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Semantics: Residuated Frames Peter Jipsen, Chapman University, Orange, California, USA joint work with Nikolaos Galatos, University of Denver, Colorado, USA ICLA, January, 2009 P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames

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Semantics: Residuated Frames

Peter Jipsen, Chapman University, Orange, California, USA joint work with Nikolaos Galatos, University of Denver, Colorado, USA ICLA, January, 2009

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 1 / 39

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Outline

Part I Universal Algebra Examples of residuated lattices Congruences and normal filters The lattice of subvarieties Varieties generated by positive universal classes Direct decompositions and poset products Part II Residuated Frames Decidability Poset products of residuated lattices

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 2 / 39

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Outline

Part I Universal Algebra Examples of residuated lattices Congruences and normal filters The lattice of subvarieties Varieties generated by positive universal classes Direct decompositions and poset products Part II Residuated Frames Decidability Poset products of residuated lattices

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 2 / 39

SLIDE 4

Outline

Part I Universal Algebra Examples of residuated lattices Congruences and normal filters The lattice of subvarieties Varieties generated by positive universal classes Direct decompositions and poset products Part II Residuated Frames Decidability Poset products of residuated lattices

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 2 / 39

SLIDE 5

Outline

Part I Universal Algebra Examples of residuated lattices Congruences and normal filters The lattice of subvarieties Varieties generated by positive universal classes Direct decompositions and poset products Part II Residuated Frames Decidability Poset products of residuated lattices

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 2 / 39

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Part II: Frames of residuated lattices

We now consider semantics for residuated lattices Kripke frames for modal logics are a useful tool for building counter models They also led to many interesting notions such as frame completeness, canonical frames, and correspondence results Aim: To present frames for arbitrary residuated lattices and connect them with the proof theory of substructural logic

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 3 / 39

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Part II: Frames of residuated lattices

We now consider semantics for residuated lattices Kripke frames for modal logics are a useful tool for building counter models They also led to many interesting notions such as frame completeness, canonical frames, and correspondence results Aim: To present frames for arbitrary residuated lattices and connect them with the proof theory of substructural logic

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 3 / 39

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Part II: Frames of residuated lattices

We now consider semantics for residuated lattices Kripke frames for modal logics are a useful tool for building counter models They also led to many interesting notions such as frame completeness, canonical frames, and correspondence results Aim: To present frames for arbitrary residuated lattices and connect them with the proof theory of substructural logic

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 3 / 39

SLIDE 9

Part II: Frames of residuated lattices

We now consider semantics for residuated lattices Kripke frames for modal logics are a useful tool for building counter models They also led to many interesting notions such as frame completeness, canonical frames, and correspondence results Aim: To present frames for arbitrary residuated lattices and connect them with the proof theory of substructural logic

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 3 / 39

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Part II: Frames of residuated lattices

We now consider semantics for residuated lattices Kripke frames for modal logics are a useful tool for building counter models They also led to many interesting notions such as frame completeness, canonical frames, and correspondence results Aim: To present frames for arbitrary residuated lattices and connect them with the proof theory of substructural logic

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 3 / 39

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Galois connections and closure operators

For posets P, Q, maps f : P → Q, g : Q → P are a Galois connection if y ≤ f (x) ⇐ ⇒ x ≤ g(y), for all x ∈ P, y ∈ Q A map c : P → P is a closure operator if x ≤ y implies c(x) ≤ c(y), x ≤ c(x) and c(c(x)) = c(x) Exercise: If f , g are a Galois connection then c(x) = g(f (x)) is a closure

perator on P

A lattice frame is a structure W = (W , W ′, N) where W and W ′ are sets and N is a binary relation from W to W ′ E.g. If L is a lattice, WL = (L, L, ≤) is a lattice frame Let J(L) be the set completely join irreducibles and M(L) be the set completely meet irreducibles of L. Then L+ = (J(L), M(L), ≤) is a lattice frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 4 / 39

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Galois connections and closure operators

For posets P, Q, maps f : P → Q, g : Q → P are a Galois connection if y ≤ f (x) ⇐ ⇒ x ≤ g(y), for all x ∈ P, y ∈ Q A map c : P → P is a closure operator if x ≤ y implies c(x) ≤ c(y), x ≤ c(x) and c(c(x)) = c(x) Exercise: If f , g are a Galois connection then c(x) = g(f (x)) is a closure

perator on P

A lattice frame is a structure W = (W , W ′, N) where W and W ′ are sets and N is a binary relation from W to W ′ E.g. If L is a lattice, WL = (L, L, ≤) is a lattice frame Let J(L) be the set completely join irreducibles and M(L) be the set completely meet irreducibles of L. Then L+ = (J(L), M(L), ≤) is a lattice frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 4 / 39

SLIDE 13

Galois connections and closure operators

For posets P, Q, maps f : P → Q, g : Q → P are a Galois connection if y ≤ f (x) ⇐ ⇒ x ≤ g(y), for all x ∈ P, y ∈ Q A map c : P → P is a closure operator if x ≤ y implies c(x) ≤ c(y), x ≤ c(x) and c(c(x)) = c(x) Exercise: If f , g are a Galois connection then c(x) = g(f (x)) is a closure

perator on P

A lattice frame is a structure W = (W , W ′, N) where W and W ′ are sets and N is a binary relation from W to W ′ E.g. If L is a lattice, WL = (L, L, ≤) is a lattice frame Let J(L) be the set completely join irreducibles and M(L) be the set completely meet irreducibles of L. Then L+ = (J(L), M(L), ≤) is a lattice frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 4 / 39

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Galois connections and closure operators

For posets P, Q, maps f : P → Q, g : Q → P are a Galois connection if y ≤ f (x) ⇐ ⇒ x ≤ g(y), for all x ∈ P, y ∈ Q A map c : P → P is a closure operator if x ≤ y implies c(x) ≤ c(y), x ≤ c(x) and c(c(x)) = c(x) Exercise: If f , g are a Galois connection then c(x) = g(f (x)) is a closure

perator on P

A lattice frame is a structure W = (W , W ′, N) where W and W ′ are sets and N is a binary relation from W to W ′ E.g. If L is a lattice, WL = (L, L, ≤) is a lattice frame Let J(L) be the set completely join irreducibles and M(L) be the set completely meet irreducibles of L. Then L+ = (J(L), M(L), ≤) is a lattice frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 4 / 39

SLIDE 15

Galois connections and closure operators

For posets P, Q, maps f : P → Q, g : Q → P are a Galois connection if y ≤ f (x) ⇐ ⇒ x ≤ g(y), for all x ∈ P, y ∈ Q A map c : P → P is a closure operator if x ≤ y implies c(x) ≤ c(y), x ≤ c(x) and c(c(x)) = c(x) Exercise: If f , g are a Galois connection then c(x) = g(f (x)) is a closure

perator on P

A lattice frame is a structure W = (W , W ′, N) where W and W ′ are sets and N is a binary relation from W to W ′ E.g. If L is a lattice, WL = (L, L, ≤) is a lattice frame Let J(L) be the set completely join irreducibles and M(L) be the set completely meet irreducibles of L. Then L+ = (J(L), M(L), ≤) is a lattice frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 4 / 39

SLIDE 16

Galois connections and closure operators

For posets P, Q, maps f : P → Q, g : Q → P are a Galois connection if y ≤ f (x) ⇐ ⇒ x ≤ g(y), for all x ∈ P, y ∈ Q A map c : P → P is a closure operator if x ≤ y implies c(x) ≤ c(y), x ≤ c(x) and c(c(x)) = c(x) Exercise: If f , g are a Galois connection then c(x) = g(f (x)) is a closure

perator on P

A lattice frame is a structure W = (W , W ′, N) where W and W ′ are sets and N is a binary relation from W to W ′ E.g. If L is a lattice, WL = (L, L, ≤) is a lattice frame Let J(L) be the set completely join irreducibles and M(L) be the set completely meet irreducibles of L. Then L+ = (J(L), M(L), ≤) is a lattice frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 4 / 39

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Lattice frames

For X ⊆ W and Y ⊆ W ′ we define the polarities X ⊲ = {b ∈ W ′ : x N b, for all x ∈ X} Y ⊳ = {a ∈ W : a N y, for all y ∈ Y } Exercise: The maps ⊲: P(W ) → P(W ′) and ⊳: P(W ′) → P(W ) form a Galois connection If γN(X) = X ⊲⊳ then γN : P(W ) → P(W ) is a closure operator

Lemma. If L = (L, ∧, ∨) is a lattice and γ is a closure operator on L, then

(γ[L], ∧, ∨γ) is a lattice where x ∨γ y = γ(x ∨ y)

Corollary. If W is a lattice frame then the Galois algebra

W+ = (γN[P(W )], ∩, ∪γN) is a complete lattice If L is a lattice, W+

L is the Dedekind-MacNeille completion of L and

x → {x}⊳ is an embedding

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 5 / 39

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Lattice frames

For X ⊆ W and Y ⊆ W ′ we define the polarities X ⊲ = {b ∈ W ′ : x N b, for all x ∈ X} Y ⊳ = {a ∈ W : a N y, for all y ∈ Y } Exercise: The maps ⊲: P(W ) → P(W ′) and ⊳: P(W ′) → P(W ) form a Galois connection If γN(X) = X ⊲⊳ then γN : P(W ) → P(W ) is a closure operator

Lemma. If L = (L, ∧, ∨) is a lattice and γ is a closure operator on L, then

(γ[L], ∧, ∨γ) is a lattice where x ∨γ y = γ(x ∨ y)

Corollary. If W is a lattice frame then the Galois algebra

W+ = (γN[P(W )], ∩, ∪γN) is a complete lattice If L is a lattice, W+

L is the Dedekind-MacNeille completion of L and

x → {x}⊳ is an embedding

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 5 / 39

SLIDE 19

Lattice frames

For X ⊆ W and Y ⊆ W ′ we define the polarities X ⊲ = {b ∈ W ′ : x N b, for all x ∈ X} Y ⊳ = {a ∈ W : a N y, for all y ∈ Y } Exercise: The maps ⊲: P(W ) → P(W ′) and ⊳: P(W ′) → P(W ) form a Galois connection If γN(X) = X ⊲⊳ then γN : P(W ) → P(W ) is a closure operator

Lemma. If L = (L, ∧, ∨) is a lattice and γ is a closure operator on L, then

(γ[L], ∧, ∨γ) is a lattice where x ∨γ y = γ(x ∨ y)

Corollary. If W is a lattice frame then the Galois algebra

W+ = (γN[P(W )], ∩, ∪γN) is a complete lattice If L is a lattice, W+

L is the Dedekind-MacNeille completion of L and

x → {x}⊳ is an embedding

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 5 / 39

SLIDE 20

Lattice frames

For X ⊆ W and Y ⊆ W ′ we define the polarities X ⊲ = {b ∈ W ′ : x N b, for all x ∈ X} Y ⊳ = {a ∈ W : a N y, for all y ∈ Y } Exercise: The maps ⊲: P(W ) → P(W ′) and ⊳: P(W ′) → P(W ) form a Galois connection If γN(X) = X ⊲⊳ then γN : P(W ) → P(W ) is a closure operator

Lemma. If L = (L, ∧, ∨) is a lattice and γ is a closure operator on L, then

(γ[L], ∧, ∨γ) is a lattice where x ∨γ y = γ(x ∨ y)

Corollary. If W is a lattice frame then the Galois algebra

W+ = (γN[P(W )], ∩, ∪γN) is a complete lattice If L is a lattice, W+

L is the Dedekind-MacNeille completion of L and

x → {x}⊳ is an embedding

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 5 / 39

SLIDE 21

Lattice frames

For X ⊆ W and Y ⊆ W ′ we define the polarities X ⊲ = {b ∈ W ′ : x N b, for all x ∈ X} Y ⊳ = {a ∈ W : a N y, for all y ∈ Y } Exercise: The maps ⊲: P(W ) → P(W ′) and ⊳: P(W ′) → P(W ) form a Galois connection If γN(X) = X ⊲⊳ then γN : P(W ) → P(W ) is a closure operator

Lemma. If L = (L, ∧, ∨) is a lattice and γ is a closure operator on L, then

(γ[L], ∧, ∨γ) is a lattice where x ∨γ y = γ(x ∨ y)

Corollary. If W is a lattice frame then the Galois algebra

W+ = (γN[P(W )], ∩, ∪γN) is a complete lattice If L is a lattice, W+

L is the Dedekind-MacNeille completion of L and

x → {x}⊳ is an embedding

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 5 / 39

SLIDE 22

Residuated frames

A residuated frame is a structure W = (W , W ′, N, ◦, , ) where W and W ′ are sets, N ⊆ W × W ′, ◦ ⊆ W 3, ⊆ W × W ′ × W and ⊆ W ′ × W × W such that for all x, y ∈ W , w ∈ W ′ (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y) Here x ◦ y = {z | (x, y, z) ∈ ◦} and similarly for , We also use X N y to abbreviate x N y for all x ∈ X and likewise for x N Y A ternary relation structure W = (W , ◦) is said to be associative if it satisfies (x ◦ y) ◦ z = x ◦ (y ◦ z), i.e., if it satisfies the following equivalence ∃u[(x, y, u) ∈ ◦ and (u, z, w) ∈ ◦] ⇐ ⇒ ∃v[(x, v, w) ∈ ◦ and (y, z, v) ∈ ◦] It is said to have a unit E ⊆ W if x ◦ E = {x} = E ◦ x, i.e., if ∃e ∈ E[(x, e, y) ∈ ◦] ⇐ ⇒ x = y ⇐ ⇒ ∃e ∈ E[(e, x, y) ∈ ◦]

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 6 / 39

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Residuated frames

A residuated frame is a structure W = (W , W ′, N, ◦, , ) where W and W ′ are sets, N ⊆ W × W ′, ◦ ⊆ W 3, ⊆ W × W ′ × W and ⊆ W ′ × W × W such that for all x, y ∈ W , w ∈ W ′ (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y) Here x ◦ y = {z | (x, y, z) ∈ ◦} and similarly for , We also use X N y to abbreviate x N y for all x ∈ X and likewise for x N Y A ternary relation structure W = (W , ◦) is said to be associative if it satisfies (x ◦ y) ◦ z = x ◦ (y ◦ z), i.e., if it satisfies the following equivalence ∃u[(x, y, u) ∈ ◦ and (u, z, w) ∈ ◦] ⇐ ⇒ ∃v[(x, v, w) ∈ ◦ and (y, z, v) ∈ ◦] It is said to have a unit E ⊆ W if x ◦ E = {x} = E ◦ x, i.e., if ∃e ∈ E[(x, e, y) ∈ ◦] ⇐ ⇒ x = y ⇐ ⇒ ∃e ∈ E[(e, x, y) ∈ ◦]

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 6 / 39

SLIDE 24

Residuated frames

A residuated frame is a structure W = (W , W ′, N, ◦, , ) where W and W ′ are sets, N ⊆ W × W ′, ◦ ⊆ W 3, ⊆ W × W ′ × W and ⊆ W ′ × W × W such that for all x, y ∈ W , w ∈ W ′ (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y) Here x ◦ y = {z | (x, y, z) ∈ ◦} and similarly for , We also use X N y to abbreviate x N y for all x ∈ X and likewise for x N Y A ternary relation structure W = (W , ◦) is said to be associative if it satisfies (x ◦ y) ◦ z = x ◦ (y ◦ z), i.e., if it satisfies the following equivalence ∃u[(x, y, u) ∈ ◦ and (u, z, w) ∈ ◦] ⇐ ⇒ ∃v[(x, v, w) ∈ ◦ and (y, z, v) ∈ ◦] It is said to have a unit E ⊆ W if x ◦ E = {x} = E ◦ x, i.e., if ∃e ∈ E[(x, e, y) ∈ ◦] ⇐ ⇒ x = y ⇐ ⇒ ∃e ∈ E[(e, x, y) ∈ ◦]

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 6 / 39

SLIDE 25

Residuated frames

A residuated frame is a structure W = (W , W ′, N, ◦, , ) where W and W ′ are sets, N ⊆ W × W ′, ◦ ⊆ W 3, ⊆ W × W ′ × W and ⊆ W ′ × W × W such that for all x, y ∈ W , w ∈ W ′ (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y) Here x ◦ y = {z | (x, y, z) ∈ ◦} and similarly for , We also use X N y to abbreviate x N y for all x ∈ X and likewise for x N Y A ternary relation structure W = (W , ◦) is said to be associative if it satisfies (x ◦ y) ◦ z = x ◦ (y ◦ z), i.e., if it satisfies the following equivalence ∃u[(x, y, u) ∈ ◦ and (u, z, w) ∈ ◦] ⇐ ⇒ ∃v[(x, v, w) ∈ ◦ and (y, z, v) ∈ ◦] It is said to have a unit E ⊆ W if x ◦ E = {x} = E ◦ x, i.e., if ∃e ∈ E[(x, e, y) ∈ ◦] ⇐ ⇒ x = y ⇐ ⇒ ∃e ∈ E[(e, x, y) ∈ ◦]

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 6 / 39

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Nuclei

A nucleus γ on a residuated lattice L is a closure operator on L such that γ(x)γ(y) ≤ γ(xy) (or γ(γ(x)γ(y)) = γ(xy)).

Theorem. Given a RL L = (L, ∧, ∨, ·, \, /, 1) and a nucleus on L, the

algebra Lγ = (Lγ, ∧, ∨γ, ·γ, \, /, γ(1)), is a residuated lattice, where x ·γ y = γ(x · y), x ∨γ y = γ(x ∨ y).

Theorem. For a frame W, γN is a nucleus on (P(W ), ∩, ∪, ◦, \, /, {1})
Corollary. If W is a residuated frame then the Galois algebra

W+ = (P(W ), ∩, ∪, ◦, \, /, 1)γN is a complete residuated lattice. Moreover, for WL, x → {x}⊳ is an embedding. If L is a RL, WL = (L, L, ≤, ·, \, /) is a residuated frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39

SLIDE 27

Nuclei

A nucleus γ on a residuated lattice L is a closure operator on L such that γ(x)γ(y) ≤ γ(xy) (or γ(γ(x)γ(y)) = γ(xy)).

Theorem. Given a RL L = (L, ∧, ∨, ·, \, /, 1) and a nucleus on L, the

algebra Lγ = (Lγ, ∧, ∨γ, ·γ, \, /, γ(1)), is a residuated lattice, where x ·γ y = γ(x · y), x ∨γ y = γ(x ∨ y).

Theorem. For a frame W, γN is a nucleus on (P(W ), ∩, ∪, ◦, \, /, {1})
Corollary. If W is a residuated frame then the Galois algebra

W+ = (P(W ), ∩, ∪, ◦, \, /, 1)γN is a complete residuated lattice. Moreover, for WL, x → {x}⊳ is an embedding. If L is a RL, WL = (L, L, ≤, ·, \, /) is a residuated frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39

SLIDE 28

Nuclei

A nucleus γ on a residuated lattice L is a closure operator on L such that γ(x)γ(y) ≤ γ(xy) (or γ(γ(x)γ(y)) = γ(xy)).

Theorem. Given a RL L = (L, ∧, ∨, ·, \, /, 1) and a nucleus on L, the

algebra Lγ = (Lγ, ∧, ∨γ, ·γ, \, /, γ(1)), is a residuated lattice, where x ·γ y = γ(x · y), x ∨γ y = γ(x ∨ y).

Theorem. For a frame W, γN is a nucleus on (P(W ), ∩, ∪, ◦, \, /, {1})
Corollary. If W is a residuated frame then the Galois algebra

W+ = (P(W ), ∩, ∪, ◦, \, /, 1)γN is a complete residuated lattice. Moreover, for WL, x → {x}⊳ is an embedding. If L is a RL, WL = (L, L, ≤, ·, \, /) is a residuated frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39

SLIDE 29

Nuclei

A nucleus γ on a residuated lattice L is a closure operator on L such that γ(x)γ(y) ≤ γ(xy) (or γ(γ(x)γ(y)) = γ(xy)).

Theorem. Given a RL L = (L, ∧, ∨, ·, \, /, 1) and a nucleus on L, the

algebra Lγ = (Lγ, ∧, ∨γ, ·γ, \, /, γ(1)), is a residuated lattice, where x ·γ y = γ(x · y), x ∨γ y = γ(x ∨ y).

Theorem. For a frame W, γN is a nucleus on (P(W ), ∩, ∪, ◦, \, /, {1})
Corollary. If W is a residuated frame then the Galois algebra

W+ = (P(W ), ∩, ∪, ◦, \, /, 1)γN is a complete residuated lattice. Moreover, for WL, x → {x}⊳ is an embedding. If L is a RL, WL = (L, L, ≤, ·, \, /) is a residuated frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39

SLIDE 30

Nuclei

A nucleus γ on a residuated lattice L is a closure operator on L such that γ(x)γ(y) ≤ γ(xy) (or γ(γ(x)γ(y)) = γ(xy)).

Theorem. Given a RL L = (L, ∧, ∨, ·, \, /, 1) and a nucleus on L, the

algebra Lγ = (Lγ, ∧, ∨γ, ·γ, \, /, γ(1)), is a residuated lattice, where x ·γ y = γ(x · y), x ∨γ y = γ(x ∨ y).

Theorem. For a frame W, γN is a nucleus on (P(W ), ∩, ∪, ◦, \, /, {1})
Corollary. If W is a residuated frame then the Galois algebra

W+ = (P(W ), ∩, ∪, ◦, \, /, 1)γN is a complete residuated lattice. Moreover, for WL, x → {x}⊳ is an embedding. If L is a RL, WL = (L, L, ≤, ·, \, /) is a residuated frame.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39

SLIDE 31

Frames of complete perfect lattices

A lattice L is perfect if every element is a join of elements of J(L) and a meet of elements of M(L) E.g. a Boolean algebra is perfect iff it is atomic For a perfect residuated lattice A, let A+ = (J(A), M(A), ≤, ◦, , , E) where x ◦ y = {z ∈ J(A) | z ≤ xy} and E = {z ∈ J(A) | z ≤ 1}

Theorem

A+ is a residuated frame and if A is complete then (A+)+ ∼ = A In particular, any finite lattice is complete and perfect So for finite residuated lattices, residuated frames give a compact representation analogous to atom structures for relation algebras

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 8 / 39

SLIDE 32

Frames of complete perfect lattices

A lattice L is perfect if every element is a join of elements of J(L) and a meet of elements of M(L) E.g. a Boolean algebra is perfect iff it is atomic For a perfect residuated lattice A, let A+ = (J(A), M(A), ≤, ◦, , , E) where x ◦ y = {z ∈ J(A) | z ≤ xy} and E = {z ∈ J(A) | z ≤ 1}

Theorem

A+ is a residuated frame and if A is complete then (A+)+ ∼ = A In particular, any finite lattice is complete and perfect So for finite residuated lattices, residuated frames give a compact representation analogous to atom structures for relation algebras

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 8 / 39

SLIDE 33

Frames of complete perfect lattices

A lattice L is perfect if every element is a join of elements of J(L) and a meet of elements of M(L) E.g. a Boolean algebra is perfect iff it is atomic For a perfect residuated lattice A, let A+ = (J(A), M(A), ≤, ◦, , , E) where x ◦ y = {z ∈ J(A) | z ≤ xy} and E = {z ∈ J(A) | z ≤ 1}

Theorem

A+ is a residuated frame and if A is complete then (A+)+ ∼ = A In particular, any finite lattice is complete and perfect So for finite residuated lattices, residuated frames give a compact representation analogous to atom structures for relation algebras

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 8 / 39

SLIDE 34

Frames of complete perfect lattices

A lattice L is perfect if every element is a join of elements of J(L) and a meet of elements of M(L) E.g. a Boolean algebra is perfect iff it is atomic For a perfect residuated lattice A, let A+ = (J(A), M(A), ≤, ◦, , , E) where x ◦ y = {z ∈ J(A) | z ≤ xy} and E = {z ∈ J(A) | z ≤ 1}

Theorem

A+ is a residuated frame and if A is complete then (A+)+ ∼ = A In particular, any finite lattice is complete and perfect So for finite residuated lattices, residuated frames give a compact representation analogous to atom structures for relation algebras

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 8 / 39

SLIDE 35

Frames of complete perfect lattices

A lattice L is perfect if every element is a join of elements of J(L) and a meet of elements of M(L) E.g. a Boolean algebra is perfect iff it is atomic For a perfect residuated lattice A, let A+ = (J(A), M(A), ≤, ◦, , , E) where x ◦ y = {z ∈ J(A) | z ≤ xy} and E = {z ∈ J(A) | z ≤ 1}

Theorem

A+ is a residuated frame and if A is complete then (A+)+ ∼ = A In particular, any finite lattice is complete and perfect So for finite residuated lattices, residuated frames give a compact representation analogous to atom structures for relation algebras

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 8 / 39

SLIDE 36

Frames of complete perfect lattices

A lattice L is perfect if every element is a join of elements of J(L) and a meet of elements of M(L) E.g. a Boolean algebra is perfect iff it is atomic For a perfect residuated lattice A, let A+ = (J(A), M(A), ≤, ◦, , , E) where x ◦ y = {z ∈ J(A) | z ≤ xy} and E = {z ∈ J(A) | z ≤ 1}

Theorem

A+ is a residuated frame and if A is complete then (A+)+ ∼ = A In particular, any finite lattice is complete and perfect So for finite residuated lattices, residuated frames give a compact representation analogous to atom structures for relation algebras

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 8 / 39

SLIDE 37

All (dually-)nonisomorphic lattices with ≤ 7 elements

K1,1 Kn,k = selfdual lattice of size n Ln,k = nonselfdual lattices of size n K2,1 K3,1 K4,1 K4,2 L5,1 K5,1 K5,2 K5,3 L6,1 L6,2 L6,3 L6,4 K6,1 K6,2 K6,3 K6,4 K6,5 K6,6 K6,7 L7,1 L7,2 L7,3 L7,4 L7,5 L7,6 L7,7 L7,8 L7,9 L7,10 L7,11 L7,12 L7,13 L7,14 L7,15 L7,16 L7,17 L7,18 L7,19 L7,20 K7,1 K7,2 K7,3 K7,4 K7,5 K7,6 K7,7 K7,8 K7,9 K7,10 K7,11 K7,12K7,13

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 9 / 39

SLIDE 38

FL

x ⇒ a y◦a◦z ⇒ c y◦x◦z ⇒ c (cut) a ⇒ a (Id) y◦a◦z ⇒ c y◦a ∧ b◦z ⇒ c (∧Lℓ) y◦b◦z ⇒ c y◦a ∧ b◦z ⇒ c (∧Lr) x ⇒ a x ⇒ b x ⇒ a ∧ b (∧R) y◦a◦z ⇒ c y◦b◦z ⇒ c y◦a ∨ b◦z ⇒ c (∨L) x ⇒ a x ⇒ a ∨ b (∨Rℓ) x ⇒ b x ⇒ a ∨ b (∨Rr) x ⇒ a y◦b◦z ⇒ c y◦x ◦ (a\b)◦z ⇒ c (\L) a ◦ x ⇒ b x ⇒ a\b (\R) x ⇒ a y◦b◦z ⇒ c y◦(b/a) ◦ x◦z ⇒ c (/L) x ◦ a ⇒ b x ⇒ b/a (/R) y◦a ◦ b◦z ⇒ c y◦a · b◦z ⇒ c (·L) x ⇒ a y ⇒ b x ◦ y ⇒ a · b (·R) y ◦ z ⇒ a y◦1◦z ⇒ a (1L) ε ⇒ 1 (1R) where a, b, c ∈ Fm, x, y, z ∈ Fm∗.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 10 / 39

SLIDE 39

FL with context notation

x ⇒ a u[a] ⇒ c u[x] ⇒ c (cut) a ⇒ a (Id) u[a] ⇒ c u[a ∧ b] ⇒ c (∧Lℓ) u[b] ⇒ c u[a ∧ b] ⇒ c (∧Lr) x ⇒ a x ⇒ b x ⇒ a ∧ b (∧R) u[a] ⇒ c u[b] ⇒ c u[a ∨ b] ⇒ c (∨L) x ⇒ a x ⇒ a ∨ b (∨Rℓ) x ⇒ b x ⇒ a ∨ b (∨Rr) x ⇒ a u[b] ⇒ c u[x ◦ (a\b)] ⇒ c (\L) a ◦ x ⇒ b x ⇒ a\b (\R) x ⇒ a u[b] ⇒ c u[(b/a) ◦ x] ⇒ c (/L) x ◦ a ⇒ b x ⇒ b/a (/R) u[a ◦ b] ⇒ c u[a · b] ⇒ c (·L) x ⇒ a y ⇒ b x ◦ y ⇒ a · b (·R) |u| ⇒ a u[1] ⇒ a (1L) ε ⇒ 1 (1R)

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 11 / 39

SLIDE 40

Basic substructural logics

If the sequent s is provable in FL from the set of sequents S, we write S ⊢FL s. u[x ◦ y] ⇒ c u[y ◦ x] ⇒ c (e) (exchange) xy ≤ yx u[x ◦ x] ⇒ c u[x] ⇒ c (c) (contraction) x ≤ x2 |u| ⇒ c u[x] ⇒ c (i) (integrality) x ≤ 1 We write FLec for FL + (e) + (c).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39

SLIDE 41

Basic substructural logics

If the sequent s is provable in FL from the set of sequents S, we write S ⊢FL s. u[x ◦ y] ⇒ c u[y ◦ x] ⇒ c (e) (exchange) xy ≤ yx u[x ◦ x] ⇒ c u[x] ⇒ c (c) (contraction) x ≤ x2 |u| ⇒ c u[x] ⇒ c (i) (integrality) x ≤ 1 We write FLec for FL + (e) + (c).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39

SLIDE 42

Basic substructural logics

If the sequent s is provable in FL from the set of sequents S, we write S ⊢FL s. u[x ◦ y] ⇒ c u[y ◦ x] ⇒ c (e) (exchange) xy ≤ yx u[x ◦ x] ⇒ c u[x] ⇒ c (c) (contraction) x ≤ x2 |u| ⇒ c u[x] ⇒ c (i) (integrality) x ≤ 1 We write FLec for FL + (e) + (c).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39

SLIDE 43

Basic substructural logics

If the sequent s is provable in FL from the set of sequents S, we write S ⊢FL s. u[x ◦ y] ⇒ c u[y ◦ x] ⇒ c (e) (exchange) xy ≤ yx u[x ◦ x] ⇒ c u[x] ⇒ c (c) (contraction) x ≤ x2 |u| ⇒ c u[x] ⇒ c (i) (integrality) x ≤ 1 We write FLec for FL + (e) + (c).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39

SLIDE 44

Basic substructural logics

If the sequent s is provable in FL from the set of sequents S, we write S ⊢FL s. u[x ◦ y] ⇒ c u[y ◦ x] ⇒ c (e) (exchange) xy ≤ yx u[x ◦ x] ⇒ c u[x] ⇒ c (c) (contraction) x ≤ x2 |u| ⇒ c u[x] ⇒ c (i) (integrality) x ≤ 1 We write FLec for FL + (e) + (c).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39

SLIDE 45

Examples of frames (FL)

Consider the Gentzen system FL (full Lambek calculus). We define the frame WFL, where (W , ◦, ε) to be the free monoid over the set Fm of all formulas W ′ = SW × Fm, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of W , and x N (u, a) iff ⊢FL u[x] ⇒ a. For (u, a) x = {(u[ ◦ x], a)} and x (u, a) = {(u[x ◦ ], a)}, we have x ◦ yN(u, a) iff ⊢FL u[x ◦ y] ⇒ a iff ⊢FL u[x◦y] ⇒ a iff xN(u[ ◦ y], a) iff yN(u[x ◦ ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 13 / 39

SLIDE 46

Examples of frames (FL)

Consider the Gentzen system FL (full Lambek calculus). We define the frame WFL, where (W , ◦, ε) to be the free monoid over the set Fm of all formulas W ′ = SW × Fm, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of W , and x N (u, a) iff ⊢FL u[x] ⇒ a. For (u, a) x = {(u[ ◦ x], a)} and x (u, a) = {(u[x ◦ ], a)}, we have x ◦ yN(u, a) iff ⊢FL u[x ◦ y] ⇒ a iff ⊢FL u[x◦y] ⇒ a iff xN(u[ ◦ y], a) iff yN(u[x ◦ ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 13 / 39

SLIDE 47

Examples of frames (FL)

Consider the Gentzen system FL (full Lambek calculus). We define the frame WFL, where (W , ◦, ε) to be the free monoid over the set Fm of all formulas W ′ = SW × Fm, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of W , and x N (u, a) iff ⊢FL u[x] ⇒ a. For (u, a) x = {(u[ ◦ x], a)} and x (u, a) = {(u[x ◦ ], a)}, we have x ◦ yN(u, a) iff ⊢FL u[x ◦ y] ⇒ a iff ⊢FL u[x◦y] ⇒ a iff xN(u[ ◦ y], a) iff yN(u[x ◦ ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 13 / 39

SLIDE 48

Examples of frames (FL)

Consider the Gentzen system FL (full Lambek calculus). We define the frame WFL, where (W , ◦, ε) to be the free monoid over the set Fm of all formulas W ′ = SW × Fm, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of W , and x N (u, a) iff ⊢FL u[x] ⇒ a. For (u, a) x = {(u[ ◦ x], a)} and x (u, a) = {(u[x ◦ ], a)}, we have x ◦ yN(u, a) iff ⊢FL u[x ◦ y] ⇒ a iff ⊢FL u[x◦y] ⇒ a iff xN(u[ ◦ y], a) iff yN(u[x ◦ ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 13 / 39

SLIDE 49

Examples of frames (FL)

Consider the Gentzen system FL (full Lambek calculus). We define the frame WFL, where (W , ◦, ε) to be the free monoid over the set Fm of all formulas W ′ = SW × Fm, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of W , and x N (u, a) iff ⊢FL u[x] ⇒ a. For (u, a) x = {(u[ ◦ x], a)} and x (u, a) = {(u[x ◦ ], a)}, we have x ◦ yN(u, a) iff ⊢FL u[x ◦ y] ⇒ a iff ⊢FL u[x◦y] ⇒ a iff xN(u[ ◦ y], a) iff yN(u[x ◦ ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 13 / 39

SLIDE 50

Examples of frames (FEP)

Let A be a residuated lattice and B a partial subalgebra of A. We define the frame WA,B, where (W , ·, 1) to be the submonoid of A generated by B, W ′ = SB × B, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of (W , ·, 1), and x N (u, b) by u[x] ≤A b. For (u, a) x = {(u[ · x], a)} and x (u, a) = {(u[x · ], a)}, we have x · yN(u, a) iff u[x · y] ≤ a iff xN(u[ · y], a) iff yN(u[x · ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 14 / 39

SLIDE 51

Examples of frames (FEP)

Let A be a residuated lattice and B a partial subalgebra of A. We define the frame WA,B, where (W , ·, 1) to be the submonoid of A generated by B, W ′ = SB × B, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of (W , ·, 1), and x N (u, b) by u[x] ≤A b. For (u, a) x = {(u[ · x], a)} and x (u, a) = {(u[x · ], a)}, we have x · yN(u, a) iff u[x · y] ≤ a iff xN(u[ · y], a) iff yN(u[x · ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 14 / 39

SLIDE 52

Examples of frames (FEP)

Let A be a residuated lattice and B a partial subalgebra of A. We define the frame WA,B, where (W , ·, 1) to be the submonoid of A generated by B, W ′ = SB × B, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of (W , ·, 1), and x N (u, b) by u[x] ≤A b. For (u, a) x = {(u[ · x], a)} and x (u, a) = {(u[x · ], a)}, we have x · yN(u, a) iff u[x · y] ≤ a iff xN(u[ · y], a) iff yN(u[x · ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 14 / 39

SLIDE 53

Examples of frames (FEP)

Let A be a residuated lattice and B a partial subalgebra of A. We define the frame WA,B, where (W , ·, 1) to be the submonoid of A generated by B, W ′ = SB × B, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of (W , ·, 1), and x N (u, b) by u[x] ≤A b. For (u, a) x = {(u[ · x], a)} and x (u, a) = {(u[x · ], a)}, we have x · yN(u, a) iff u[x · y] ≤ a iff xN(u[ · y], a) iff yN(u[x · ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 14 / 39

SLIDE 54

Examples of frames (FEP)

Let A be a residuated lattice and B a partial subalgebra of A. We define the frame WA,B, where (W , ·, 1) to be the submonoid of A generated by B, W ′ = SB × B, where SW is the set of all unary linear polynomials u[x] = y◦x◦z of (W , ·, 1), and x N (u, b) by u[x] ≤A b. For (u, a) x = {(u[ · x], a)} and x (u, a) = {(u[x · ], a)}, we have x · yN(u, a) iff u[x · y] ≤ a iff xN(u[ · y], a) iff yN(u[x · ], a).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 14 / 39

SLIDE 55

GN

xNa aNz xNz (CUT) aNa (Id) xNa bNz x ◦ (a\b)Nz (\L) a ◦ xNb xNa\b (\R) xNa bNz (b/a) ◦ xNz (/L) x ◦ aNb xNb/a (/R) a ◦ bNz a · bNz (·L) xNa yNb x ◦ yNa · b (·R) aNz a ∧ bNz (∧Lℓ) bNz a ∧ bNz (∧Lr) xNa xNb xNa ∧ b (∧R) aNz bNz a ∨ bNz (∨L) xNa xNa ∨ b (∨Rℓ) xNb xNa ∨ b (∨Rr) εNz 1Nz (1L) εN1 (1R)

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 15 / 39

SLIDE 56

Gentzen frames

The following properties hold for WL, WFL and WA,B:

1 W is a residuated frame 2 B is a (partial) algebra of the same type, (B = L, Fm, B) 3 B generates (W , ◦, ε) (as a monoid) 4 W ′ contains a copy of B (b ↔ (id, b)) 5 N satisfies GN, for all a, b ∈ B, x, y ∈ W , z ∈ W ′.

We call such pairs (W, B) Gentzen frames. A cut-free Gentzen frame is not assumed to satisfy the (CUT)-rule.

Theorem. Given a Gentzen frame (W, B), the map

{}⊳ : B → W+, b → {b}⊳ is a (partial) homomorphism. (Namely, if a, b ∈ B and a • b ∈ B (• is a connective) then {a •B b}⊳ = {a}⊳ •W+ {b}⊳).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 16 / 39

SLIDE 57

Gentzen frames

The following properties hold for WL, WFL and WA,B:

We call such pairs (W, B) Gentzen frames. A cut-free Gentzen frame is not assumed to satisfy the (CUT)-rule.

Theorem. Given a Gentzen frame (W, B), the map

{}⊳ : B → W+, b → {b}⊳ is a (partial) homomorphism. (Namely, if a, b ∈ B and a • b ∈ B (• is a connective) then {a •B b}⊳ = {a}⊳ •W+ {b}⊳).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 16 / 39

SLIDE 58

Gentzen frames

The following properties hold for WL, WFL and WA,B:

We call such pairs (W, B) Gentzen frames. A cut-free Gentzen frame is not assumed to satisfy the (CUT)-rule.

Theorem. Given a Gentzen frame (W, B), the map

{}⊳ : B → W+, b → {b}⊳ is a (partial) homomorphism. (Namely, if a, b ∈ B and a • b ∈ B (• is a connective) then {a •B b}⊳ = {a}⊳ •W+ {b}⊳).

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 16 / 39

SLIDE 59

Proof

Key Lemma. Let (W, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ W+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then

1 a •B b ∈ X •W+ Y ⊆ {a •B b}⊳ (1B ∈ 1W+ ⊆ {1B}⊳ ) 2 In particular, a •B b ∈ {a}⊳ •W+ {b}⊳ ⊆ {a •B b}⊳. 3 Furthermore, because of (CUT), we have equality.

Proof Let • = ∨. If x ∈ X, then x ∈ {a}⊳; so xNa and xNa ∨ b, by (∨Rℓ); hence x ∈ {a ∨ b}⊳ and X ⊆ {a ∨ b}⊳. Likewise Y ⊆ {a ∨ b}⊳, so X ∪ Y ⊆ {a ∨ b}⊳ and X ∨ Y = γ(X ∪ Y ) ⊆ {a ∨ b}⊳. On the other hand, let X ∨ Y ⊆ {z}⊳, for some z ∈ W . Then, a ∈ X ⊆ X ∨ Y ⊆ {z}⊳, so aNz. Similarly, bNz, so a ∨ bNz by (∨L), hence a ∨ b ∈ {z}⊳. Thus, a ∨ b ∈ X ∨ Y . We used that every closed set is an intersection of basic closed sets {z}⊳, for z ∈ W .

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 17 / 39

SLIDE 60

Proof

Key Lemma. Let (W, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ W+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then

1 a •B b ∈ X •W+ Y ⊆ {a •B b}⊳ (1B ∈ 1W+ ⊆ {1B}⊳ ) 2 In particular, a •B b ∈ {a}⊳ •W+ {b}⊳ ⊆ {a •B b}⊳. 3 Furthermore, because of (CUT), we have equality.

Proof Let • = ∨. If x ∈ X, then x ∈ {a}⊳; so xNa and xNa ∨ b, by (∨Rℓ); hence x ∈ {a ∨ b}⊳ and X ⊆ {a ∨ b}⊳. Likewise Y ⊆ {a ∨ b}⊳, so X ∪ Y ⊆ {a ∨ b}⊳ and X ∨ Y = γ(X ∪ Y ) ⊆ {a ∨ b}⊳. On the other hand, let X ∨ Y ⊆ {z}⊳, for some z ∈ W . Then, a ∈ X ⊆ X ∨ Y ⊆ {z}⊳, so aNz. Similarly, bNz, so a ∨ bNz by (∨L), hence a ∨ b ∈ {z}⊳. Thus, a ∨ b ∈ X ∨ Y . We used that every closed set is an intersection of basic closed sets {z}⊳, for z ∈ W .

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 17 / 39

SLIDE 61

Proof

Key Lemma. Let (W, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ W+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then

1 a •B b ∈ X •W+ Y ⊆ {a •B b}⊳ (1B ∈ 1W+ ⊆ {1B}⊳ ) 2 In particular, a •B b ∈ {a}⊳ •W+ {b}⊳ ⊆ {a •B b}⊳. 3 Furthermore, because of (CUT), we have equality.

Proof Let • = ∨. If x ∈ X, then x ∈ {a}⊳; so xNa and xNa ∨ b, by (∨Rℓ); hence x ∈ {a ∨ b}⊳ and X ⊆ {a ∨ b}⊳. Likewise Y ⊆ {a ∨ b}⊳, so X ∪ Y ⊆ {a ∨ b}⊳ and X ∨ Y = γ(X ∪ Y ) ⊆ {a ∨ b}⊳. On the other hand, let X ∨ Y ⊆ {z}⊳, for some z ∈ W . Then, a ∈ X ⊆ X ∨ Y ⊆ {z}⊳, so aNz. Similarly, bNz, so a ∨ bNz by (∨L), hence a ∨ b ∈ {z}⊳. Thus, a ∨ b ∈ X ∨ Y . We used that every closed set is an intersection of basic closed sets {z}⊳, for z ∈ W .

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 17 / 39

SLIDE 62

DM-completion

For a residuated lattice L, we associated the Gentzen frame (WL, L). The underlying poset of W+

L is the Dedekind-MacNeille completion of the

underlying poset reduct of L.

Theorem. The map x → x⊳ is an embedding of L into W+

L .

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 18 / 39

SLIDE 63

DM-completion

For a residuated lattice L, we associated the Gentzen frame (WL, L). The underlying poset of W+

L is the Dedekind-MacNeille completion of the

underlying poset reduct of L.

Theorem. The map x → x⊳ is an embedding of L into W+

L .

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 18 / 39

SLIDE 64

Completeness - Cut elimination

For every homomorphism f : Fm → B, let ¯ f : FmL → W+ be the homomorphism that extends ¯ f (p) = {f (p)}⊳ for any variable p

Corollary. If (W, B) is a cf Gentzen frame then for every homomorphism

f : Fm → B, we have f (a) ∈ ¯ f (a) ⊆ {f (a)}⊳. With CUT ¯ f (a)={f (a)}⊳ We define WFL | = x ⇒ c if f (x) N f (c) for all f : Fm → Fm

Theorem. If W+

FL |

= x· ≤ c, then WFL | = x ⇒ c. Idea: For f : Fm → B, f (x) ∈ ¯ f (x) ⊆ ¯ f (c) ⊆ {f (c)}⊳, so f (x) N f (c).

Corollary. FL is complete with respect to W+

FL.

Corollary. The algebra W+

FL generates RL.

The frame WFLf corresponds to cut-free FL. Corollary (CE). FL and FLf prove the same sequents.

Corollary. FL and the equational theory of RL are decidable.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 19 / 39

SLIDE 65

Completeness - Cut elimination

For every homomorphism f : Fm → B, let ¯ f : FmL → W+ be the homomorphism that extends ¯ f (p) = {f (p)}⊳ for any variable p

Corollary. If (W, B) is a cf Gentzen frame then for every homomorphism

f : Fm → B, we have f (a) ∈ ¯ f (a) ⊆ {f (a)}⊳. With CUT ¯ f (a)={f (a)}⊳ We define WFL | = x ⇒ c if f (x) N f (c) for all f : Fm → Fm

Theorem. If W+

FL |

= x· ≤ c, then WFL | = x ⇒ c. Idea: For f : Fm → B, f (x) ∈ ¯ f (x) ⊆ ¯ f (c) ⊆ {f (c)}⊳, so f (x) N f (c).

Corollary. FL is complete with respect to W+

FL.

Corollary. The algebra W+

FL generates RL.

The frame WFLf corresponds to cut-free FL. Corollary (CE). FL and FLf prove the same sequents.

Corollary. FL and the equational theory of RL are decidable.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 19 / 39

SLIDE 66

Completeness - Cut elimination

For every homomorphism f : Fm → B, let ¯ f : FmL → W+ be the homomorphism that extends ¯ f (p) = {f (p)}⊳ for any variable p

Corollary. If (W, B) is a cf Gentzen frame then for every homomorphism

f : Fm → B, we have f (a) ∈ ¯ f (a) ⊆ {f (a)}⊳. With CUT ¯ f (a)={f (a)}⊳ We define WFL | = x ⇒ c if f (x) N f (c) for all f : Fm → Fm

Theorem. If W+

FL |

= x· ≤ c, then WFL | = x ⇒ c. Idea: For f : Fm → B, f (x) ∈ ¯ f (x) ⊆ ¯ f (c) ⊆ {f (c)}⊳, so f (x) N f (c).

Corollary. FL is complete with respect to W+

FL.

Corollary. The algebra W+

FL generates RL.

The frame WFLf corresponds to cut-free FL. Corollary (CE). FL and FLf prove the same sequents.

Corollary. FL and the equational theory of RL are decidable.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 19 / 39

SLIDE 67

Completeness - Cut elimination

For every homomorphism f : Fm → B, let ¯ f : FmL → W+ be the homomorphism that extends ¯ f (p) = {f (p)}⊳ for any variable p

Corollary. If (W, B) is a cf Gentzen frame then for every homomorphism

f : Fm → B, we have f (a) ∈ ¯ f (a) ⊆ {f (a)}⊳. With CUT ¯ f (a)={f (a)}⊳ We define WFL | = x ⇒ c if f (x) N f (c) for all f : Fm → Fm

Theorem. If W+

FL |

= x· ≤ c, then WFL | = x ⇒ c. Idea: For f : Fm → B, f (x) ∈ ¯ f (x) ⊆ ¯ f (c) ⊆ {f (c)}⊳, so f (x) N f (c).

Corollary. FL is complete with respect to W+

FL.

Corollary. The algebra W+

FL generates RL.

The frame WFLf corresponds to cut-free FL. Corollary (CE). FL and FLf prove the same sequents.

Corollary. FL and the equational theory of RL are decidable.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 19 / 39

SLIDE 68

Completeness - Cut elimination

For every homomorphism f : Fm → B, let ¯ f : FmL → W+ be the homomorphism that extends ¯ f (p) = {f (p)}⊳ for any variable p

Corollary. If (W, B) is a cf Gentzen frame then for every homomorphism

f : Fm → B, we have f (a) ∈ ¯ f (a) ⊆ {f (a)}⊳. With CUT ¯ f (a)={f (a)}⊳ We define WFL | = x ⇒ c if f (x) N f (c) for all f : Fm → Fm

Theorem. If W+

FL |

= x· ≤ c, then WFL | = x ⇒ c. Idea: For f : Fm → B, f (x) ∈ ¯ f (x) ⊆ ¯ f (c) ⊆ {f (c)}⊳, so f (x) N f (c).

Corollary. FL is complete with respect to W+

FL.

Corollary. The algebra W+

FL generates RL.

The frame WFLf corresponds to cut-free FL. Corollary (CE). FL and FLf prove the same sequents.

Corollary. FL and the equational theory of RL are decidable.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 19 / 39

SLIDE 69

Finite model property

For WFL, given (x, z) ∈ W × W ′ (if z = (u, c), then u(x) ⇒ c is a sequent), we define (x, z)↑ as the smallest subset of W × W ′ that contains (x, z) and is closed upwards with respect to the rules of FLf. Note that (x, z)↑ is finite. The new frame W′ associated with N′ = N ∪ ((y, v)↑)c is residuated and Gentzen. (N′)c is finite, so has finite domain Dom((N′)c) and codomain Cod((N′)c) For every z ∈ Cod((N′)c), {z}⊳ = W . So, {{z}⊳ : z ∈ W } is finite and a basis for γN′. So W′+ is finite. Moreover, if u(x) ⇒ c is not provable in FL, then it is not valid in W′+.

Corollary. The system FL has the finite model property.
Corollary. The variety RL is generated by its finite members.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 20 / 39

SLIDE 70

Finite model property

For WFL, given (x, z) ∈ W × W ′ (if z = (u, c), then u(x) ⇒ c is a sequent), we define (x, z)↑ as the smallest subset of W × W ′ that contains (x, z) and is closed upwards with respect to the rules of FLf. Note that (x, z)↑ is finite. The new frame W′ associated with N′ = N ∪ ((y, v)↑)c is residuated and Gentzen. (N′)c is finite, so has finite domain Dom((N′)c) and codomain Cod((N′)c) For every z ∈ Cod((N′)c), {z}⊳ = W . So, {{z}⊳ : z ∈ W } is finite and a basis for γN′. So W′+ is finite. Moreover, if u(x) ⇒ c is not provable in FL, then it is not valid in W′+.

Corollary. The system FL has the finite model property.
Corollary. The variety RL is generated by its finite members.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 20 / 39

SLIDE 71

Finite model property

For WFL, given (x, z) ∈ W × W ′ (if z = (u, c), then u(x) ⇒ c is a sequent), we define (x, z)↑ as the smallest subset of W × W ′ that contains (x, z) and is closed upwards with respect to the rules of FLf. Note that (x, z)↑ is finite. The new frame W′ associated with N′ = N ∪ ((y, v)↑)c is residuated and Gentzen. (N′)c is finite, so has finite domain Dom((N′)c) and codomain Cod((N′)c) For every z ∈ Cod((N′)c), {z}⊳ = W . So, {{z}⊳ : z ∈ W } is finite and a basis for γN′. So W′+ is finite. Moreover, if u(x) ⇒ c is not provable in FL, then it is not valid in W′+.

Corollary. The system FL has the finite model property.
Corollary. The variety RL is generated by its finite members.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 20 / 39

SLIDE 72

Finite model property

For WFL, given (x, z) ∈ W × W ′ (if z = (u, c), then u(x) ⇒ c is a sequent), we define (x, z)↑ as the smallest subset of W × W ′ that contains (x, z) and is closed upwards with respect to the rules of FLf. Note that (x, z)↑ is finite. The new frame W′ associated with N′ = N ∪ ((y, v)↑)c is residuated and Gentzen. (N′)c is finite, so has finite domain Dom((N′)c) and codomain Cod((N′)c) For every z ∈ Cod((N′)c), {z}⊳ = W . So, {{z}⊳ : z ∈ W } is finite and a basis for γN′. So W′+ is finite. Moreover, if u(x) ⇒ c is not provable in FL, then it is not valid in W′+.

Corollary. The system FL has the finite model property.
Corollary. The variety RL is generated by its finite members.

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 20 / 39

SLIDE 73

FEP

A class of algebras K has the finite embeddability property (FEP) if for every A ∈ K, every finite partial subalgebra B of A can be (partially) embedded in a finite D ∈ K. The corresponding logic has the strong finite model property: if Φ ⊢ ψ, for finite Φ, then there is a finite counter-model, namely there is D ∈ K and a homomorphism f : Fm → D, such that f (φ) = 1, for all φ ∈ Φ, but f (ψ) = 1. For logics with finitely many axioms and rules, this implies that the deducibility relation is decidable For a finitely axiomatized variety, FEP implies the decidability of the universal theory

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 21 / 39

SLIDE 74

FEP

A class of algebras K has the finite embeddability property (FEP) if for every A ∈ K, every finite partial subalgebra B of A can be (partially) embedded in a finite D ∈ K. The corresponding logic has the strong finite model property: if Φ ⊢ ψ, for finite Φ, then there is a finite counter-model, namely there is D ∈ K and a homomorphism f : Fm → D, such that f (φ) = 1, for all φ ∈ Φ, but f (ψ) = 1. For logics with finitely many axioms and rules, this implies that the deducibility relation is decidable For a finitely axiomatized variety, FEP implies the decidability of the universal theory

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 21 / 39

SLIDE 75

FEP

A class of algebras K has the finite embeddability property (FEP) if for every A ∈ K, every finite partial subalgebra B of A can be (partially) embedded in a finite D ∈ K. The corresponding logic has the strong finite model property: if Φ ⊢ ψ, for finite Φ, then there is a finite counter-model, namely there is D ∈ K and a homomorphism f : Fm → D, such that f (φ) = 1, for all φ ∈ Φ, but f (ψ) = 1. For logics with finitely many axioms and rules, this implies that the deducibility relation is decidable For a finitely axiomatized variety, FEP implies the decidability of the universal theory

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 21 / 39

SLIDE 76

FEP

A class of algebras K has the finite embeddability property (FEP) if for every A ∈ K, every finite partial subalgebra B of A can be (partially) embedded in a finite D ∈ K. The corresponding logic has the strong finite model property: if Φ ⊢ ψ, for finite Φ, then there is a finite counter-model, namely there is D ∈ K and a homomorphism f : Fm → D, such that f (φ) = 1, for all φ ∈ Φ, but f (ψ) = 1. For logics with finitely many axioms and rules, this implies that the deducibility relation is decidable For a finitely axiomatized variety, FEP implies the decidability of the universal theory

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 21 / 39

SLIDE 77

FEP for integral RLs with {∨, ·, 1}-equations

Blok and van Alten 2002 proved FEP for integral RLs, and extended it to residuated groupoids (2005)

Theorem. Every variety of integral RL’s axiomatized by equations over

{∨, ·, 1} has the FEP. B embeds in W+

A,B via { }⊳ : B → W+

W+

A,B is finite

W+

A,B ∈ V

Corollary. These varieties are generated as quasivarieties by their finite

members.

Corollary. The corresponding logics have the strong finite model property

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 22 / 39

SLIDE 78

FEP for integral RLs with {∨, ·, 1}-equations

Blok and van Alten 2002 proved FEP for integral RLs, and extended it to residuated groupoids (2005)

Theorem. Every variety of integral RL’s axiomatized by equations over

{∨, ·, 1} has the FEP. B embeds in W+

A,B via { }⊳ : B → W+

W+

A,B is finite

W+

A,B ∈ V

Corollary. These varieties are generated as quasivarieties by their finite

members.

Corollary. The corresponding logics have the strong finite model property

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 22 / 39

SLIDE 79

FEP for integral RLs with {∨, ·, 1}-equations

Blok and van Alten 2002 proved FEP for integral RLs, and extended it to residuated groupoids (2005)

Theorem. Every variety of integral RL’s axiomatized by equations over

{∨, ·, 1} has the FEP. B embeds in W+

A,B via { }⊳ : B → W+

W+

A,B is finite

W+

A,B ∈ V

Corollary. These varieties are generated as quasivarieties by their finite

members.

Corollary. The corresponding logics have the strong finite model property

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 22 / 39

SLIDE 80

Finiteness

Idea for finiteness: Every element in W+

A,B is an intersection of basic

elements. So it suffices to prove that there are only finitely many such

elements. Replace the frame WA,B by one WM

A,B, where it is easier to work.

Let M be the free monoid with unit over the set B and f : M → W the extension of the identity map. M

f

− → W

N

− − W ′ .

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 23 / 39

SLIDE 81

Finiteness

Idea for finiteness: Every element in W+

A,B is an intersection of basic

elements. So it suffices to prove that there are only finitely many such

elements. Replace the frame WA,B by one WM

A,B, where it is easier to work.

Let M be the free monoid with unit over the set B and f : M → W the extension of the identity map. M

f

− → W

N

− − W ′ .

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 23 / 39

SLIDE 82

Finiteness

Idea for finiteness: Every element in W+

A,B is an intersection of basic

elements. So it suffices to prove that there are only finitely many such

elements. Replace the frame WA,B by one WM

A,B, where it is easier to work.

Let M be the free monoid with unit over the set B and f : M → W the extension of the identity map. M

f

− → W

N

− − W ′ .

P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 23 / 39

SLIDE 83