Twist products of bimonoids Nick Galatos and Adam Přenosil University of Denver, Denver, CO Vanderbilt University, Nashville, TN BLAST 2018 Denver, 7 August 2018 1 / 28
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, . . . ) 2 / 28
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.) 2 / 28
Problem : Can we embed each bimonoidal structure into a complemented one? Restriction : Throughout the talk we restrict to the commutative case. 3 / 28
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. 3 / 28
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. 3 / 28
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, . . . 4 / 28
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). 5 / 28
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.) 6 / 28
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 . 7 / 28
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 S 3 and the Boolean algebra B 2 have the same negative cones as RLs, but not as residuated ℓ -bimonoids. 8 / 28
Complemented MacNeille completions A set X ⊆ L is join dense in a lattice L if each x ∈ L is a join of elements of 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. 9 / 28
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 ∆ . 10 / 28
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 , ◦ , 1 L � , a monoid of meet generators R = � R , ⊕ , 0 R � , 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. 11 / 28
Complemented completions: construction The join generators will be pairs � a , b � L ∈ A 2 interpreted as a · b . The meet generators will be pairs � a , b � R ∈ A 2 interpreted as a + b . Thus: � a , b � L ◦ � c , d � L = � a · c , b + d � L 1 L = � 1 , 0 � � a , b � R ⊕ � c , d � R = � a + c , b · d � R 0 R = � 0 , 1 � The order relation will be � a , b � L ⊑ � c , d � R ⇐ ⇒ “ a · b ≤ c + d ” ⇐ ⇒ a · d ≤ b + c The isomorphisms will be � a , b � L = � b , a � R � a , b � R = � b , a � L The embeddings will be λ ( a ) = � a , 0 � L ρ ( a ) = � a , 1 � 12 / 28
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. 13 / 28
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 − 1 ( a ∧ ¬ b ) ∨ ( c ∧ ¬ d ) vs. We want to understand when the complemented extension has this form. 14 / 28
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 � ∈ A 2 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 ? 15 / 28
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 16 / 28
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