Is there a useful duality for residuated lattices? Nick Galatos - PowerPoint PPT Presentation

Lattice frames Substructural logics Lattice representation Lattices A lattice frame is a structure W = ( W, ⊑ , W ′ ) where W and W ′ are Contexts Dedekind-Birkhoff sets and ⊑ is a binary relation from W to W ′ . Lattice frames Gentzen lattice frames Cut elimination for For X ⊆ W and Y ⊆ W ′ we define lattices Residuated frames X ⊲ = { b ∈ W ′ : x ⊑ b, for all x ∈ X } Variants of frames Y ⊳ = { a ∈ W : a ⊑ y, for all y ∈ Y } References We define γ ( X ) = X ⊲⊳ . Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 10 / 44

Lattice frames Substructural logics Lattice representation Lattices A lattice frame is a structure W = ( W, ⊑ , W ′ ) where W and W ′ are Contexts Dedekind-Birkhoff sets and ⊑ is a binary relation from W to W ′ . Lattice frames Gentzen lattice frames Cut elimination for For X ⊆ W and Y ⊆ W ′ we define lattices Residuated frames X ⊲ = { b ∈ W ′ : x ⊑ b, for all x ∈ X } Variants of frames Y ⊳ = { a ∈ W : a ⊑ y, for all y ∈ Y } References We define γ ( X ) = X ⊲⊳ . Lemma. If W is a lattice frame then the Galois/dual algebra W + = ( γ [ P ( W )] , ∩ , ∪ γ ) is a complete lattice. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 10 / 44

Lattice frames Substructural logics Lattice representation Lattices A lattice frame is a structure W = ( W, ⊑ , W ′ ) where W and W ′ are Contexts Dedekind-Birkhoff sets and ⊑ is a binary relation from W to W ′ . Lattice frames Gentzen lattice frames Cut elimination for For X ⊆ W and Y ⊆ W ′ we define lattices Residuated frames X ⊲ = { b ∈ W ′ : x ⊑ b, for all x ∈ X } Variants of frames Y ⊳ = { a ∈ W : a ⊑ y, for all y ∈ Y } References We define γ ( X ) = X ⊲⊳ . Lemma. If W is a lattice frame then the Galois/dual algebra W + = ( γ [ P ( W )] , ∩ , ∪ γ ) is a complete lattice. Every γ -closed set is an intersection of basic closed sets : { z } ⊳ , where z ∈ W ′ . Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 10 / 44

Lattice frames Substructural logics Lattice representation Lattices A lattice frame is a structure W = ( W, ⊑ , W ′ ) where W and W ′ are Contexts Dedekind-Birkhoff sets and ⊑ is a binary relation from W to W ′ . Lattice frames Gentzen lattice frames Cut elimination for For X ⊆ W and Y ⊆ W ′ we define lattices Residuated frames X ⊲ = { b ∈ W ′ : x ⊑ b, for all x ∈ X } Variants of frames Y ⊳ = { a ∈ W : a ⊑ y, for all y ∈ Y } References We define γ ( X ) = X ⊲⊳ . Lemma. If W is a lattice frame then the Galois/dual algebra W + = ( γ [ P ( W )] , ∩ , ∪ γ ) is a complete lattice. Every γ -closed set is an intersection of basic closed sets : { z } ⊳ , where z ∈ W ′ . If W satisfies the condition (COM), then W + is a chain. x ⊑ z y ⊑ w x ⊑ w OR y ⊑ z (COM) Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 10 / 44

Gentzen lattice frames Substructural logics Lattice representation Lattices Contexts A Gentzen lattice frame is a pair ( W , S ) , where W is a lattice Dedekind-Birkhoff frame, S = ( S, ∧ , ∨ ) is an algebra, Lattice frames Gentzen lattice frames Cut elimination for lattices Residuated frames Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 11 / 44

Gentzen lattice frames Substructural logics Lattice representation Lattices Contexts A Gentzen lattice frame is a pair ( W , S ) , where W is a lattice Dedekind-Birkhoff frame, S = ( S, ∧ , ∨ ) is an algebra, S maps to W and W ′ Lattice frames Gentzen lattice frames Cut elimination for lattices Residuated frames Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 11 / 44

Gentzen lattice frames Substructural logics Lattice representation Lattices Contexts A Gentzen lattice frame is a pair ( W , S ) , where W is a lattice Dedekind-Birkhoff frame, S = ( S, ∧ , ∨ ) is an algebra, S maps to W and W ′ and the Lattice frames Gentzen lattice frames conditions are satisfied for all a, b ∈ S , x ∈ W and z ∈ W ′ . Cut elimination for lattices Residuated frames x ⊑ a a ⊑ z (CUT) a ⊑ a (Id) Variants of frames x ⊑ z References a ⊑ z b ⊑ z x ⊑ a x ⊑ b a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) x ⊑ a ∧ b a ⊑ z b ⊑ z x ⊑ a x ⊑ b ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) a ∨ b ⊑ z Corollary. The map q : S → W + , q ( a ) = { a } ⊳ is a homomorphism: q ( a ∧ B b ) = q ( a ) ∧ W + q ( b ) and q ( a ∨ B b ) = q ( a ) ∨ W + q ( b ) . Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 11 / 44

Gentzen lattice frames Substructural logics Lattice representation Lattices Contexts A Gentzen lattice frame is a pair ( W , S ) , where W is a lattice Dedekind-Birkhoff frame, S = ( S, ∧ , ∨ ) is an algebra, S maps to W and W ′ and the Lattice frames Gentzen lattice frames conditions are satisfied for all a, b ∈ S , x ∈ W and z ∈ W ′ . Cut elimination for lattices Residuated frames x ⊑ a a ⊑ z (CUT) a ⊑ a (Id) Variants of frames x ⊑ z References a ⊑ z b ⊑ z x ⊑ a x ⊑ b a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) x ⊑ a ∧ b a ⊑ z b ⊑ z x ⊑ a x ⊑ b ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) a ∨ b ⊑ z Corollary. The map q : S → W + , q ( a ) = { a } ⊳ is a homomorphism: q ( a ∧ B b ) = q ( a ) ∧ W + q ( b ) and q ( a ∨ B b ) = q ( a ) ∨ W + q ( b ) . If ⊑ is antisymmetric on S , then q is injective. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 11 / 44

Gentzen lattice frames Substructural logics Lattice representation Lattices Contexts A Gentzen lattice frame is a pair ( W , S ) , where W is a lattice Dedekind-Birkhoff frame, S = ( S, ∧ , ∨ ) is an algebra, S maps to W and W ′ and the Lattice frames Gentzen lattice frames conditions are satisfied for all a, b ∈ S , x ∈ W and z ∈ W ′ . Cut elimination for lattices Residuated frames x ⊑ a a ⊑ z (CUT) a ⊑ a (Id) Variants of frames x ⊑ z References a ⊑ z b ⊑ z x ⊑ a x ⊑ b a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) x ⊑ a ∧ b a ⊑ z b ⊑ z x ⊑ a x ⊑ b ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) a ∨ b ⊑ z Corollary. The map q : S → W + , q ( a ) = { a } ⊳ is a homomorphism: q ( a ∧ B b ) = q ( a ) ∧ W + q ( b ) and q ( a ∨ B b ) = q ( a ) ∨ W + q ( b ) . If ⊑ is antisymmetric on S , then q is injective. Application (DM-completion/embedding) : Given a lattice L , W L = ( L, ≤ , L ) is a lattice frame Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 11 / 44

Gentzen lattice frames Substructural logics Lattice representation Lattices Contexts A Gentzen lattice frame is a pair ( W , S ) , where W is a lattice Dedekind-Birkhoff frame, S = ( S, ∧ , ∨ ) is an algebra, S maps to W and W ′ and the Lattice frames Gentzen lattice frames conditions are satisfied for all a, b ∈ S , x ∈ W and z ∈ W ′ . Cut elimination for lattices Residuated frames x ⊑ a a ⊑ z (CUT) a ⊑ a (Id) Variants of frames x ⊑ z References a ⊑ z b ⊑ z x ⊑ a x ⊑ b a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) x ⊑ a ∧ b a ⊑ z b ⊑ z x ⊑ a x ⊑ b ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) a ∨ b ⊑ z Corollary. The map q : S → W + , q ( a ) = { a } ⊳ is a homomorphism: q ( a ∧ B b ) = q ( a ) ∧ W + q ( b ) and q ( a ∨ B b ) = q ( a ) ∨ W + q ( b ) . If ⊑ is antisymmetric on S , then q is injective. Application (DM-completion/embedding) : Given a lattice L , W L = ( L, ≤ , L ) is a lattice frame and the pair ( W L , L ) is a Genzen lattice frame. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 11 / 44

Gentzen lattice frames Substructural logics Lattice representation Lattices Contexts A Gentzen lattice frame is a pair ( W , S ) , where W is a lattice Dedekind-Birkhoff frame, S = ( S, ∧ , ∨ ) is an algebra, S maps to W and W ′ and the Lattice frames Gentzen lattice frames conditions are satisfied for all a, b ∈ S , x ∈ W and z ∈ W ′ . Cut elimination for lattices Residuated frames x ⊑ a a ⊑ z (CUT) a ⊑ a (Id) Variants of frames x ⊑ z References a ⊑ z b ⊑ z x ⊑ a x ⊑ b a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) x ⊑ a ∧ b a ⊑ z b ⊑ z x ⊑ a x ⊑ b ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) a ∨ b ⊑ z Corollary. The map q : S → W + , q ( a ) = { a } ⊳ is a homomorphism: q ( a ∧ B b ) = q ( a ) ∧ W + q ( b ) and q ( a ∨ B b ) = q ( a ) ∨ W + q ( b ) . If ⊑ is antisymmetric on S , then q is injective. Application (DM-completion/embedding) : Given a lattice L , W L = ( L, ≤ , L ) is a lattice frame and the pair ( W L , L ) is a Genzen lattice frame. W + L is the Dedekind-MacNeille completion of L and q : L → W + L is an embedding. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 11 / 44

Cut elimination for lattices Substructural logics Lattice representation Lattices Contexts a ≤ b b ≤ a a ≤ b b ≤ c Dedekind-Birkhoff a ≤ a a ≤ c Lattice frames a = b Gentzen lattice frames Cut elimination for lattices Residuated frames Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 12 / 44

Cut elimination for lattices Substructural logics Lattice representation Lattices Contexts a ≤ b b ≤ a a ≤ b b ≤ c Dedekind-Birkhoff a ≤ a a ≤ c Lattice frames a = b Gentzen lattice frames Cut elimination for lattices a ≤ c b ≤ c c ≤ a c ≤ b Residuated frames a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 12 / 44

Cut elimination for lattices Substructural logics Lattice representation Lattices Contexts a ≤ b b ≤ a a ≤ b b ≤ c Dedekind-Birkhoff a ≤ a a ≤ c Lattice frames a = b Gentzen lattice frames Cut elimination for lattices a ≤ c b ≤ c c ≤ a c ≤ b Residuated frames a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Variants of frames References c ≤ a c ≤ b a ≤ c b ≤ c c ≤ a ∨ b c ≤ a ∨ b a ∨ b ≤ c Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 12 / 44

Cut elimination for lattices Substructural logics Lattice representation Lattices Contexts a ≤ b b ≤ a a ≤ b b ≤ c Dedekind-Birkhoff a ≤ a a ≤ c Lattice frames a = b Gentzen lattice frames Cut elimination for lattices a ≤ c b ≤ c c ≤ a c ≤ b Residuated frames a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Variants of frames References c ≤ a c ≤ b a ≤ c b ≤ c c ≤ a ∨ b c ≤ a ∨ b a ∨ b ≤ c Theorem. (Cut elimination) Lat and Lat cf ( Lat without cut) prove the same sequents. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 12 / 44

Cut elimination for lattices Substructural logics Lattice representation Lattices Contexts a ≤ b b ≤ a a ≤ b b ≤ c Dedekind-Birkhoff a ≤ a a ≤ c Lattice frames a = b Gentzen lattice frames Cut elimination for lattices a ≤ c b ≤ c c ≤ a c ≤ b Residuated frames a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Variants of frames References c ≤ a c ≤ b a ≤ c b ≤ c c ≤ a ∨ b c ≤ a ∨ b a ∨ b ≤ c Theorem. (Cut elimination) Lat and Lat cf ( Lat without cut) prove the same sequents. We consider the lattice frame W , where W = Fm , W ′ = Fm and a ⊑ b iff a ≤ b is provable in Lat cf . Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 12 / 44

Cut elimination for lattices Substructural logics Lattice representation Lattices Contexts a ≤ b b ≤ a a ≤ b b ≤ c Dedekind-Birkhoff a ≤ a a ≤ c Lattice frames a = b Gentzen lattice frames Cut elimination for lattices a ≤ c b ≤ c c ≤ a c ≤ b Residuated frames a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Variants of frames References c ≤ a c ≤ b a ≤ c b ≤ c c ≤ a ∨ b c ≤ a ∨ b a ∨ b ≤ c Theorem. (Cut elimination) Lat and Lat cf ( Lat without cut) prove the same sequents. We consider the lattice frame W , where W = Fm , W ′ = Fm and a ⊑ b iff a ≤ b is provable in Lat cf . We will show that if a sequent holds in all lattices then it is provable Lat cf . Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 12 / 44

Cut elimination for lattices Substructural logics Lattice representation Lattices Contexts a ≤ b b ≤ a a ≤ b b ≤ c Dedekind-Birkhoff a ≤ a a ≤ c Lattice frames a = b Gentzen lattice frames Cut elimination for lattices a ≤ c b ≤ c c ≤ a c ≤ b Residuated frames a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Variants of frames References c ≤ a c ≤ b a ≤ c b ≤ c c ≤ a ∨ b c ≤ a ∨ b a ∨ b ≤ c Theorem. (Cut elimination) Lat and Lat cf ( Lat without cut) prove the same sequents. We consider the lattice frame W , where W = Fm , W ′ = Fm and a ⊑ b iff a ≤ b is provable in Lat cf . We will show that if a sequent holds in all lattices then it is provable Lat cf . Lemma. For all a, b ∈ S , then a ∧ B b ∈ q ( a ) ∧ W + q ( b ) ⊆ q ( a ∧ B b ) and a ∨ B b ∈ q ( a ) ∨ W + q ( b ) ⊆ q ( a ∨ B b ) . ( W , Fm ) is cf-Gentzen. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 12 / 44

Cut elimination for lattices Substructural logics Lattice representation Lattices Contexts a ≤ b b ≤ a a ≤ b b ≤ c Dedekind-Birkhoff a ≤ a a ≤ c Lattice frames a = b Gentzen lattice frames Cut elimination for lattices a ≤ c b ≤ c c ≤ a c ≤ b Residuated frames a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Variants of frames References c ≤ a c ≤ b a ≤ c b ≤ c c ≤ a ∨ b c ≤ a ∨ b a ∨ b ≤ c Theorem. (Cut elimination) Lat and Lat cf ( Lat without cut) prove the same sequents. We consider the lattice frame W , where W = Fm , W ′ = Fm and a ⊑ b iff a ≤ b is provable in Lat cf . We will show that if a sequent holds in all lattices then it is provable Lat cf . Lemma. For all a, b ∈ S , then a ∧ B b ∈ q ( a ) ∧ W + q ( b ) ⊆ q ( a ∧ B b ) and a ∨ B b ∈ q ( a ) ∨ W + q ( b ) ⊆ q ( a ∨ B b ) . ( W , Fm ) is cf-Gentzen. Corollary. The homomorphism h : Fm → W + extending the variable assignment p �→ q ( p ) satisfies a ∈ h ( a ) ⊆ q ( a ) . So, if W + | = a ≤ b , then a ∈ h ( a ) ⊆ h ( b ) ⊆ q ( b ) = { b } ⊳ , so a ⊑ b . Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 12 / 44

Substructural logics Lattice representation Residuated frames Residuated frames Simple equations Gentzen frames DM-completions Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP Residuated frames FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 13 / 44

Residuated frames Substructural logics Lattice representation Residuated frames A residuated frame is a structure W = ( W, ◦ , ε, ⊑ , W ′ ) where Residuated frames Simple equations ( W, ⊑ , W ′ ) is a lattice frame ■ Gentzen frames DM-completions ( W, ◦ , ε ) is a monoid ■ Embedding of subreducts there exist � and � such that for all x, y ∈ W and z ∈ W ′ ■ Pre-frames Embedding of subreducts using preframes Examples of frames: FL ( x ◦ y ) ⊑ z ⇔ y ⊑ ( x � z ) ⇔ x ⊑ ( z � y ) . FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 14 / 44

Residuated frames Substructural logics Lattice representation Residuated frames A residuated frame is a structure W = ( W, ◦ , ε, ⊑ , W ′ ) where Residuated frames Simple equations ( W, ⊑ , W ′ ) is a lattice frame ■ Gentzen frames DM-completions ( W, ◦ , ε ) is a monoid ■ Embedding of subreducts there exist � and � such that for all x, y ∈ W and z ∈ W ′ ■ Pre-frames Embedding of subreducts using preframes Examples of frames: FL ( x ◦ y ) ⊑ z ⇔ y ⊑ ( x � z ) ⇔ x ⊑ ( z � y ) . FL FMP FEP Combining frames Amalgamation Corollary. If W is a residuated frame then the Galois/dual algebra Gen. amalgamation Densification W + = ( γ [ P ( W )] , ∩ , ∪ γ , ◦ γ , γ (1) , \ , / ) is a residuated lattice, where Densification Interpolation X ◦ Y = { x ◦ y : x ∈ X, y ∈ Y } , Disjunction property Undecidability X \ Y = { z : X ◦ { z } ⊆ Y } Modular CE Y/X = { z : { z } ◦ X ⊆ Y } . Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 14 / 44

Simple equations Substructural logics Lattice representation Residuated frames Consider the equation ε : Residuated frames Simple equations Gentzen frames xyw ≤ x 2 ∨ yx ∨ xw 3 y 2 DM-completions Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 15 / 44

Simple equations Substructural logics Lattice representation Residuated frames Consider the equation ε : Residuated frames Simple equations Gentzen frames xyw ≤ x 2 ∨ yx ∨ xw 3 y 2 DM-completions Embedding of subreducts Pre-frames Embedding of subreducts x 2 ≤ z xw 3 y 2 ≤ z using preframes yx ≤ z Examples of frames: FL FL xyw ≤ z FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 15 / 44

Simple equations Substructural logics Lattice representation Residuated frames Consider the equation ε : Residuated frames Simple equations Gentzen frames xyw ≤ x 2 ∨ yx ∨ xw 3 y 2 DM-completions Embedding of subreducts Pre-frames Embedding of subreducts x 2 ≤ z xw 3 y 2 ≤ z using preframes yx ≤ z Examples of frames: FL FL xyw ≤ z FMP FEP Combining frames x ◦ x ⊑ z y ◦ x ⊑ z x ◦ w ◦ w ◦ w ◦ y ◦ y N z Amalgamation R ( ε ) Gen. amalgamation x ◦ y ◦ w ⊑ z Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 15 / 44

Simple equations Substructural logics Lattice representation Residuated frames Consider the equation ε : Residuated frames Simple equations Gentzen frames xyw ≤ x 2 ∨ yx ∨ xw 3 y 2 DM-completions Embedding of subreducts Pre-frames Embedding of subreducts x 2 ≤ z xw 3 y 2 ≤ z using preframes yx ≤ z Examples of frames: FL FL xyw ≤ z FMP FEP Combining frames x ◦ x ⊑ z y ◦ x ⊑ z x ◦ w ◦ w ◦ w ◦ y ◦ y N z Amalgamation R ( ε ) Gen. amalgamation x ◦ y ◦ w ⊑ z Densification Densification Interpolation Disjunction property Theorem: If W satisfies R ( ε ) iff W + satisfies ε . Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 15 / 44

Simple equations Substructural logics Lattice representation Residuated frames Consider the equation ε : Residuated frames Simple equations Gentzen frames xyw ≤ x 2 ∨ yx ∨ xw 3 y 2 DM-completions Embedding of subreducts Pre-frames Embedding of subreducts x 2 ≤ z xw 3 y 2 ≤ z using preframes yx ≤ z Examples of frames: FL FL xyw ≤ z FMP FEP Combining frames x ◦ x ⊑ z y ◦ x ⊑ z x ◦ w ◦ w ◦ w ◦ y ◦ y N z Amalgamation R ( ε ) Gen. amalgamation x ◦ y ◦ w ⊑ z Densification Densification Interpolation Disjunction property Theorem: If W satisfies R ( ε ) iff W + satisfies ε . Undecidability Modular CE Hilbert system for FL Lemma. Every equation over {∨ , · , 1 } is equivalent to a conjunction Strong separation Variants of frames of simple equations: t 0 ≤ t 1 ∨ · · · ∨ t n , where t i are {· , 1 } -terms and References t 0 is linear. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 15 / 44

Gentzen frames Substructural logics Lattice representation Residuated frames x ⊑ a a ⊑ z Residuated frames (CUT) a ⊑ a (Id) Simple equations x ⊑ z Gentzen frames DM-completions a ⊑ z b ⊑ z x ⊑ a x ⊑ b Embedding of subreducts ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) Pre-frames a ∨ b ⊑ z Embedding of subreducts using preframes a ⊑ z b ⊑ z x ⊑ a x ⊑ b Examples of frames: FL a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) FL x ⊑ a ∧ b FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 16 / 44

Gentzen frames Substructural logics Lattice representation Residuated frames x ⊑ a a ⊑ z Residuated frames (CUT) a ⊑ a (Id) Simple equations x ⊑ z Gentzen frames DM-completions a ⊑ z b ⊑ z x ⊑ a x ⊑ b Embedding of subreducts ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) Pre-frames a ∨ b ⊑ z Embedding of subreducts using preframes a ⊑ z b ⊑ z x ⊑ a x ⊑ b Examples of frames: FL a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) FL x ⊑ a ∧ b FMP FEP Combining frames x ⊑ a y ⊑ b Amalgamation a ◦ b ⊑ z ε ⊑ z a · b ⊑ z ( · L) x ◦ y ⊑ a · b ( · R) 1 ⊑ z ( 1 L) ε ⊑ 1 ( 1 R) Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 16 / 44

Gentzen frames Substructural logics Lattice representation Residuated frames x ⊑ a a ⊑ z Residuated frames (CUT) a ⊑ a (Id) Simple equations x ⊑ z Gentzen frames DM-completions a ⊑ z b ⊑ z x ⊑ a x ⊑ b Embedding of subreducts ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) Pre-frames a ∨ b ⊑ z Embedding of subreducts using preframes a ⊑ z b ⊑ z x ⊑ a x ⊑ b Examples of frames: FL a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) FL x ⊑ a ∧ b FMP FEP Combining frames x ⊑ a y ⊑ b Amalgamation a ◦ b ⊑ z ε ⊑ z a · b ⊑ z ( · L) x ◦ y ⊑ a · b ( · R) 1 ⊑ z ( 1 L) ε ⊑ 1 ( 1 R) Gen. amalgamation Densification Densification Interpolation x ⊑ a � b x ⊑ b � a x ⊑ a b ⊑ z x ⊑ a b ⊑ z Disjunction property a \ b ⊑ x � z ( \ L) ( \ R) b/a ⊑ z � x ( / L) ( / R) x ⊑ a \ b x ⊑ b/a Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 16 / 44

Gentzen frames Substructural logics Lattice representation Residuated frames x ⊑ a a ⊑ z Residuated frames (CUT) a ⊑ a (Id) Simple equations x ⊑ z Gentzen frames DM-completions a ⊑ z b ⊑ z x ⊑ a x ⊑ b Embedding of subreducts ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) Pre-frames a ∨ b ⊑ z Embedding of subreducts using preframes a ⊑ z b ⊑ z x ⊑ a x ⊑ b Examples of frames: FL a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) FL x ⊑ a ∧ b FMP FEP Combining frames x ⊑ a y ⊑ b Amalgamation a ◦ b ⊑ z ε ⊑ z a · b ⊑ z ( · L) x ◦ y ⊑ a · b ( · R) 1 ⊑ z ( 1 L) ε ⊑ 1 ( 1 R) Gen. amalgamation Densification Densification Interpolation x ⊑ a � b x ⊑ b � a x ⊑ a b ⊑ z x ⊑ a b ⊑ z Disjunction property a \ b ⊑ x � z ( \ L) ( \ R) b/a ⊑ z � x ( / L) ( / R) x ⊑ a \ b x ⊑ b/a Undecidability Modular CE Hilbert system for FL Strong separation If we have a common subset S of W and W ′ that supports a (partial) Variants of frames References algebra S = ( S, ∧ , ∨ , · , \ , /, 1) , and for a, b, c ∈ S , x, y ∈ W , z ∈ W ′ , Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 16 / 44

Gentzen frames Substructural logics Lattice representation Residuated frames x ⊑ a a ⊑ z Residuated frames (CUT) a ⊑ a (Id) Simple equations x ⊑ z Gentzen frames DM-completions a ⊑ z b ⊑ z x ⊑ a x ⊑ b Embedding of subreducts ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) Pre-frames a ∨ b ⊑ z Embedding of subreducts using preframes a ⊑ z b ⊑ z x ⊑ a x ⊑ b Examples of frames: FL a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) FL x ⊑ a ∧ b FMP FEP Combining frames x ⊑ a y ⊑ b Amalgamation a ◦ b ⊑ z ε ⊑ z a · b ⊑ z ( · L) x ◦ y ⊑ a · b ( · R) 1 ⊑ z ( 1 L) ε ⊑ 1 ( 1 R) Gen. amalgamation Densification Densification Interpolation x ⊑ a � b x ⊑ b � a x ⊑ a b ⊑ z x ⊑ a b ⊑ z Disjunction property a \ b ⊑ x � z ( \ L) ( \ R) b/a ⊑ z � x ( / L) ( / R) x ⊑ a \ b x ⊑ b/a Undecidability Modular CE Hilbert system for FL Strong separation If we have a common subset S of W and W ′ that supports a (partial) Variants of frames References algebra S = ( S, ∧ , ∨ , · , \ , /, 1) , and for a, b, c ∈ S , x, y ∈ W , z ∈ W ′ , then we call ( W , S ) a Gentzen frame and we call W an S -frame. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 16 / 44

Gentzen frames Substructural logics Lattice representation Residuated frames x ⊑ a a ⊑ z Residuated frames (CUT) a ⊑ a (Id) Simple equations x ⊑ z Gentzen frames DM-completions a ⊑ z b ⊑ z x ⊑ a x ⊑ b Embedding of subreducts ( ∨ L) x ⊑ a ∨ b ( ∨ R ℓ ) x ⊑ a ∨ b ( ∨ R r ) Pre-frames a ∨ b ⊑ z Embedding of subreducts using preframes a ⊑ z b ⊑ z x ⊑ a x ⊑ b Examples of frames: FL a ∧ b ⊑ z ( ∧ L ℓ ) a ∧ b ⊑ z ( ∧ L r ) ( ∧ R) FL x ⊑ a ∧ b FMP FEP Combining frames x ⊑ a y ⊑ b Amalgamation a ◦ b ⊑ z ε ⊑ z a · b ⊑ z ( · L) x ◦ y ⊑ a · b ( · R) 1 ⊑ z ( 1 L) ε ⊑ 1 ( 1 R) Gen. amalgamation Densification Densification Interpolation x ⊑ a � b x ⊑ b � a x ⊑ a b ⊑ z x ⊑ a b ⊑ z Disjunction property a \ b ⊑ x � z ( \ L) ( \ R) b/a ⊑ z � x ( / L) ( / R) x ⊑ a \ b x ⊑ b/a Undecidability Modular CE Hilbert system for FL Strong separation If we have a common subset S of W and W ′ that supports a (partial) Variants of frames References algebra S = ( S, ∧ , ∨ , · , \ , /, 1) , and for a, b, c ∈ S , x, y ∈ W , z ∈ W ′ , then we call ( W , S ) a Gentzen frame and we call W an S -frame. Again, q : S → W + is a homomorphism (in the full signature). Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 16 / 44

DM-completions Substructural logics Lattice representation Residuated frames To a residuated lattice A , we associate the Gentzen frame ( W A , A ) , Residuated frames Simple equations where W A = ( A, · , 1 , ≤ , A ) . Gentzen frames DM-completions Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 17 / 44

DM-completions Substructural logics Lattice representation Residuated frames To a residuated lattice A , we associate the Gentzen frame ( W A , A ) , Residuated frames Simple equations where W A = ( A, · , 1 , ≤ , A ) . We define x � z = x \ z and z � x = z/x . Gentzen frames DM-completions Embedding of subreducts Theorem. The map x �→ x ⊳ is an embedding of A into W + A . Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 17 / 44

DM-completions Substructural logics Lattice representation Residuated frames To a residuated lattice A , we associate the Gentzen frame ( W A , A ) , Residuated frames Simple equations where W A = ( A, · , 1 , ≤ , A ) . We define x � z = x \ z and z � x = z/x . Gentzen frames DM-completions Embedding of subreducts Theorem. The map x �→ x ⊳ is an embedding of A into W + A . Pre-frames Embedding of subreducts using preframes Examples of frames: FL Corollary. The variety of residuated lattices is closed under FL FMP DM-completions. FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 17 / 44

Embedding of subreducts Substructural logics Lattice representation Residuated frames To a partially-odrered semigroup A = ( A, ≤ , · ) , Residuated frames Simple equations Gentzen frames DM-completions Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 18 / 44

Embedding of subreducts Substructural logics Lattice representation Residuated frames To a partially-odrered semigroup A = ( A, ≤ , · ) , we associate the Residuated frames Simple equations Gentzen frame ( W A , A ) , where W A = ( A ε , · , ⊑ , A ε × A × A ε ) , Gentzen frames DM-completions A ε = A ∪ { ε } for ε �∈ A , where a ◦ b = ab for a, b ∈ A and Embedding of subreducts ε ◦ a = a ◦ ε = a . Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 18 / 44

Embedding of subreducts Substructural logics Lattice representation Residuated frames To a partially-odrered semigroup A = ( A, ≤ , · ) , we associate the Residuated frames Simple equations Gentzen frame ( W A , A ) , where W A = ( A ε , · , ⊑ , A ε × A × A ε ) , Gentzen frames DM-completions A ε = A ∪ { ε } for ε �∈ A , where a ◦ b = ab for a, b ∈ A and Embedding of subreducts ε ◦ a = a ◦ ε = a . Also, Pre-frames Embedding of subreducts using preframes x ⊑ ( y, a, z ) iff y ◦ x ◦ z ≤ a . Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 18 / 44

Embedding of subreducts Substructural logics Lattice representation Residuated frames To a partially-odrered semigroup A = ( A, ≤ , · ) , we associate the Residuated frames Simple equations Gentzen frame ( W A , A ) , where W A = ( A ε , · , ⊑ , A ε × A × A ε ) , Gentzen frames DM-completions A ε = A ∪ { ε } for ε �∈ A , where a ◦ b = ab for a, b ∈ A and Embedding of subreducts ε ◦ a = a ◦ ε = a . Also, Pre-frames Embedding of subreducts using preframes x ⊑ ( y, a, z ) iff y ◦ x ◦ z ≤ a . Examples of frames: FL FL FMP FEP Combining frames This is an A -frame, where the maps from A are a �→ a and Amalgamation Gen. amalgamation a �→ ( ε, a, ε ) . Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 18 / 44

Embedding of subreducts Substructural logics Lattice representation Residuated frames To a partially-odrered semigroup A = ( A, ≤ , · ) , we associate the Residuated frames Simple equations Gentzen frame ( W A , A ) , where W A = ( A ε , · , ⊑ , A ε × A × A ε ) , Gentzen frames DM-completions A ε = A ∪ { ε } for ε �∈ A , where a ◦ b = ab for a, b ∈ A and Embedding of subreducts ε ◦ a = a ◦ ε = a . Also, Pre-frames Embedding of subreducts using preframes x ⊑ ( y, a, z ) iff y ◦ x ◦ z ≤ a . Examples of frames: FL FL FMP FEP Combining frames This is an A -frame, where the maps from A are a �→ a and Amalgamation Gen. amalgamation a �→ ( ε, a, ε ) . Densification Densification Interpolation Theorem. The map x �→ x ⊳ is an embedding of A into W + A . If A Disjunction property Undecidability has a multiplicative unit then the embeding preserves it. The embedding preserves exising joins � X for which Modular CE Hilbert system for FL y ( � X ) z = � ( yx i z ) for all y, z ∈ A . The embedding preserves all Strong separation Variants of frames existing residuals. References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 18 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts := ( x, z, y ) ∈ W × W ′ × W =: � using preframes W ′ Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts := ( x, z, y ) ∈ W × W ′ × W =: � using preframes W ′ Examples of frames: FL FL FMP So we define: w � ⊑ ( x, z, y ) iff x ◦ w ◦ y ⊑ z . FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts := ( x, z, y ) ∈ W × W ′ × W =: � using preframes W ′ Examples of frames: FL FL FMP So we define: w � ⊑ ( x, z, y ) iff x ◦ w ◦ y ⊑ z . FEP Combining frames Amalgamation We now check if the new frame is residuated: Gen. amalgamation Densification w 1 ◦ w 2 � ⊑ ( x, z, y ) iff x ◦ w 1 ◦ w 2 ◦ y ⊑ z Densification Interpolation iff w 1 � ⊑ ( x, z, w 2 ◦ y ) Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts := ( x, z, y ) ∈ W × W ′ × W =: � using preframes W ′ Examples of frames: FL FL FMP So we define: w � ⊑ ( x, z, y ) iff x ◦ w ◦ y ⊑ z . FEP Combining frames Amalgamation We now check if the new frame is residuated: Gen. amalgamation Densification w 1 ◦ w 2 � ⊑ ( x, z, y ) iff x ◦ w 1 ◦ w 2 ◦ y ⊑ z Densification Interpolation iff w 1 � ⊑ ( x, z, w 2 ◦ y ) = ( x, z, y ) � w 2 Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts := ( x, z, y ) ∈ W × W ′ × W =: � using preframes W ′ Examples of frames: FL FL FMP So we define: w � ⊑ ( x, z, y ) iff x ◦ w ◦ y ⊑ z . FEP Combining frames Amalgamation We now check if the new frame is residuated: Gen. amalgamation Densification w 1 ◦ w 2 � ⊑ ( x, z, y ) iff x ◦ w 1 ◦ w 2 ◦ y ⊑ z Densification Interpolation iff w 1 � ⊑ ( x, z, w 2 ◦ y ) = ( x, z, y ) � w 2 Disjunction property Undecidability iff w 2 � ⊑ ( x ◦ w 1 , z ) Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts := ( x, z, y ) ∈ W × W ′ × W =: � using preframes W ′ Examples of frames: FL FL FMP So we define: w � ⊑ ( x, z, y ) iff x ◦ w ◦ y ⊑ z . FEP Combining frames Amalgamation We now check if the new frame is residuated: Gen. amalgamation Densification w 1 ◦ w 2 � ⊑ ( x, z, y ) iff x ◦ w 1 ◦ w 2 ◦ y ⊑ z Densification Interpolation iff w 1 � ⊑ ( x, z, w 2 ◦ y ) = ( x, z, y ) � w 2 Disjunction property Undecidability iff w 2 � ⊑ ( x ◦ w 1 , z ) = w 1 � ( x, z, y ) Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts := ( x, z, y ) ∈ W × W ′ × W =: � using preframes W ′ Examples of frames: FL FL FMP So we define: w � ⊑ ( x, z, y ) iff x ◦ w ◦ y ⊑ z . FEP Combining frames Amalgamation We now check if the new frame is residuated: Gen. amalgamation Densification w 1 ◦ w 2 � ⊑ ( x, z, y ) iff x ◦ w 1 ◦ w 2 ◦ y ⊑ z Densification Interpolation iff w 1 � ⊑ ( x, z, w 2 ◦ y ) = ( x, z, y ) � w 2 Disjunction property Undecidability iff w 2 � ⊑ ( x ◦ w 1 , z ) = w 1 � ( x, z, y ) Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Pre-frames Substructural logics Lattice representation Residuated frames Given a frame W = ( W, ◦ , ε, ⊑ , W ′ ) which might not be residuated, Residuated frames Simple equations we can construct a residuated frame � W = ( W, ◦ , ε, � ⊑ , � W ′ ) out of it. Gentzen frames DM-completions Embedding of subreducts We have x ◦ w ◦ y ⊑ z iff w ⊑ x � z � y Pre-frames Embedding of subreducts := ( x, z, y ) ∈ W × W ′ × W =: � using preframes W ′ Examples of frames: FL FL FMP So we define: w � ⊑ ( x, z, y ) iff x ◦ w ◦ y ⊑ z . FEP Combining frames Amalgamation We now check if the new frame is residuated: Gen. amalgamation Densification w 1 ◦ w 2 � ⊑ ( x, z, y ) iff x ◦ w 1 ◦ w 2 ◦ y ⊑ z Densification Interpolation iff w 1 � ⊑ ( x, z, w 2 ◦ y ) = ( x, z, y ) � w 2 Disjunction property Undecidability iff w 2 � ⊑ ( x ◦ w 1 , z ) = w 1 � ( x, z, y ) Modular CE Hilbert system for FL Strong separation Often we will write ⊑ for the extension � ⊑ . Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 19 / 44

Embedding of subreducts using preframes Substructural logics Lattice representation Residuated frames To a partially-odrered semigroup A = ( A, ≤ , · ) , we associate the Residuated frames Simple equations Gentzen pre-frame ( W A , A ) , where W A = ( A ε , · , ⊑ , A ) , Gentzen frames DM-completions A ε = A ∪ { ε } for ε �∈ A , where a ◦ b = ab for a, b ∈ A and Embedding of subreducts ε ◦ a = a ◦ ε = a . Also, Pre-frames Embedding of subreducts using preframes x ⊑ a iff x ≤ a . Examples of frames: FL FL FMP This is an A -frame, where the maps from A are a �→ a and FEP Combining frames a �→ ( ε, a, ε ) . Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 20 / 44

Embedding of subreducts using preframes Substructural logics Lattice representation Residuated frames To a partially-odrered semigroup A = ( A, ≤ , · ) , we associate the Residuated frames Simple equations Gentzen pre-frame ( W A , A ) , where W A = ( A ε , · , ⊑ , A ) , Gentzen frames DM-completions A ε = A ∪ { ε } for ε �∈ A , where a ◦ b = ab for a, b ∈ A and Embedding of subreducts ε ◦ a = a ◦ ε = a . Also, Pre-frames Embedding of subreducts using preframes x ⊑ a iff x ≤ a . Examples of frames: FL FL FMP This is an A -frame, where the maps from A are a �→ a and FEP Combining frames a �→ ( ε, a, ε ) . Amalgamation Gen. amalgamation Densification Theorem. The map x �→ x ⊳ is an embedding of A into W + A . If A Densification Interpolation has a multiplicative unit then the embeding preserves it. The Disjunction property embedding preserves exising joins � X for which Undecidability y ( � X ) z = � ( yx i z ) for all y, z ∈ A . The embedding preserves all Modular CE Hilbert system for FL Strong separation existing residuals. Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 20 / 44

Examples of frames: FL Substructural logics Lattice representation Residuated frames Based on the Gentzen system FL , we define the residuated frame Residuated frames Simple equations W FL based on the preframe: Gentzen frames DM-completions Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 21 / 44

Examples of frames: FL Substructural logics Lattice representation Residuated frames Based on the Gentzen system FL , we define the residuated frame Residuated frames Simple equations W FL based on the preframe: Gentzen frames DM-completions ( W, ◦ , ε ) is the free monoid over the set Fm of all formulas Embedding of subreducts ■ Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 21 / 44

Examples of frames: FL Substructural logics Lattice representation Residuated frames Based on the Gentzen system FL , we define the residuated frame Residuated frames Simple equations W FL based on the preframe: Gentzen frames DM-completions ( W, ◦ , ε ) is the free monoid over the set Fm of all formulas Embedding of subreducts ■ W ′ = Fm , and Pre-frames ■ Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 21 / 44

Examples of frames: FL Substructural logics Lattice representation Residuated frames Based on the Gentzen system FL , we define the residuated frame Residuated frames Simple equations W FL based on the preframe: Gentzen frames DM-completions ( W, ◦ , ε ) is the free monoid over the set Fm of all formulas Embedding of subreducts ■ W ′ = Fm , and Pre-frames ■ Embedding of subreducts using preframes x N a iff ⊢ FL x ⇒ a . ■ Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 21 / 44

Examples of frames: FL Substructural logics Lattice representation Residuated frames Based on the Gentzen system FL , we define the residuated frame Residuated frames Simple equations W FL based on the preframe: Gentzen frames DM-completions ( W, ◦ , ε ) is the free monoid over the set Fm of all formulas Embedding of subreducts ■ W ′ = Fm , and Pre-frames ■ Embedding of subreducts using preframes x N a iff ⊢ FL x ⇒ a . ■ Examples of frames: FL FL FMP It is easy to see that ( W FL , Fm ) is a Gentzen frame. For example, FEP Combining frames consider Amalgamation x ⊑ a b ⊑ z Gen. amalgamation a \ b ⊑ x � z ( \ L) Densification Densification Interpolation Where a, b, c ∈ Fm , x, u, v ∈ W = Fm ∗ , z ∈ W × Fm × W . Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 21 / 44

Examples of frames: FL Substructural logics Lattice representation Residuated frames Based on the Gentzen system FL , we define the residuated frame Residuated frames Simple equations W FL based on the preframe: Gentzen frames DM-completions ( W, ◦ , ε ) is the free monoid over the set Fm of all formulas Embedding of subreducts ■ W ′ = Fm , and Pre-frames ■ Embedding of subreducts using preframes x N a iff ⊢ FL x ⇒ a . ■ Examples of frames: FL FL FMP It is easy to see that ( W FL , Fm ) is a Gentzen frame. For example, FEP Combining frames consider Amalgamation x ⊑ a b ⊑ z Gen. amalgamation a \ b ⊑ x � z ( \ L) Densification Densification Interpolation Where a, b, c ∈ Fm , x, u, v ∈ W = Fm ∗ , z ∈ W × Fm × W . The Disjunction property Undecidability rule can be rewritten as Modular CE Hilbert system for FL x ⊑ a b ⊑ z Strong separation Variants of frames x ◦ ( a \ b ) ⊑ z References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 21 / 44

Examples of frames: FL Substructural logics Lattice representation Residuated frames Based on the Gentzen system FL , we define the residuated frame Residuated frames Simple equations W FL based on the preframe: Gentzen frames DM-completions ( W, ◦ , ε ) is the free monoid over the set Fm of all formulas Embedding of subreducts ■ W ′ = Fm , and Pre-frames ■ Embedding of subreducts using preframes x N a iff ⊢ FL x ⇒ a . ■ Examples of frames: FL FL FMP It is easy to see that ( W FL , Fm ) is a Gentzen frame. For example, FEP Combining frames consider Amalgamation x ⊑ a b ⊑ z Gen. amalgamation a \ b ⊑ x � z ( \ L) Densification Densification Interpolation Where a, b, c ∈ Fm , x, u, v ∈ W = Fm ∗ , z ∈ W × Fm × W . The Disjunction property Undecidability rule can be rewritten as Modular CE Hilbert system for FL x ⊑ a b ⊑ ( v, c, u ) x ⊑ a b ⊑ z Strong separation Variants of frames x ◦ ( a \ b ) ⊑ z x ◦ ( a \ b ) ⊑ ( v, c, u ) References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 21 / 44

Examples of frames: FL Substructural logics Lattice representation Residuated frames Based on the Gentzen system FL , we define the residuated frame Residuated frames Simple equations W FL based on the preframe: Gentzen frames DM-completions ( W, ◦ , ε ) is the free monoid over the set Fm of all formulas Embedding of subreducts ■ W ′ = Fm , and Pre-frames ■ Embedding of subreducts using preframes x N a iff ⊢ FL x ⇒ a . ■ Examples of frames: FL FL FMP It is easy to see that ( W FL , Fm ) is a Gentzen frame. For example, FEP Combining frames consider Amalgamation x ⊑ a b ⊑ z Gen. amalgamation a \ b ⊑ x � z ( \ L) Densification Densification Interpolation Where a, b, c ∈ Fm , x, u, v ∈ W = Fm ∗ , z ∈ W × Fm × W . The Disjunction property Undecidability rule can be rewritten as Modular CE Hilbert system for FL x ⊑ a b ⊑ ( v, c, u ) x ⊑ a b ⊑ z x ⊑ a v ◦ b ◦ u ⊑ c Strong separation Variants of frames x ◦ ( a \ b ) ⊑ z x ◦ ( a \ b ) ⊑ ( v, c, u ) v ◦ x ◦ ( a \ b ) ◦ u ⊑ c References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 21 / 44

FL Substructural logics Lattice representation Residuated frames x ⇒ a y ◦ a ◦ z ⇒ c Residuated frames (cut) a ⇒ a (Id) Simple equations y ◦ x ◦ z ⇒ c Gentzen frames DM-completions y ◦ a ◦ z ⇒ c y ◦ b ◦ z ⇒ c x ⇒ a x ⇒ b Embedding of subreducts y ◦ a ∧ b ◦ z ⇒ c ( ∧ L ℓ ) y ◦ a ∧ b ◦ z ⇒ c ( ∧ L r ) ( ∧ R) x ⇒ a ∧ b Pre-frames Embedding of subreducts using preframes y ◦ a ◦ z ⇒ c y ◦ b ◦ z ⇒ c x ⇒ a x ⇒ b Examples of frames: FL ( ∨ L) x ⇒ a ∨ b ( ∨ R ℓ ) x ⇒ a ∨ b ( ∨ R r ) FL y ◦ a ∨ b ◦ z ⇒ c FMP FEP x ⇒ a y ◦ b ◦ z ⇒ c a ◦ x ⇒ b Combining frames y ◦ x ◦ ( a \ b ) ◦ z ⇒ c ( \ L) x ⇒ a \ b ( \ R) Amalgamation Gen. amalgamation Densification x ⇒ a y ◦ b ◦ z ⇒ c x ◦ a ⇒ b Densification y ◦ ( b/a ) ◦ x ◦ z ⇒ c ( / L) x ⇒ b/a ( / R) Interpolation Disjunction property Undecidability y ◦ a ◦ b ◦ z ⇒ c x ⇒ a y ⇒ b Modular CE y ◦ a · b ◦ z ⇒ c ( · L) ( · R) Hilbert system for FL x ◦ y ⇒ a · b Strong separation y ◦ z ⇒ a Variants of frames y ◦ 1 ◦ z ⇒ a ( 1 L) ε ⇒ 1 ( 1 R) References where a, b, c ∈ Fm , x, y, z ∈ Fm ∗ . Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 22 / 44

Finite model property Substructural logics Lattice representation Residuated frames Given a sequent s which is not provable in FL we construct a finite Residuated frames Simple equations countermodel of it. Gentzen frames DM-completions Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 23 / 44

Finite model property Substructural logics Lattice representation Residuated frames Given a sequent s which is not provable in FL we construct a finite Residuated frames Simple equations countermodel of it. Gentzen frames DM-completions Recall the residuated frame W FL based on x ⊑ a iff x ⇒ a is Embedding of subreducts Pre-frames provable in FL cf . Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 23 / 44

Finite model property Substructural logics Lattice representation Residuated frames Given a sequent s which is not provable in FL we construct a finite Residuated frames Simple equations countermodel of it. Gentzen frames DM-completions Recall the residuated frame W FL based on x ⊑ a iff x ⇒ a is Embedding of subreducts Pre-frames provable in FL cf . Embedding of subreducts using preframes Examples of frames: FL Even though s is not provable we consider all the sequents that FL appear in all failed proof attempts if s . We define s ↑ the set of pairs FMP ( w, ( x, c, y )) in W × W ′ such that x, w, y ⇒ c is one of those FEP Combining frames sequents. Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 23 / 44

Finite model property Substructural logics Lattice representation Residuated frames Given a sequent s which is not provable in FL we construct a finite Residuated frames Simple equations countermodel of it. Gentzen frames DM-completions Recall the residuated frame W FL based on x ⊑ a iff x ⇒ a is Embedding of subreducts Pre-frames provable in FL cf . Embedding of subreducts using preframes Examples of frames: FL Even though s is not provable we consider all the sequents that FL appear in all failed proof attempts if s . We define s ↑ the set of pairs FMP ( w, ( x, c, y )) in W × W ′ such that x, w, y ⇒ c is one of those FEP Combining frames sequents. Amalgamation Gen. amalgamation Densification We also define a new relation ⊑ s = � ⊑ ∪ ( s ↑ ) c . Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 23 / 44

Finite model property Substructural logics Lattice representation Residuated frames Given a sequent s which is not provable in FL we construct a finite Residuated frames Simple equations countermodel of it. Gentzen frames DM-completions Recall the residuated frame W FL based on x ⊑ a iff x ⇒ a is Embedding of subreducts Pre-frames provable in FL cf . Embedding of subreducts using preframes Examples of frames: FL Even though s is not provable we consider all the sequents that FL appear in all failed proof attempts if s . We define s ↑ the set of pairs FMP ( w, ( x, c, y )) in W × W ′ such that x, w, y ⇒ c is one of those FEP Combining frames sequents. Amalgamation Gen. amalgamation Densification We also define a new relation ⊑ s = � ⊑ ∪ ( s ↑ ) c . The resulting frame Densification W s is residuated. Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 23 / 44

Finite model property Substructural logics Lattice representation Residuated frames Given a sequent s which is not provable in FL we construct a finite Residuated frames Simple equations countermodel of it. Gentzen frames DM-completions Recall the residuated frame W FL based on x ⊑ a iff x ⇒ a is Embedding of subreducts Pre-frames provable in FL cf . Embedding of subreducts using preframes Examples of frames: FL Even though s is not provable we consider all the sequents that FL appear in all failed proof attempts if s . We define s ↑ the set of pairs FMP ( w, ( x, c, y )) in W × W ′ such that x, w, y ⇒ c is one of those FEP Combining frames sequents. Amalgamation Gen. amalgamation Densification We also define a new relation ⊑ s = � ⊑ ∪ ( s ↑ ) c . The resulting frame Densification W s is residuated. Interpolation Disjunction property Using the finiteness of ( ⊑ s ) c we get that W + s is finite. Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 23 / 44

Finite model property Substructural logics Lattice representation Residuated frames Given a sequent s which is not provable in FL we construct a finite Residuated frames Simple equations countermodel of it. Gentzen frames DM-completions Recall the residuated frame W FL based on x ⊑ a iff x ⇒ a is Embedding of subreducts Pre-frames provable in FL cf . Embedding of subreducts using preframes Examples of frames: FL Even though s is not provable we consider all the sequents that FL appear in all failed proof attempts if s . We define s ↑ the set of pairs FMP ( w, ( x, c, y )) in W × W ′ such that x, w, y ⇒ c is one of those FEP Combining frames sequents. Amalgamation Gen. amalgamation Densification We also define a new relation ⊑ s = � ⊑ ∪ ( s ↑ ) c . The resulting frame Densification W s is residuated. Interpolation Disjunction property Using the finiteness of ( ⊑ s ) c we get that W + s is finite. Moreover Undecidability Modular CE ( W s , Fm ) is a cut-free Gentzen frame and s is not valid in W + s . Hilbert system for FL Strong separation Variants of frames Corollary. The system FL has the finite model property. The same References holds for reducing simple extensions. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 23 / 44

Finite model property Substructural logics Lattice representation Residuated frames Given a sequent s which is not provable in FL we construct a finite Residuated frames Simple equations countermodel of it. Gentzen frames DM-completions Recall the residuated frame W FL based on x ⊑ a iff x ⇒ a is Embedding of subreducts Pre-frames provable in FL cf . Embedding of subreducts using preframes Examples of frames: FL Even though s is not provable we consider all the sequents that FL appear in all failed proof attempts if s . We define s ↑ the set of pairs FMP ( w, ( x, c, y )) in W × W ′ such that x, w, y ⇒ c is one of those FEP Combining frames sequents. Amalgamation Gen. amalgamation Densification We also define a new relation ⊑ s = � ⊑ ∪ ( s ↑ ) c . The resulting frame Densification W s is residuated. Interpolation Disjunction property Using the finiteness of ( ⊑ s ) c we get that W + s is finite. Moreover Undecidability Modular CE ( W s , Fm ) is a cut-free Gentzen frame and s is not valid in W + s . Hilbert system for FL Strong separation Variants of frames Corollary. The system FL has the finite model property. The same References holds for reducing simple extensions. The corresponding varieties of residuated lattices are generated by their finite members. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 23 / 44

FEP Substructural logics Lattice representation Residuated frames A class of algebras K has the finite embeddability property (FEP) if Residuated frames Simple equations for every A ∈ K , every finite partial subalgebra B of A can be Gentzen frames DM-completions (partially) embedded in a finite D ∈ K . Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 24 / 44

FEP Substructural logics Lattice representation Residuated frames A class of algebras K has the finite embeddability property (FEP) if Residuated frames Simple equations for every A ∈ K , every finite partial subalgebra B of A can be Gentzen frames DM-completions (partially) embedded in a finite D ∈ K . Embedding of subreducts Pre-frames We define W based on the preframe Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 24 / 44

FEP Substructural logics Lattice representation Residuated frames A class of algebras K has the finite embeddability property (FEP) if Residuated frames Simple equations for every A ∈ K , every finite partial subalgebra B of A can be Gentzen frames DM-completions (partially) embedded in a finite D ∈ K . Embedding of subreducts Pre-frames We define W based on the preframe Embedding of subreducts using preframes ( W, · , 1) is the submonoid of A generated by B , ■ Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 24 / 44

FEP Substructural logics Lattice representation Residuated frames A class of algebras K has the finite embeddability property (FEP) if Residuated frames Simple equations for every A ∈ K , every finite partial subalgebra B of A can be Gentzen frames DM-completions (partially) embedded in a finite D ∈ K . Embedding of subreducts Pre-frames We define W based on the preframe Embedding of subreducts using preframes ( W, · , 1) is the submonoid of A generated by B , ■ Examples of frames: FL W ′ = B , and FL ■ FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 24 / 44

FEP Substructural logics Lattice representation Residuated frames A class of algebras K has the finite embeddability property (FEP) if Residuated frames Simple equations for every A ∈ K , every finite partial subalgebra B of A can be Gentzen frames DM-completions (partially) embedded in a finite D ∈ K . Embedding of subreducts Pre-frames We define W based on the preframe Embedding of subreducts using preframes ( W, · , 1) is the submonoid of A generated by B , ■ Examples of frames: FL W ′ = B , and FL ■ FMP x ⊑ b by x ≤ A b . ■ FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 24 / 44

FEP Substructural logics Lattice representation Residuated frames A class of algebras K has the finite embeddability property (FEP) if Residuated frames Simple equations for every A ∈ K , every finite partial subalgebra B of A can be Gentzen frames DM-completions (partially) embedded in a finite D ∈ K . Embedding of subreducts Pre-frames We define W based on the preframe Embedding of subreducts using preframes ( W, · , 1) is the submonoid of A generated by B , ■ Examples of frames: FL W ′ = B , and FL ■ FMP x ⊑ b by x ≤ A b . ■ FEP Combining frames Amalgamation Theorem. Every variety of integral (alt., by commutative and Gen. amalgamation Densification knotted) RL’s axiomatized by equations over {∨ , · , 1 } has the FEP. Densification Interpolation q : B → W + is an embedding Disjunction property ■ Undecidability Modular CE W + ∈ V ■ Hilbert system for FL Strong separation W + is finite ■ Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 24 / 44

FEP Substructural logics Lattice representation Residuated frames A class of algebras K has the finite embeddability property (FEP) if Residuated frames Simple equations for every A ∈ K , every finite partial subalgebra B of A can be Gentzen frames DM-completions (partially) embedded in a finite D ∈ K . Embedding of subreducts Pre-frames We define W based on the preframe Embedding of subreducts using preframes ( W, · , 1) is the submonoid of A generated by B , ■ Examples of frames: FL W ′ = B , and FL ■ FMP x ⊑ b by x ≤ A b . ■ FEP Combining frames Amalgamation Theorem. Every variety of integral (alt., by commutative and Gen. amalgamation Densification knotted) RL’s axiomatized by equations over {∨ , · , 1 } has the FEP. Densification Interpolation q : B → W + is an embedding Disjunction property ■ Undecidability Modular CE W + ∈ V ■ Hilbert system for FL Strong separation W + is finite ■ Variants of frames References Corollary. These varieties are generated as quasivarieties by their finite members. Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 24 / 44

FEP Substructural logics Lattice representation Residuated frames A class of algebras K has the finite embeddability property (FEP) if Residuated frames Simple equations for every A ∈ K , every finite partial subalgebra B of A can be Gentzen frames DM-completions (partially) embedded in a finite D ∈ K . Embedding of subreducts Pre-frames We define W based on the preframe Embedding of subreducts using preframes ( W, · , 1) is the submonoid of A generated by B , ■ Examples of frames: FL W ′ = B , and FL ■ FMP x ⊑ b by x ≤ A b . ■ FEP Combining frames Amalgamation Theorem. Every variety of integral (alt., by commutative and Gen. amalgamation Densification knotted) RL’s axiomatized by equations over {∨ , · , 1 } has the FEP. Densification Interpolation q : B → W + is an embedding Disjunction property ■ Undecidability Modular CE W + ∈ V ■ Hilbert system for FL Strong separation W + is finite ■ Variants of frames References Corollary. These varieties are generated as quasivarieties by their finite members. The corresponding logics have the strong finite model property . Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 24 / 44

Combining frames Substructural logics Lattice representation Residuated frames Given two commutative residuated frames Residuated frames Simple equations Gentzen frames W B = ( B, ◦ , ε, ⊑ B , B ′ ) and W C = ( C, ◦ , ε, ⊑ C , C ′ ) , DM-completions Embedding of subreducts Pre-frames Embedding of subreducts using preframes Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 25 / 44

Combining frames Substructural logics Lattice representation Residuated frames Given two commutative residuated frames Residuated frames Simple equations Gentzen frames W B = ( B, ◦ , ε, ⊑ B , B ′ ) and W C = ( C, ◦ , ε, ⊑ C , C ′ ) , DM-completions Embedding of subreducts Pre-frames and also given relations Embedding of subreducts using preframes ⊑ BC ′ ⊆ B × C ′ and ⊑ CB ′ ⊆ C × B ′ , Examples of frames: FL FL FMP FEP Combining frames Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 25 / 44

Combining frames Substructural logics Lattice representation Residuated frames Given two commutative residuated frames Residuated frames Simple equations Gentzen frames W B = ( B, ◦ , ε, ⊑ B , B ′ ) and W C = ( C, ◦ , ε, ⊑ C , C ′ ) , DM-completions Embedding of subreducts Pre-frames and also given relations Embedding of subreducts using preframes ⊑ BC ′ ⊆ B × C ′ and ⊑ CB ′ ⊆ C × B ′ , Examples of frames: FL FL FMP we define the relation ⊑ from B ∪ C to B ′ ∪ C ′ as FEP Combining frames ⊑ B ∪ ⊑ C ∪ ⊑ BC ′ ∪ ⊑ CB ′ . Amalgamation Gen. amalgamation Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 25 / 44

Combining frames Substructural logics Lattice representation Residuated frames Given two commutative residuated frames Residuated frames Simple equations Gentzen frames W B = ( B, ◦ , ε, ⊑ B , B ′ ) and W C = ( C, ◦ , ε, ⊑ C , C ′ ) , DM-completions Embedding of subreducts Pre-frames and also given relations Embedding of subreducts using preframes ⊑ BC ′ ⊆ B × C ′ and ⊑ CB ′ ⊆ C × B ′ , Examples of frames: FL FL FMP we define the relation ⊑ from B ∪ C to B ′ ∪ C ′ as FEP Combining frames ⊑ B ∪ ⊑ C ∪ ⊑ BC ′ ∪ ⊑ CB ′ . We consider BC , the free commutative Amalgamation Gen. amalgamation monoid generated by B ∪ C , where ( bc ) ◦ ( b ′ c ′ ) = ( b ◦ b ′ )( c ◦ c ′ ) , Densification Densification Interpolation Disjunction property Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 25 / 44

Combining frames Substructural logics Lattice representation Residuated frames Given two commutative residuated frames Residuated frames Simple equations Gentzen frames W B = ( B, ◦ , ε, ⊑ B , B ′ ) and W C = ( C, ◦ , ε, ⊑ C , C ′ ) , DM-completions Embedding of subreducts Pre-frames and also given relations Embedding of subreducts using preframes ⊑ BC ′ ⊆ B × C ′ and ⊑ CB ′ ⊆ C × B ′ , Examples of frames: FL FL FMP we define the relation ⊑ from B ∪ C to B ′ ∪ C ′ as FEP Combining frames ⊑ B ∪ ⊑ C ∪ ⊑ BC ′ ∪ ⊑ CB ′ . We consider BC , the free commutative Amalgamation Gen. amalgamation monoid generated by B ∪ C , where ( bc ) ◦ ( b ′ c ′ ) = ( b ◦ b ′ )( c ◦ c ′ ) , and Densification we extend ⊑ from BC to B ′ ∪ C ′ : Densification Interpolation Disjunction property bc ⊑ b ′ iff c ⊑ b � b ′ Undecidability Modular CE Hilbert system for FL Strong separation Variants of frames References Nick Galatos, SYSMICS, Chapman, September 2018 Duality for residuated lattices – 25 / 44