Convexity and the Kalmbach monad Gejza Jena August 10, 2018 Gejza - - PowerPoint PPT Presentation

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 varieties Examples of monads Kalmbach monad on bounded posets Effect algebras Convex effect algebras The product [0, 1]-actions Distributive laws The composite monad and its algebras

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 2 / 42

Monads generalize varieties of algebras

The ‘free algebra‘ endofunctor

Let V be a variety of universal algebras.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 3 / 42

Monads generalize varieties of algebras

The ‘free algebra‘ endofunctor

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

• ver X.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 3 / 42

Monads generalize varieties of algebras

The ‘free algebra‘ endofunctor

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

• ver X.

Then TV : Set → Set is a functor: TV (X

f

− → Y ): F(X) → F(Y ) replaces variable x in terms by the variable f (x).

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 3 / 42

Monads generalize varieties of algebras

The unit and the multiplication

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 4 / 42

Monads generalize varieties of algebras

The unit and the multiplication

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 4 / 42

Monads generalize varieties of algebras

The unit and the multiplication

For every set X there is a natural mapping ηX : X → TV (X), given by ηX(x) = x. η is the unit of the monad For every set X there is a natural mapping µX : TV (TV (X)) → TV (X), given by ‘flattening of terms over terms’

• r ‘evaluation in the free algebra’.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 4 / 42

Monads generalize varieties of algebras

The unit and the multiplication

For every set X there is a natural mapping ηX : X → TV (X), given by ηX(x) = x. η is the unit of the monad For every set X there is a natural mapping µX : TV (TV (X)) → TV (X), given by ‘flattening of terms over terms’

• r ‘evaluation in the free algebra’.

µ is the multiplication of the monad

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 4 / 42

Monads generalize varieties of algebras

Example: the free monoid monad

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

Monads generalize varieties of algebras

Example: the free monoid monad

T(X) is the set of all words over alphabet X: X = {a, b, c} [], [a], [babbca] ∈ T(X)

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

Monads generalize varieties of algebras

Example: 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)]

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

Monads generalize varieties of algebras

Example: 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]

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

Monads generalize varieties of algebras

Example: 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]

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

(Set, η, µ)

. So we have data of the following type:

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

(Set, η, µ)

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

(Set, η, µ)

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

(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,

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

(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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

The monad laws

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 7 / 42

Axioms of a monad

Right unit axiom

T(X)

T(ηX ) idT(X)

• T 2(X)

µX

• T(X)

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 8 / 42

Axioms of a monad

Right unit axiom

T(X)

T(ηX ) idT(X)

• T 2(X)

µX

• T(X)

[abac] ✤

• ✌
• [[a][b][a][c]]

❴

• [abac]

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 8 / 42

Axioms

Left unit axiom

T 2(X)

µX

• T(X)

ηT(X)

• idT(X)
• T(X)

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 9 / 42

Axioms

Left unit axiom

T 2(X)

µX

• T(X)

ηT(X)

• idT(X)
• T(X)

[[abac]]

❴

• [abac]

✤

• ✹
• [abac]

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 9 / 42

Axioms of a monad

Associativity axiom

T 3(X)

T(µX ) µT(X)

• T 2(X)

µX

• T 2(X)

µX

T(X)

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

Axioms of a monad

Associativity axiom

T 3(X)

T(µX ) µT(X)

• T 2(X)

µX

• T 2(X)

µX

T(X)

• [ab][bc]
• [ca]

✤

• ❴
• [abbc][ca]
• ❴
• [ab][bc][ca]

✤

[abbcca]

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

The definition

Definition

Let 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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 11 / 42

Algebras for a monad

Definition

Let (T, η, µ) be a monad over C. Then an algebra for T (or T-algebra) is a pair (X, α), where X is an object of C and

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 12 / 42

Algebras for a monad

Definition

Let (T, η, µ) be a monad over C. Then an algebra for T (or T-algebra) is a pair (X, α), where X is an object of C and α : T(X) → X

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 12 / 42

Algebras for a monad

Definition

Let (T, η, µ) be a monad over C. Then an algebra for T (or T-algebra) is a pair (X, α), where X is an object of C and α : T(X) → X such that the following diagrams commute X

ηX idX

• T(X)

α

• X

T 2(X)

µX

• T(α)
• T(X)

α

• T(X)

α

X

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 12 / 42

Intuition/meaning of all this

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 13 / 42

Morphisms of algebras

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 14 / 42

Morphisms of algebras

Definition

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

T(f ) α

• T(B)

β

• A

f

B

commutes.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 14 / 42

The Eilenberg-Moore category

Definition

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 15 / 42

Algebras are algebras

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 16 / 42

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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 16 / 42

Algebras over other categories

Example

For every ring A, A-modules are algebras for a monad on the category

• f abelian groups.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 17 / 42

Algebras over other categories

Example

For every ring A, A-modules are algebras for a monad on the category

• f abelian groups.

Graphs equipped with a perfect matching are algebras for a monad on the category of graphs.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 17 / 42

Algebras over other categories

Example

For every ring A, A-modules are algebras for a monad on the category

• f abelian groups.

Graphs equipped with a perfect matching are algebras for a monad on the category of graphs. Bounded posets are algebras for a monad on the category of posets.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 17 / 42

Algebras over other categories

Example

For every ring A, A-modules are algebras for a monad on the category

• f abelian groups.

Graphs equipped with a perfect matching are algebras for a monad on the category of graphs. Bounded posets are algebras for a monad on the category of posets. Involutive posets are algebras for a monad on posets.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 17 / 42

Algebras over other categories

Example

For every ring A, A-modules are algebras for a monad on the category

• f abelian groups.

Graphs equipped with a perfect matching are algebras for a monad on the 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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 17 / 42

Algebras over other categories

Example

For every ring A, A-modules are algebras for a monad on the category

• f abelian groups.

Graphs equipped with a perfect matching are algebras for a monad on the 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 C has coproducts.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 17 / 42

Algebras over other categories

Example

For every ring A, A-modules are algebras for a monad on the category

• f abelian groups.

Graphs equipped with a perfect matching are algebras for a monad on the 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 C has coproducts. Compact Haussdorff spaces are algebras for a monad on Set.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 17 / 42

Algebras over other categories

Example

For every ring A, A-modules are algebras for a monad on the category

• f abelian groups.

Graphs equipped with a perfect matching are algebras for a monad on the 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 C has coproducts. Compact Haussdorff spaces are algebras for a monad on Set. Small categories are algebras for a monad on the category of directed multigraphs.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 17 / 42

The Kalmbach embedding

[Kalmbach, 1977] proved the following

Theorem

Every bounded lattice can be embedded into an orthomodular lattice.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 18 / 42

The Kalmbach embedding

[Kalmbach, 1977] proved the following

Theorem

Every bounded lattice can be embedded into an orthomodular lattice.

Corollary

Orthomodular lattices do not satisfy any special lattice equation.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 18 / 42

The Kalmbach embedding

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 19 / 42

The Kalmbach embedding

Let L be a bounded lattice. Let K(L) be the set of all finite chains in L 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 that b2j−1 ≤ a2i−1 < a2i ≤ b2j.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 19 / 42

Then K(L) is a bounded lattice. Moreover, it is an orthomodular lattice: the orthocomplementation is ({ai}2n

i=1)′ := {ai}2n i=1∆{0, 1},

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 20 / 42

The Kalmbach embedding

K cannot be made to a functor from the category of lattices into the category of orthomodular lattices. However, K can be extended to a functor from the category of bounded 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]) = ∆2n

i=1{f (ai)}.

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 21 / 42

The Kalmbach monad

Definition

The Kalmbach monad (T, η, µ) on the category BPos is given as follows T : BPos → BPos is the Kalmbach embedding K : BPos → OMP composed 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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 22 / 42

What are algebras for the Kalmbach monad?

Answer: effect algebras

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 23 / 42

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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 24 / 42

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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 25 / 42

Morphisms of effect algebras

A morphism of effect algebras f : A → B is a mapping of the underlying sets such that f (0) = 0, f (1) = 1 and whenever a + b exists in A, f (a) + f (b) exists in B and f (a + b) = f (a) + f (b).

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 26 / 42

Examples of effect algebras

Example

Any 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 then p + q = p ∪ q. Any other orthomodular lattice.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 27 / 42

Convex effect algebras

Definition

A convex effect algebra is an effect algebra E equipped with a multiplication by real numbers from interval [0, 1] such that, for all ρ, 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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 28 / 42

Theorem

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 29 / 42

Problem

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 30 / 42

The product on BPos

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 31 / 42

The product on BPos

Let A, B, C be bounded posets. We say that a BPos-morphism H : A × B → C is a 0-bimorphism if and only if, for all a ∈ A and b ∈ B, h(0, b) = h(b, 0) = 0. Let us write AB for the poset A × B/ ∼, where ∼ is the equivalence

• n A × B generated by the relations (a, 0) ∼ (0, b), for all a ∈ A

b ∈ B.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 31 / 42

The product on BPos

Let A, B, C be bounded posets. We say that a BPos-morphism H : A × B → C is a 0-bimorphism if and only if, for all a ∈ A and b ∈ B, h(0, b) = h(b, 0) = 0. Let us write AB for the poset A × B/ ∼, where ∼ is the equivalence

• n A × B generated by the relations (a, 0) ∼ (0, b), for all a ∈ A

b ∈ B. All the elements of A × B that have 0 in first or second coordinate form one of the equivalence classes of ∼, all the other elements form singleton equivalence classes.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 31 / 42

The product on BPos

Clearly, the mapping : A × B → AB that takes an element of A × B to its equivalence class is a 0-bimorphism. Moreover, it is an universal 0-bimorphism in the following sense: for every 0-bimorphism h : A × B → C, there is a unique morphism of bounded posets f : AB → C such that A × B

• h
• AB

f

C

commutes.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 32 / 42

Fact

The category (BPos, , 2) is a monoidal category.a

aHere, 2 is a 2-element chain Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 33 / 42

[0, 1] is a monoid

Proposition

The real interval [0, 1], equipped with multiplication of reals is a monoid in the monoidal category (BPos, , 1).

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 34 / 42

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 rule S(A) = A[0, 1]

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 35 / 42

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 rule S(A) = A[0, 1] ηS : idBPos → S is a natural transformations given by ηS

A(x) = x1

and

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 35 / 42

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 rule S(A) = A[0, 1] ηS : idBPos → S is a natural transformations given by ηS

A(x) = x1

and µS : S ◦ S → S is a natural transformation given by µS

A(xρψ) = x(ρ.ψ).

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 35 / 42

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 rule S(A) = A[0, 1] ηS : idBPos → S is a natural transformations given by ηS

A(x) = x1

and µS : S ◦ S → S is a natural transformation given by µS

A(xρψ) = x(ρ.ψ).

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 35 / 42

Distributive laws [Beck, 1969]

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 36 / 42

Distributive laws [Beck, 1969]

S

SηT

• ηT S
• ST

λ

TS

T

ηST

• TηS
• ST

λ

TS

SST

µST

• Sλ STS

λS TSS TµS

• ST

λ

TS

STT

SµT

• λT TST

Tλ TTS µT S

• ST

λ

TS

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 37 / 42

Example

There is a distributive law between the ‘free abelian group’ monad on Set and the ‘free monoid’ monad on Set. The composite monad is the ‘free ring’ monad.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 38 / 42

Results

Theorem

There is a distributive law between the Kalmbach monad T on BPos and the free [0, 1]-action monad on BPos.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 39 / 42

Results

Theorem

The category of algebras for the composite monad TS is equivalent for the category of effect algebras equipped with multiplication with a scalar, satisfying the following conditions: (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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 40 / 42

Results

Theorem

The category of algebras for the composite monad TS is equivalent for the category of effect algebras equipped with multiplication with a scalar, satisfying the following conditions: (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 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.

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 40 / 42

What next?

Call the algebras on the previous slide weak effect algebras.

Problem

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

Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 41 / 42

Jon Beck. Distributive laws. In Seminar on triples and categorical homology 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.
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