI every homset Rel(A ,B) is a poset under ⊆.I That means, that Rel is enriched in Pos, in other wordsI Rel a (locally posetal/thin) 2-category.

Morphisms of monoids in Rel

There are several meaningful notions of morphisms of monoids inRel. In this talk, we shall deal with two of them.

M ×M h×h //

��

M′ ×M′

��

M h // M′

Lax morphism

(h ◦ ∗) ⊆ (∗ ◦ (h × h))

h ◦ e ⊆ e

M ×M h×h //

��

M′ ×M′

��

M h // M′

Oplax morphism

(h ◦ ∗) ⊇ (∗ ◦ (h × h))

e ⊆ h ◦ e

Category of relational monoids RelMon

By Category of relational monoids we mean a 2-categoryI 0-cells are relational monoids,I 1-cells are lax morphisms of relational monoids,I 2-cells are the ⊆ of relations, inherited from Rel.

I The category of small categories is a 1-subcategory of thiscategory.

I The category of effect algebras is a subcategory of thiscategory.

Category of relational monoids RelMon

By Category of relational monoids we mean a 2-categoryI 0-cells are relational monoids,I 1-cells are lax morphisms of relational monoids,I 2-cells are the ⊆ of relations, inherited from Rel.

I The category of small categories is a 1-subcategory of thiscategory.

I The category of effect algebras is a subcategory of thiscategory.

Effect algebras

An effect algebra (Foulis and Bennett [1994], Kôpka and Chovanec[1994], Giuntini and Greuling [1989]) is a partial algebra(E; +, 0, 1) with a binary partial operation + and two nullaryoperations 0, 1 such that + is commutative, associative and thefollowing pair of conditions is satisfied:

(E3) For every a ∈ E there is a unique a′ ∈ E such that a + a′

exists and a + a′ = 1.

(E4) If a + 1 is defined, then a = 0.

The + operation is then cancellative and 0 is a neutral element.

I Because RelMon is a 2-category, so it has a lot of structure.

I We can take the standard definitions of 2-categorial thingsfrom RelMon and examine what they mean for effectalgebras.

I We rediscover well-known notions, but now we know wherethey are coming from.

I Because RelMon is a 2-category, so it has a lot of structure.I We can take the standard definitions of 2-categorial things

from RelMon and examine what they mean for effectalgebras.

I Because RelMon is a 2-category, so it has a lot of structure.I We can take the standard definitions of 2-categorial things

from RelMon and examine what they mean for effectalgebras.

Easy and nice

Let E be an effect algebra.

I ≤ : E → E is a left Kan extension of + along the projectionp1 : E × E → E.

I An effect algebra E satisfies the Riesz decompositionproperty iff ≥ is an endomorphism of E.

Easy and nice

Let E be an effect algebra.I ≤ : E → E is a left Kan extension of + along the projection

p1 : E × E → E.

I An effect algebra E satisfies the Riesz decompositionproperty iff ≥ is an endomorphism of E.

Easy and nice

Let E be an effect algebra.I ≤ : E → E is a left Kan extension of + along the projection

p1 : E × E → E.I An effect algebra E satisfies the Riesz decomposition

property iff ≥ is an endomorphism of E.

Adjoint pairs of morphisms in RelMon

Since RelMon is a 2-category, we may speak about adjoint pairs ofmorphisms in RelMon. Unwinding the definition, it turns out everyleft adjoint in RelMon is a mapping.LetI A ,B be relational monoids,I f : A → B,I g : B → A .

Then the morphism f is left adjoint to the morphism g, if and only iff is a mapping and g = f−1.

Left adjoints in RelMon

From this, we obtain a characterization of left adjoints:

PropositionA morphism f : A → B in RelMon is a left adjoint if and only if f isa mapping and

I for all b1, b2 ∈ B and a ∈ A such that (b1, b2)∗7−→ f(a),

I there exist a1, a2 ∈ A such that b1 = f(a1), b2 = f(a2) and(a1, a2)

∗7−→ a.

What if A and B are effect algebras?

TheoremLet A ,B be effect algebras, let f : A → B be a morphism of effectalgebras. Then f is a left adjoint in RelMon iff

I f is surjective andI the equivalence on A induced by f is an effect algebra

congruence in the sense of (Gudder and Pulmannová [1998]).

What if A and B are effect algebras?

TheoremLet A ,B be effect algebras, let f : A → B be a morphism of effectalgebras. Then f is a left adjoint in RelMon iffI f is surjective andI the equivalence on A induced by f is an effect algebra

congruence in the sense of (Gudder and Pulmannová [1998]).

Monads in RelMon

Since RelMon is a 2-category, we may speak about monads inRelMon.A monad in RelMon on a relational monoid A can be characterizedas a preorder relation ≤: A → A such that

A × A ≤×≤ //

��

A × A

��A

≤// A

AI1 e //

e ��

��A

commute.

Monads arising from adjunctions in 2-categories

I In every 2-category, an adjoint pair of morphisms gives rise toa monad.

I In Cat, every monad arises from an adjoint pair(Eilenberg-Moore, Kleisli).

I But this is not true in every 2-category.I In particular, in Rel the monads (=preorders) arising from

adjunctions can be characterized as equivalence relations.

Monads arising from adjunctions in RelMon

If a monad ∼: A → A arises from an adjunction, then

I ∼ is an equivalence relation,I the diagram

A × A ∼×∼ //

��

A × A

��A ∼

commutes andI if x is a unit of A and x ∼ y, then y is a unit of A .

Monads arising from adjunctions in RelMon

If the multiplication is actually a partial operation, we obtainanother property of a monad arising from an adjunction:I If a1 ∼ b1, a2 ∼ b2 and both a1 ∗ a2 and b1 ∗ b2 exist, then

a1 ∗ a2 ∼ b1 ∗ b2.

“Dimension equivalences” on effect algebras

For an effect algebra E, we may characterize monads ∼: E → Earising from adjunctions in RelMon as follows:I ∼ is an equivalence.I If a1 ∼ b1, a2 ∼ b2 and both a1 + a2 and b1 + b2 exist, then

a1 + a2 ∼ b1 + b2.I If a ∼ b1 + b2, then there are a1, a2 such that a = a1 + a2,

a1 ∼ b1, a2 ∼ b2.

E/ ∼ is then a partial monoid.

Example

I Take a Boolean algebra B; this is an effect algebra with +being the disjoint join.

I Introduce a equivalence on B by the rule

a ∼ b ⇔ [0, a] ' [0, b]

Then this is a dimension equivalence.

A more fancy example

I Let A be an involutive ring with unit, in which

x∗x + y∗y = 0 =⇒ x = y = 0.

I Let P(A) be the set of all self-adjoint idempotents in A . Fore, f ∈ P(A), write e ⊕ f = e + f iff ef = 0, otherwise let e ⊕ fbe undefined. Then (P(A);⊕, 0, 1) is an effect algebra.

I For e, f in P(A), write e ∼ f iff there is w ∈ A such thate = w∗w and f = ww∗.

I Then this is a dimension equivalence.

Theorem(Dvurecenskij and Pulmannová [2000]) For every cancellativepositive partial abelian monoid E and every dimension equivalence∼ on P, P/ ∼ is a positive partial abelian monoid.

A new perspective:I P/ ∼ is the EM-object for the monad ∼.

A new perspective:

I P/ ∼ is the EM-object for the monad ∼.

A new perspective:I P/ ∼ is the EM-object for the monad ∼.

Comonads

Let E be an effect algebra. A relation i : E → E is a comonad inRelMon iff

i = {(x, x) : x ∈ I}

where I is an order ideal of E.

I is then the EM object for i.

Comonads

Let E be an effect algebra. A relation i : E → E is a comonad inRelMon iff

i = {(x, x) : x ∈ I}

where I is an order ideal of E. I is then the EM object for i.

Test spaces

I A test space is a pair (X ,T ), where X is a set

I and T is a system of subsets of X , called tests,I such that no two tests are comparable.I A subset of a test is an event.

Test spaces

I A test space is a pair (X ,T ), where X is a setI and T is a system of subsets of X , called tests,

I such that no two tests are comparable.I A subset of a test is an event.

Test spaces

I A test space is a pair (X ,T ), where X is a setI and T is a system of subsets of X , called tests,I such that no two tests are comparable.

I A subset of a test is an event.

Test spaces

I A test space is a pair (X ,T ), where X is a setI and T is a system of subsets of X , called tests,I such that no two tests are comparable.I A subset of a test is an event.

Relations on events

I Two events a, b are said to be orthogonal (in symbols a ⊥ b)if they are disjoint and their union is an event.

I It is obvious that the set of all events of a test space equippedwith the disjoint union of orthogonal events + and ∅ is apartial commutative monoid.

I If a ⊥ b and a ∪ b is a test, then they are complements ofeach other (in symbols a co b).

Relations on events

Algebraic test spaces

I A test space is algebraic if co = co ◦ co ◦ co.

I For an algebraic test space, the relation ∼:= co ◦ co, calledperspectivity is an equivalence on events and

I the partial abelian monoid of events, factored by ∼ is anorthoalgebra, i.e. and effect algebra with a ∧ a′ = 0.

Algebraic test spaces

I A test space is algebraic if co = co ◦ co ◦ co.I For an algebraic test space, the relation ∼:= co ◦ co, called

perspectivity is an equivalence on events andI the partial abelian monoid of events, factored by ∼ is an

orthoalgebra, i.e. and effect algebra with a ∧ a′ = 0.

A characterization of algebraic test spaces

PropositionLet (X ,T ) be a test space. Let us write (A ,+, ∅) for the partialcommutative monoid of the events of (X ,T ). The following areequivalent.

1. (X ,T ) is algebraic.

2. ∼ is an equivalence relation and an oplax endomorphism of A.

3. ∼ is a preorder and an oplax endomorphism of A.

4. (A , +) is associative, where + = ∼ ◦+.

Moreover, if ∼ is a preorder and ∼ is an lax and oplaxendomorphism of A , then A/ ∼ is a Boolean algebra.

Thank you for your attention.

A. Dvurecenskij and S. Pulmannová. New Trends in QuantumStructures. Kluwer, Dordrecht and Ister Science, Bratislava,2000.

D.J. Foulis and M.K. Bennett. Effect algebras and unsharpquantum logics. Found. Phys., 24:1325–1346, 1994.

R. Giuntini and H. Greuling. Toward a formal language for unsharpproperties. Found. Phys., 19:931–945, 1989.

S. Gudder and S. Pulmannová. Quotients of partial abelianmonoids. Algebra univers., 38:395–421, 1998.

F. Kôpka and F. Chovanec. D-posets. Math. Slovaca, 44:21–34,1994.

quantum logics as relational...

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