Twist products of bimonoids Nick Galatos and Adam Penosil - - PowerPoint PPT Presentation

Twist products of bimonoids

Nick Galatos and Adam Přenosil

University of Denver, Denver, CO Vanderbilt University, Nashville, TN

BLAST 2018 Denver, 7 August 2018

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Basic themes of the talk: Embedding ordered algebraic structures into complemented ones. Reconstructing ordered algebraic structures from their negative cones. (Compare: embeddings into complete structures, into dense structures, . . . )

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Basic themes of the talk: Embedding ordered algebraic structures into complemented ones. Reconstructing ordered algebraic structures from their negative cones. (Compare: embeddings into complete structures, into dense structures, . . . ) We will be interested in bimonoidal structures. Informally, these are to involutive residuated lattices as distributive lattices are to Boolean algebras. (Each distributive lattice embeds into a complemented one, i.e. a Boolean

• algebra. Conversely, distributive lattices are the lattice subreducts of BAs.)

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Problem: Can we embed each bimonoidal structure into a complemented one? Restriction: Throughout the talk we restrict to the commutative case.

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Problem: Can we embed each bimonoidal structure into a complemented one? Restriction: Throughout the talk we restrict to the commutative case. General answer: We can construct a complemented Dedekind–MacNeille completion.

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Problem: Can we embed each bimonoidal structure into a complemented one? Restriction: Throughout the talk we restrict to the commutative case. General answer: We can construct a complemented Dedekind–MacNeille completion. In some cases: Sitting inside this completion we can find a twist product. In the best cases: Equivalences between integral and involutive residuated structures.

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Residuated lattices

A partially ordered monoid (pomonoid) A, ≤, 1, · is a poset as well as a monoid such that multiplication is isotone. We assume commutativity. A residuated lattice A, ∧, ∨, 1, ·, → is a lattice as well as a pomonoid such that the multiplication has a residual (division operation) x → y, i.e. a ≤ b → c ⇐ ⇒ a · b ≤ c ⇐ ⇒ b ≤ a → c. A residuated lattice is integral if the monoidal unit 1 is its order maximum. An involutive residuated lattice is a pointed residuated lattice such that (a → 0) → 0 = a. Involutive RLs (InRLs) can be thought of as complemented RLs. Examples include Boolean algebras, MV-algebras, ℓ-groups, Sugihara monoids, . . .

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Bimonoids

A bimonoid ≤, 1, ·, 0, + is a pair of pomonoids over the same poset connected by the hemidistributive law: x · (y + z) ≤ (x · y) + z A lattice-ordered bimonoid (ℓ-bimonoid) is moreover required to satisfy x · (y ∨ z) = (x · y) ∨ (x · z) x + (y ∧ z) = (x + y) ∧ (x + z) A residuated ℓ-bimonoid is an ℓ-bimonoid as well as a residuated lattice (with respect to multiplication only). “Hemidistributivity” is due to Dunn & Hardegree, also considered by Grishin (in Lambek calculus) and Cockett & Seely (weakly distributive categories).

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Examples of bimonoids

Partially ordered monoids: x + y = x · y and 0 = 1 Bounded distributive lattices: x · y = x ∧ y (1 = ⊤) and x + y = x ∨ y (0 = ⊥) Bounded integral residuated lattices: x + y = x ∨ y and 0 = ⊥ Pointed Brouwerian algebras: x + y = (0 → x ∧ y) ∧ (x ∨ y) (Brouwerian algebras are integral idempotent RLs. Heyting algebras are precisely pointed Brouwerian algebras such that 0 is the smallest element.)

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Complementation in bimonoids

Bimonoids are an appropriate framework for the study of complementation. Elements a and b of a bimonoid are called complements if a · b ≤ 0 and 1 ≤ a + b. In particular 0 and 1 are complements. A bimonoid is complemented if each element has a complement. Fact: complements are unique whenever they exist. Proof: if b and c are complements of a, then b ≤ b · 1 ≤ b · (a + c) ≤ (b · a) + c ≤ 0 + c ≤ c. Notation: the complement of a, if it exists, will be denoted a.

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Negative cones of involutive RLs

Complemented ℓ-bimonoids are termwise equivalent to involutive RLs: x → y := x + y, x := x → 0 and x + y := x · y. The negative cone A− of an InRL A is an integral residuated ℓ-bimonoid which inherits all the operations of A except for the residual: x →A− y = 1 ∧ (x →A y)

(Here and in the following we are assuming 0 · 0 = 0, i.e. 1 + 1 = 1.)

Note: keeping track of the additive monoid is crucial for reconstructing A! Example: the odd Sugihara monoid S3 and the Boolean algebra B2 have the same negative cones as RLs, but not as residuated ℓ-bimonoids.

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Complemented MacNeille completions

A set X ⊆ L is join dense in a lattice L if each x ∈ L is a join of elements

• f X. It is meet dense in L if each x ∈ L is a meet of elements of X.

Equivalently, the join density of X and the meet density of Y amount to: a ≤ b if and only if x ≤ a = ⇒ x ≤ b for each x ∈ X, a ≤ b if and only if b ≤ y = ⇒ a ≤ y for each y ∈ Y . Let B be a complemented ℓ-bimonoid and A be its sub-bimonoid. Then B is a complemented ∆1-extension of A if the set {a · b | a, b ∈ X} is join dense and the set {a + b | a, b ∈ X} is meet dense in B. A complemented MacNeille completion of a bimonoid A is a complete complemented ∆1-extension.

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Complemented MacNeille completions

Theorem (existence): each (commutative) bimonoid has a (commutative) complemented MacNeille completion. Theorem (universality): each complemented ∆1-extension of a bimonoid A embeds into such a completion (via a unique embedding which fixes A). Corollary (uniqueness): complemented completions are unique up to iso. Notation: the complemented MacNeille completion of A is denoted A∆.

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Complemented completions: construction

We use the machinery of residuated frames (due to Galatos & Jipsen) to construct the complemented MacNeille completion of an (ℓ-)bimonoid A. These allow us to construct an involutive residuated lattice given a monoid of join generators L = L, ◦, 1L, a monoid of meet generators R = R, ⊕, 0R, an order relation between the two ⊑, an isomorphism between the two x ∈ L → x ∈ R and x ∈ R → x ∈ L, satisfying a suitable residuation law (nuclearity). Moreover, we can embed a A into it given a map λ : A → L, a map ρ : A → R, satisfying suitable Gentzen-style conditions.

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Complemented completions: construction

The join generators will be pairs a, bL ∈ A2 interpreted as a · b. The meet generators will be pairs a, bR ∈ A2 interpreted as a + b. Thus: a, bL ◦ c, dL = a · c, b + dL 1L = 1, 0 a, bR ⊕ c, dR = a + c, b · dR 0R = 0, 1 The order relation will be a, bL ⊑ c, dR ⇐ ⇒ “a · b ≤ c + d” ⇐ ⇒ a · d ≤ b + c The isomorphisms will be a, bL = b, aR a, bR = b, aL The embeddings will be λ(a) = a, 0L ρ(a) = a, 1

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Sidenote: ℓ-bimonoidal subreducts

Problem: Axiomatize the ℓ-bimonoidal subreducts of a given variety of InRLs. Fact: ℓ-bimonoids are precisely the ℓ-bimonoidal subreducts of InRLs. Theorem: there is an algorithm to do so this for varieties (in fact, positive universal classes) axiomatized in the signature {∨, ·, 1}. Proof: uses the fact that such equations can be linearized and it suffices to verify linear equations on a meet dense set (elements of form ab). Open problem: Axiomatize the ℓ-bimonoidal subreducts of MV-algebras.

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Complemented completions: examples

The complemented completion of a Boolean algebra is its MacNeille completion, a distributive lattice is the completion of its Boolean envelope, a suitable cancellative RL is the completion of its ℓ-group envelope. These are not necessarily the most natural complemented envelopes. In both cases, sitting inside the complemented completion there is a perhaps more natural complemented extension, not necessarily complete. In the latter case, this extension is generated in a much simpler way: (a ∧ ¬b) ∨ (c ∧ ¬d) vs. a · b−1 We want to understand when the complemented extension has this form.

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Twist products

In the following, let A be an integral residuated ℓ-bimonoid. Let B be a complemented ℓ-bimonoid with A as a sub-ℓ-bimonoid. A pair a, b ∈ A2 is an A-representation of x ∈ B if x = a · b. B is called a twist product of A if each x ∈ B is A-representable. Problems: When does a bimonoid A have a twist product? Can we describe it explicitly in terms of A?

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Example: integers and naturals

Consider the lattice-ordered group of integers Z = Z, ∧, ∨, +, 0, −. The positive cone of Z are the naturals N = N, ∧, ∨, +, 0, . −. Here . − denotes truncated subtraction: a . − b = (a − b) ∨ 0. There are essentially two ways of constructing Z from N: the group of fractions (differences) construction the twist product construction

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Example: integers and naturals

Group of differences: take the set of all pairs a, b ∈ N2, interpreted as a − b define an algebraic structure on this set: a, b + c, d = a + c, b + d 0 = 0, 0 −a, b = b, a define a pre-order on the set of all pairs: a, b ≦ c, d ⇐ ⇒ a + d ≤ b + c factor the pre-order down to a partial order This construction yields a partially ordered Abelian group (sometimes an Abelian ℓ-group) starting from an order-cancellative commutative monoid.

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Example: integers and naturals

Twist product: take only certain pairs a, b ∈ N2, namely a, 0 or 0, a these normal pairs are equationally definable: a ∧ b = 0 figure out how to transform arbitrary pair into normal pairs: a, b → πa, b = a . − b, b . − a define an algebraic structure on the set of all normal pairs: a, b + c, d = πa + c, b + d a, b ∧ c, d = πa ∧ c, b ∨ d −a, b = b, a a, b ∨ c, d = πa ∨ c, b ∧ d

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Example: integers and naturals, Sugihara style

We can also consider a radically different structure on the integers. Consider the algebra S = Z, ∧, ∨, ·, 0, +, 0, − with: x · y =      y if |x| < |y| x if |x| > |y| x ∧ y if |x| = |y| x + y =      y if |x| < |y| x if |x| > |y| x ∨ y if |x| = |y| Observe that the restrictions of these to the negative cone are: x · y = x ∧ y = x + y This is a Sugihara monoid, i.e. a distributive idempotent (commutative) involutive RL. It is odd, i.e. the additive and multiplicative unit coincide.

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Example: integers and naturals, Sugihara style

The negative cone of S is the algebra G = −N, ∧, ∨, 0, → with x → y =

• 0 if x ≤ y

y if y < x The odd Sugihara monoid S can be reconstructed as a twist product of G with the defining equation a ∨ b = 0 and the normalizing term: πa, b = (a → b) → a, a → b This construction generalizes to arbitrary Sugihara monoids. In that case we add to the negative cone a constant 0 such that [0, 1] is a Boolean lattice.

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Twist products

To look for twist products it suffices to search inside of A∆. Fact: A has a twist product if and only if the A-representable elements

• f A∆ form a subalgebra (a complemented sub-ℓ-bimonoid) of A∆.

Fact: this is then the unique twist product of A up to isomorphism. Notation: if it exists, the twist product of A is denoted A⊲

⊳.

Notation: we write x ∼A a, b for a · b = x ∈ A∆.

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Comparing representations

To describe A⊲

⊳, it suffices to pick a certain normal representation of each

element and then compute the normal representations of x, 1, x · y, and x ∧ y (hence 0, x + y, x ∨ y) from the normal representations of x and y. Computing products and comparing two representations is not difficult. Fact: if x ∼A a, b and y ∼A c, d, then x ≤ y if and only if c → (d + e) ≤ a → (b + e) for each e ∈ A. Fact: if x ∼A a, b and y ∼A c, d, then x · y ∼A a · c, b + d. However, it is difficult to compute the representations of x and x ∧ y.

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Normal representations

A representation a, b of x is normal if a = 1 ∧ x and b = 1 ∧ x. Fact: if a, b is a normal representation, then b = 1 ∧ (a + b) = a → b. Fact: if if x ∼A a, b and y ∼A c, d normally, then x ≤ y if and only if a ≤ c and d ≤ b. Any x can be recovered from 1 ∧ x, 1 ∧ x provided that A∆ satisfies x = (1 ∧ x) · (0 ∨ x), i.e. x = (1 ∧ x) · (1 ∧ x). Each pair of form 1 ∧ x, 1 ∧ x is then a normal representation (of x). Fact: suppose that A∆ satisfies the x = (1 ∧ x) · (0 ∨ x). Then x ∼A a, b normally = ⇒ x ∼A b, a normally.

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Normalizing functions

How to ensure enough normal representations? Find a normalizing term. A function π : A2 → A2 is a normalizing function if πa, b ∼A a, b and πa, b is normal. A is normalizable if it has a normalizing function. Fact: πa, b = π+(a, b), a → b. Fact: if A is normalizable, then A∆ satisfies x = (1 ∧ x) · (0 ∨ x). Sufficient condition for normality: Suppose that A satisfies the equation x → ((x → 0) + y) = x → y, e.g. suppose that A is odd. Then a pair a, b is normal whenever it satisfies a → b = b b → a = a a · b ≤ 0 (To verify that term π+(x, y) defines a normalizing function, it thus suffices to verify four equations in two variables.)

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Normal twist products

Theorem: If an integral residuated ℓ-bimonoid A is normalizable via π, then A⊲

⊳ exists

and is isomorphic to the algebra of all normal A-pairs with the constants 1 = 1, 0 and 0 = 0, 1, complementation a, b = b, a, and a, b · c, d = πa · c, b + d a, b ∧ c, d = πa ∧ c, b ∨ d a, b + c, d = πb · d, a + c a, b ∨ c, d = πb ∧ d, a ∨ c Theorem: If K is a variety of integral residuated ℓ-bimonoids with a complemented interval [0, 1] and a normalizing term, then K is equivalent to the variety K⊲

⊳ of involutive RLs via the twist product and negative cone functors.

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Examples

The following categorical equivalences can be derived from this theorem. Abelian ℓ-groups ∼ integral cancellative divisible RLs: y → x is a normalizing term for cancellative divisible RLs, witnessing their equivalence with Abelian ℓ-groups. (Here x + y = x · y and 0 = 1). Boolean-pointed Brouwerian algebras ∼ strongly idempotent InRLs (x → y0) → x0 is a normalizing term for Boolean-pointed Brouwerian algebras, witnessing equivalence with idempotent InRLs with x = x(1 ∧ x). (Boolean-pointed Brouwerian algebras: the interval [0, 1] is Boolean.)

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Equivalences between integral and involutive structures

Theorem: there are categorical equivalences between the varieties below via a negative functor and a twist product functor: Abelian ℓ-groups and integral cancellative divisible residuated lattices (’03 Bahls, Cole, Galatos, Jipsen, Tsinakis)

• dd Sugihara monoids and semilinear Brouwerian alg’s

(’12 Galatos & Raftery) Sugihara monoids and semilinear Boolean-pointed Brouwerian alg’s (’17 Fussner & Galatos)

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Equivalences between integral and involutive structures

Theorem: there are categorical equivalences between the varieties below via a negative functor and a twist product functor: Abelian ℓ-groups and integral cancellative divisible residuated lattices (’03 Bahls, Cole, Galatos, Jipsen, Tsinakis)

• dd Sugihara monoids and semilinear Brouwerian alg’s

(’12 Galatos & Raftery) Sugihara monoids and semilinear Boolean-pointed Brouwerian alg’s (’17 Fussner & Galatos) “non-distributive Sugihara monoids” and B.-pointed Brouwerian alg’s

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Equivalences between integral and involutive structures

Theorem: there are categorical equivalences between the varieties below via a negative functor and a twist product functor: Abelian ℓ-groups and integral cancellative divisible residuated lattices (’03 Bahls, Cole, Galatos, Jipsen, Tsinakis)

• dd Sugihara monoids and semilinear Brouwerian alg’s

(’12 Galatos & Raftery) Sugihara monoids and semilinear Boolean-pointed Brouwerian alg’s (’17 Fussner & Galatos) “non-distributive Sugihara monoids” and B.-pointed Brouwerian alg’s Theorem: the above are instances of a single categorical equivalence between certain involutive and integral structures. Proof: (x → xy0) → x is a normalizing term in both the Abelian ℓ-group case and the (non-distributive) Sugihara case.

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Review: A general notion of complementation was introduced for bimonoids. Each bimonoid has a complemented Dedekind–MacNeille completion. This complemented completion sometimes contains a twist product. Lessons: Existence of twist products is closely tied to definability of 1 ∧ ab. It is helpful to view negative cones of InRLs as residuated ℓ-bimonoids. To investigate: complemented completions in the non-commutative case ℓ-bimonoidal subreducts of varieties of involutive RLs

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Review: A general notion of complementation was introduced for bimonoids. Each bimonoid has a complemented Dedekind–MacNeille completion. This complemented completion sometimes contains a twist product. Lessons: Existence of twist products is closely tied to definability of 1 ∧ ab. It is helpful to view negative cones of InRLs as residuated ℓ-bimonoids. To investigate: complemented completions in the non-commutative case ℓ-bimonoidal subreducts of varieties of involutive RLs

Thank you for your attention.

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