Convexity and the Kalmbach monadwesfussn/slides/Jenca.pdfConvexityandtheKalmbachmonad GejzaJenča...

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Convexity and the Kalmbach monad

Gejza Jenča

August 10, 2018

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 1 / 42

The Plan

Monads as generalized varietiesExamples of monadsKalmbach monad on bounded posetsEffect algebrasConvex effect algebrasThe � product[0, 1]-actionsDistributive lawsThe composite monad and its algebras

Monads generalize varieties of algebrasThe ‘free algebra‘ endofunctor

Let V be a variety of universal algebras.

Write TV (X ) for the the underlying set of the free algebra generatedby the set X , so elements of TV (X ) are (equivalence classes) of termsover X .Then TV : Set→ Set is a functor:

TV (Xf−→ Y ) : F (X )→ F (Y )

replaces variable x in terms by the variable f (x).

Let V be a variety of universal algebras.Write TV (X ) for the the underlying set of the free algebra generatedby the set X , so elements of TV (X ) are (equivalence classes) of termsover X .

Then TV : Set→ Set is a functor:

TV (Xf−→ Y ) : F (X )→ F (Y )

Let V be a variety of universal algebras.Write TV (X ) for the the underlying set of the free algebra generatedby the set X , so elements of TV (X ) are (equivalence classes) of termsover X .Then TV : Set→ Set is a functor:

TV (Xf−→ Y ) : F (X )→ F (Y )

Monads generalize varieties of algebrasThe unit and the multiplication

For every set X there is a natural mapping ηX : X → TV (X ), given byηX (x) = x .

η is the unit of the monadFor every set X there is a natural mappingµX : TV (TV (X ))→ TV (X ), given by ‘flattening of terms over terms’or ‘evaluation in the free algebra’.µ is the multiplication of the monad

For every set X there is a natural mapping ηX : X → TV (X ), given byηX (x) = x .η is the unit of the monad

For every set X there is a natural mappingµX : TV (TV (X ))→ TV (X ), given by ‘flattening of terms over terms’or ‘evaluation in the free algebra’.µ is the multiplication of the monad

For every set X there is a natural mapping ηX : X → TV (X ), given byηX (x) = x .η is the unit of the monadFor every set X there is a natural mappingµX : TV (TV (X ))→ TV (X ), given by ‘flattening of terms over terms’or ‘evaluation in the free algebra’.

µ is the multiplication of the monad

For every set X there is a natural mapping ηX : X → TV (X ), given byηX (x) = x .η is the unit of the monadFor every set X there is a natural mappingµX : TV (TV (X ))→ TV (X ), given by ‘flattening of terms over terms’or ‘evaluation in the free algebra’.µ is the multiplication of the monad

Monads generalize varieties of algebrasExample: the free monoid monad

T (X ) is the set of all words over alphabet X :

X = {a, b, c} [], [a], [babbca] ∈ T (X )

For a mapping f : X → Y , T (f ) : T (X )→ T (Y ) is given by

T (f )([x1x2 . . . xn]) = [f (x1)f (x2) . . . f (xn)]

For a set X , ηX : X → T (X ) is given by

ηX (x) = [x ]

For a set X , µX : T (T (X ))→ T (X ) concatenates the words:

µX ([[aba][acd ][][da]]) = [abaacdda]

X = {a, b, c} [], [a], [babbca] ∈ T (X )

T (f )([x1x2 . . . xn]) = [f (x1)f (x2) . . . f (xn)]

ηX (x) = [x ]

X = {a, b, c} [], [a], [babbca] ∈ T (X )

T (f )([x1x2 . . . xn]) = [f (x1)f (x2) . . . f (xn)]

ηX (x) = [x ]

X = {a, b, c} [], [a], [babbca] ∈ T (X )

T (f )([x1x2 . . . xn]) = [f (x1)f (x2) . . . f (xn)]

ηX (x) = [x ]

X = {a, b, c} [], [a], [babbca] ∈ T (X )

T (f )([x1x2 . . . xn]) = [f (x1)f (x2) . . . f (xn)]

ηX (x) = [x ]

(Set, η, µ)

. So we have data of the following type:

a category Set,a functor T : Set→ Set,a natural transformation η : idSet → T ,a natural transformation µ : T 2 → T .

(Set, η, µ)

. So we have data of the following type:a category Set,

a functor T : Set→ Set,a natural transformation η : idSet → T ,a natural transformation µ : T 2 → T .

(Set, η, µ)

. So we have data of the following type:a category Set,a functor T : Set→ Set,

a natural transformation η : idSet → T ,a natural transformation µ : T 2 → T .

(Set, η, µ)

. So we have data of the following type:a category Set,a functor T : Set→ Set,a natural transformation η : idSet → T ,

a natural transformation µ : T 2 → T .

(Set, η, µ)

. So we have data of the following type:a category Set,a functor T : Set→ Set,a natural transformation η : idSet → T ,a natural transformation µ : T 2 → T .

The monad laws

Axioms of a monadRight unit axiom

T (X )T (ηX )//

idT (X ) $$

T 2(X )

��T (X )

[abac] � //

[[a][b][a][c]]_

��[abac]

Axioms of a monadRight unit axiom

T (X )T (ηX )//

idT (X ) $$

T 2(X )

��T (X )

[abac] � //

[[a][b][a][c]]_

��[abac]

AxiomsLeft unit axiom

T 2(X )

��

T (X )ηT (X )oo

idT (X )zzT (X )

[[abac]]_

��

[abac]�oo4

zz[abac]

AxiomsLeft unit axiom

T 2(X )

��

T (X )ηT (X )oo

idT (X )zzT (X )

[[abac]]_

��

[abac]�oo4

zz[abac]

Axioms of a monadAssociativity axiom

T 3(X )T (µX )//

µT (X )

��

T 2(X )

��T 2(X )

µX // T (X )

[[[ab][bc]

][[ca]]] � //

��

[[abbc][ca]

��[[ab][bc][ca]

] � // [abbcca]

Axioms of a monadAssociativity axiom

T 3(X )T (µX )//

µT (X )

��

T 2(X )

��T 2(X )

µX // T (X )[[[ab][bc]

][[ca]]] � //

��

[[abbc][ca]

��[[ab][bc][ca]

] � // [abbcca]

The definition

DefinitionLet C be a category. A monad over C is a triple (T , η, µ) such that

T : C → C,η : idC → T ,µ : T 2 → T

such that the unit and associativity axioms hold.

Algebras for a monad

DefinitionLet (T , η, µ) be a monad over C. Then an algebra for T (or T -algebra) isa pair (X , α), where

X is an object of C and

α : T (X )→ X

such that the following diagrams commute

XηX //

idX ""

T (X )

��X

T 2(X )µX //

T (α)

��

T (X )

��T (X ) α

X is an object of C andα : T (X )→ X

XηX //

idX ""

T (X )

��X

T 2(X )µX //

T (α)

��

T (X )

��T (X ) α

X is an object of C andα : T (X )→ X

XηX //

idX ""

T (X )

��X

T 2(X )µX //

T (α)

��

T (X )

��T (X ) α

Intuition/meaning of all this

An algebra α : T (X )→ X equips the set X with evaluation of terms overX .

Morphisms of algebras

DefinitionIf (A, α), (B, β) are algebras for a monad T , then a morphism of algebrasf : (A, α)→ (B, β) is a morphism f : A→ B in the underlying categorysuch that the square

T (A)T (f ) //

��

β��

commutes.

Morphisms of algebras

DefinitionIf (A, α), (B, β) are algebras for a monad T , then a morphism of algebrasf : (A, α)→ (B, β) is a morphism f : A→ B in the underlying categorysuch that the square

T (A)T (f ) //

��

β��

commutes.

The Eilenberg-Moore category

DefinitionLet (T , η, µ) be a monad on a category C. The category CT of T -algebrasand their morphisms is called the Eilenberg-Moore category of the monadT .

Algebras are algebras

If V is a variety of algebras and TV is the monad associated with V, then

SetTV ' V.

So every variety is faithfully represented by its associated monad.

Algebras are algebras

If V is a variety of algebras and TV is the monad associated with V, then

SetTV ' V.

So every variety is faithfully represented by its associated monad.

Algebras over other categories

ExampleFor every ring A, A-modules are algebras for a monad on the categoryof abelian groups.

Graphs equipped with a perfect matching are algebras for a monad onthe category of graphs.Bounded posets are algebras for a monad on the category of posets.Involutive posets are algebras for a monad on posets.Closure operators are algebras for a monad on posets.Retractions are algebras for a monad on the category C→, whenever Chas coproducts.Compact Haussdorff spaces are algebras for a monad on Set.Small categories are algebras for a monad on the category of directedmultigraphs.

ExampleFor every ring A, A-modules are algebras for a monad on the categoryof abelian groups.Graphs equipped with a perfect matching are algebras for a monad onthe category of graphs.

Bounded posets are algebras for a monad on the category of posets.Involutive posets are algebras for a monad on posets.Closure operators are algebras for a monad on posets.Retractions are algebras for a monad on the category C→, whenever Chas coproducts.Compact Haussdorff spaces are algebras for a monad on Set.Small categories are algebras for a monad on the category of directedmultigraphs.

ExampleFor every ring A, A-modules are algebras for a monad on the categoryof abelian groups.Graphs equipped with a perfect matching are algebras for a monad onthe category of graphs.Bounded posets are algebras for a monad on the category of posets.

Involutive posets are algebras for a monad on posets.Closure operators are algebras for a monad on posets.Retractions are algebras for a monad on the category C→, whenever Chas coproducts.Compact Haussdorff spaces are algebras for a monad on Set.Small categories are algebras for a monad on the category of directedmultigraphs.

ExampleFor every ring A, A-modules are algebras for a monad on the categoryof abelian groups.Graphs equipped with a perfect matching are algebras for a monad onthe category of graphs.Bounded posets are algebras for a monad on the category of posets.Involutive posets are algebras for a monad on posets.

Closure operators are algebras for a monad on posets.Retractions are algebras for a monad on the category C→, whenever Chas coproducts.Compact Haussdorff spaces are algebras for a monad on Set.Small categories are algebras for a monad on the category of directedmultigraphs.

ExampleFor every ring A, A-modules are algebras for a monad on the categoryof abelian groups.Graphs equipped with a perfect matching are algebras for a monad onthe category of graphs.Bounded posets are algebras for a monad on the category of posets.Involutive posets are algebras for a monad on posets.Closure operators are algebras for a monad on posets.

Retractions are algebras for a monad on the category C→, whenever Chas coproducts.Compact Haussdorff spaces are algebras for a monad on Set.Small categories are algebras for a monad on the category of directedmultigraphs.

ExampleFor every ring A, A-modules are algebras for a monad on the categoryof abelian groups.Graphs equipped with a perfect matching are algebras for a monad onthe category of graphs.Bounded posets are algebras for a monad on the category of posets.Involutive posets are algebras for a monad on posets.Closure operators are algebras for a monad on posets.Retractions are algebras for a monad on the category C→, whenever Chas coproducts.

Compact Haussdorff spaces are algebras for a monad on Set.Small categories are algebras for a monad on the category of directedmultigraphs.

ExampleFor every ring A, A-modules are algebras for a monad on the categoryof abelian groups.Graphs equipped with a perfect matching are algebras for a monad onthe category of graphs.Bounded posets are algebras for a monad on the category of posets.Involutive posets are algebras for a monad on posets.Closure operators are algebras for a monad on posets.Retractions are algebras for a monad on the category C→, whenever Chas coproducts.Compact Haussdorff spaces are algebras for a monad on Set.

Small categories are algebras for a monad on the category of directedmultigraphs.

ExampleFor every ring A, A-modules are algebras for a monad on the categoryof abelian groups.Graphs equipped with a perfect matching are algebras for a monad onthe category of graphs.Bounded posets are algebras for a monad on the category of posets.Involutive posets are algebras for a monad on posets.Closure operators are algebras for a monad on posets.Retractions are algebras for a monad on the category C→, whenever Chas coproducts.Compact Haussdorff spaces are algebras for a monad on Set.Small categories are algebras for a monad on the category of directedmultigraphs.

The Kalmbach embedding

[Kalmbach, 1977] proved the following

TheoremEvery bounded lattice can be embedded into an orthomodular lattice.

CorollaryOrthomodular lattices do not satisfy any special lattice equation.

[Kalmbach, 1977] proved the following

TheoremEvery bounded lattice can be embedded into an orthomodular lattice.

CorollaryOrthomodular lattices do not satisfy any special lattice equation.

Let L be a bounded lattice. Let K (L) be the set of all finite chains inL with even number of elements.

Introduce a partial order on the set K (L) by the following rule:

[a1 < a2 < · · · < a2n−1 < a2n] ≤ [b1 < b2 < · · · < b2n−1 < b2k ]

if and only if for every 1 ≤ i ≤ n there exists 1 ≤ j ≤ k such thatb2j−1 ≤ a2i−1 < a2i ≤ b2j .

Let L be a bounded lattice. Let K (L) be the set of all finite chains inL with even number of elements.Introduce a partial order on the set K (L) by the following rule:

[a1 < a2 < · · · < a2n−1 < a2n] ≤ [b1 < b2 < · · · < b2n−1 < b2k ]

if and only if for every 1 ≤ i ≤ n there exists 1 ≤ j ≤ k such thatb2j−1 ≤ a2i−1 < a2i ≤ b2j .

Then K (L) is a bounded lattice.Moreover, it is an orthomodular lattice: the orthocomplementation is

({ai}2ni=1)′ := {ai}2ni=1∆{0, 1},

where ∆ denotes the symmetric difference andthe mapping ηL : L→ K (L) given by ηL(x) = {0, x} for x > 0 andηL(0) = ∅ is a injective morphism of lattices.

K cannot be made to a functor from the category of lattices into thecategory of orthomodular lattices.However, K can be extended to a functor from the category ofbounded posets to the category of orthomodular posets;for f : P → Q is BPos, K (f ) : K (P)→ K (Q) is given by the rule

K (f )([a1 < a2 < · · · < a2n−1 < a2n]) = ∆2ni=1{f (ai )}.

[Harding, 2004] K is left adjoint to the forgetful functor U from thecategory of orthomodular posets OMP to the category of boundedposets BPos.

The Kalmbach monad

DefinitionThe Kalmbach monad (T , η, µ) on the category BPos is given as follows

T : BPos→ BPos is the Kalmbach embedding K : BPos→ OMPcomposed with the forgetful functor U : OMP→ BPos, that means,T = U ◦ K ;ηP : P → T (P) is given by

ηP(x) =

{{0, x} x > 0∅ x = 0

µP : T 2(P)→ T (P) is given by

µP([C1 < C2 < · · · < C2n−1 < C2n]) = C1∆C2∆ . . .∆C2n,

where ∆ denotes the symmetric difference of sets.

What are algebras for the Kalmbach monad?

Answer: effect algebras

Effect algebras

An effect algebra [Foulis and Bennett, 1994, Kôpka and Chovanec, 1994,Giuntini and Greuling, 1989]

(A; +, 0, 1)

+ is a binary partial operation.0, 1 are constants.

(E1) If a + b is defined, then b + a is defined and a + b = b + a.(E2) If a + b and (a + b) + c are defined, then b + c and a + (b + c) are

defined and (a + b) + c = a + (b + c).(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.

Properties

Cancellability: a + x = a + y =⇒ x = y .E is a poset under a partial order given by a ≤ b iff (∃x)a = b + x .This poset is bounded by 0 and 1.In general (E ,≤) is not a lattice.

Morphisms of effect algebras

A morphism of effect algebras f : A→ B is a mapping of the underlyingsets such that

f (0) = 0, f (1) = 1 andwhenever a + b exists in A, f (a) + f (b) exists in B andf (a + b) = f (a) + f (b).

Examples of effect algebras

ExampleAny powerset of a set; + is the disjoint union.Any other Boolean algebra.The real interval [0, 1]; a + b exists iff a + b ≤ 1 and a + b := a + b.Any other MV-algebra.Closed subspaces of a Hilbert space; p + q exists iff p ⊥ q and thenp + q = p ∪ q.Any other orthomodular lattice.

Convex effect algebras

DefinitionA convex effect algebra is an effect algebra E equipped with amultiplication by real numbers from interval [0, 1] such that, for allρ, ψ ∈ [0, 1] and a, b ∈ E ,(C1) a.1 = a

(C2) (a.ρ).ψ = a.(ρ.ψ)

(C3) If a + b is defined, then a.ρ+ b.ρ is defined and (a + b).ρ = a.ρ+ b.ρ

(C4) If ρ+ ψ < 1, then a.ρ+ a.ψ is defined and a.(ρ+ ψ) = a.ρ+ a.ψ.

Theorem[Jacobs, 2010] The category of convex effect algebras ConvEA is anEilenberg-Moore category for a monad on EA.

ProblemIs ConvEA a category of algebras for some monad on BPos?

The � product on BPos

Let A,B,C be bounded posets. We say that a BPos-morphismH : A× B → C is a 0-bimorphism if and only if, for all a ∈ A andb ∈ B , h(0, b) = h(b, 0) = 0.

Let us write A�B for the poset A× B/ ∼, where ∼ is the equivalenceon A× B generated by the relations (a, 0) ∼ (0, b), for all a ∈ Ab ∈ B .All the elements of A× B that have 0 in first or second coordinateform one of the equivalence classes of ∼, all the other elements formsingleton equivalence classes.

Let A,B,C be bounded posets. We say that a BPos-morphismH : A× B → C is a 0-bimorphism if and only if, for all a ∈ A andb ∈ B , h(0, b) = h(b, 0) = 0.Let us write A�B for the poset A× B/ ∼, where ∼ is the equivalenceon A× B generated by the relations (a, 0) ∼ (0, b), for all a ∈ Ab ∈ B .

All the elements of A× B that have 0 in first or second coordinateform one of the equivalence classes of ∼, all the other elements formsingleton equivalence classes.

Let A,B,C be bounded posets. We say that a BPos-morphismH : A× B → C is a 0-bimorphism if and only if, for all a ∈ A andb ∈ B , h(0, b) = h(b, 0) = 0.Let us write A�B for the poset A× B/ ∼, where ∼ is the equivalenceon A× B generated by the relations (a, 0) ∼ (0, b), for all a ∈ Ab ∈ B .All the elements of A× B that have 0 in first or second coordinateform one of the equivalence classes of ∼, all the other elements formsingleton equivalence classes.

Clearly, the mapping � : A× B → A�B that takes an element of A× B toits equivalence class is a 0-bimorphism. Moreover, it is an universal0-bimorphism in the following sense: for every 0-bimorphismh : A× B → C , there is a unique morphism of bounded posetsf : A�B → C such that

��

""A�B

commutes.

FactThe category (BPos,�, 2) is a monoidal category.a

aHere, 2 is a 2-element chain

[0, 1] is a monoid

PropositionThe real interval [0, 1], equipped with multiplication of reals is a monoid inthe monoidal category (BPos,�, 1).

Every monoid induces a monad

There is a monad (S , µS , ηS) on BPos associated with [0, 1]. Explicitly,S : BPos→ BPos is an endofunctor given by the ruleS(A) = A�[0, 1]

ηS : idBPos → S is a natural transformations given by ηSA(x) = x�1andµS : S ◦ S → S is a natural transformation given byµSA(x�ρ�ψ) = x�(ρ.ψ).

This monad is called the free [0, 1]-action monad on BPos.

ηS : idBPos → S is a natural transformations given by ηSA(x) = x�1and

µS : S ◦ S → S is a natural transformation given byµSA(x�ρ�ψ) = x�(ρ.ψ).

Distributive laws [Beck, 1969]

If S ,T are monads on a category, it may happen that T ◦ S can bemade to a monad.The additional data needed to do that is a natural transformationλ : ST → TS , satisfying certain conditions.

Distributive laws [Beck, 1969]

λ// TS

µST��

Sλ // STSλS // TSS

��ST

λ// TS

��

λT // TSTTλ // TTS

µTS��

ExampleThere is a distributive law between

the ‘free abelian group’ monad on Set andthe ‘free monoid’ monad on Set.

The composite monad is the ‘free ring’ monad.

Results

TheoremThere is a distributive law between

the Kalmbach monad T on BPos andthe free [0, 1]-action monad on BPos.

Results

TheoremThe category of algebras for the composite monad TS is equivalent for thecategory of effect algebras equipped with multiplication with a scalar,satisfying the following conditions:(C1) a.1 = a

(C2) (a.ρ).ψ = a.(ρ.ψ)

Note: these are not convex effect algebras, the axiom(C4) If ρ+ ψ < 1, then a.ρ+ a.ψ is defined and a.(ρ+ ψ) = a.ρ+ a.ψ.is missing.

Results

TheoremThe category of algebras for the composite monad TS is equivalent for thecategory of effect algebras equipped with multiplication with a scalar,satisfying the following conditions:(C1) a.1 = a

(C2) (a.ρ).ψ = a.(ρ.ψ)

Note: these are not convex effect algebras, the axiom(C4) If ρ+ ψ < 1, then a.ρ+ a.ψ is defined and a.(ρ+ ψ) = a.ρ+ a.ψ.is missing.

What next?

Call the algebras on the previous slide weak effect algebras.

ProblemAre convex effect algebras algebras for a monad over weak effect algebras?

Jon Beck. Distributive laws. In Seminar on triples and categoricalhomology theory, pages 119–140. Springer, 1969.

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

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

John Harding. Remarks on concrete orthomodular lattices. InternationalJournal of Theoretical Physics, 43(10):2149–2168, 2004.

Bart Jacobs. Convexity, duality and effects. In IFIP InternationalConference on Theoretical Computer Science, pages 1–19. Springer,2010.

Gudrun Kalmbach. Orthomodular lattices do not satisfy any special latticeequation. Archiv der Mathematik, 28(1):7–8, 1977.

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

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