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 γ N ( X ) = X ⊲⊳ then γ N : 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 + = ( γ N [ P ( W )] , ∩ , ∪ γ N ) 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
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 γ N ( X ) = X ⊲⊳ then γ N : 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 + = ( γ N [ P ( W )] , ∩ , ∪ γ N ) 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
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 γ N ( X ) = X ⊲⊳ then γ N : 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 + = ( γ N [ P ( W )] , ∩ , ∪ γ N ) 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
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 γ N ( X ) = X ⊲⊳ then γ N : 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 + = ( γ N [ P ( W )] , ∩ , ∪ γ N ) 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
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 γ N ( X ) = X ⊲⊳ then γ N : 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 + = ( γ N [ P ( W )] , ∩ , ∪ γ N ) 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
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
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
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
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
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 L γ = ( L γ , ∧ , ∨ 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 , γ N is a nucleus on ( P ( W ) , ∩ , ∪ , ◦ , \ , /, { 1 } ) Corollary. If W is a residuated frame then the Galois algebra W + = ( P ( W ) , ∩ , ∪ , ◦ , \ , /, 1) γ N is a complete residuated lattice. Moreover, for W L , x �→ { x } ⊳ is an embedding. If L is a RL, W L = ( L , L , ≤ , · , \ , / ) is a residuated frame. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39
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 L γ = ( L γ , ∧ , ∨ 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 , γ N is a nucleus on ( P ( W ) , ∩ , ∪ , ◦ , \ , /, { 1 } ) Corollary. If W is a residuated frame then the Galois algebra W + = ( P ( W ) , ∩ , ∪ , ◦ , \ , /, 1) γ N is a complete residuated lattice. Moreover, for W L , x �→ { x } ⊳ is an embedding. If L is a RL, W L = ( L , L , ≤ , · , \ , / ) is a residuated frame. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39
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 L γ = ( L γ , ∧ , ∨ 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 , γ N is a nucleus on ( P ( W ) , ∩ , ∪ , ◦ , \ , /, { 1 } ) Corollary. If W is a residuated frame then the Galois algebra W + = ( P ( W ) , ∩ , ∪ , ◦ , \ , /, 1) γ N is a complete residuated lattice. Moreover, for W L , x �→ { x } ⊳ is an embedding. If L is a RL, W L = ( L , L , ≤ , · , \ , / ) is a residuated frame. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39
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 L γ = ( L γ , ∧ , ∨ 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 , γ N is a nucleus on ( P ( W ) , ∩ , ∪ , ◦ , \ , /, { 1 } ) Corollary. If W is a residuated frame then the Galois algebra W + = ( P ( W ) , ∩ , ∪ , ◦ , \ , /, 1) γ N is a complete residuated lattice. Moreover, for W L , x �→ { x } ⊳ is an embedding. If L is a RL, W L = ( L , L , ≤ , · , \ , / ) is a residuated frame. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39
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 L γ = ( L γ , ∧ , ∨ 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 , γ N is a nucleus on ( P ( W ) , ∩ , ∪ , ◦ , \ , /, { 1 } ) Corollary. If W is a residuated frame then the Galois algebra W + = ( P ( W ) , ∩ , ∪ , ◦ , \ , /, 1) γ N is a complete residuated lattice. Moreover, for W L , x �→ { x } ⊳ is an embedding. If L is a RL, W L = ( L , L , ≤ , · , \ , / ) is a residuated frame. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 7 / 39
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
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
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
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
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
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
All (dually-)nonisomorphic lattices with ≤ 7 elements K n , k = selfdual lattice of size n L n , k = nonselfdual lattices of size n K 1 , 1 K 2 , 1 K 3 , 1 K 4 , 1 K 4 , 2 L 5 , 1 K 5 , 1 K 5 , 2 K 5 , 3 L 6 , 1 L 6 , 2 L 6 , 3 L 6 , 4 K 6 , 1 K 6 , 2 K 6 , 3 K 6 , 4 K 6 , 5 K 6 , 6 K 6 , 7 L 7 , 1 L 7 , 2 L 7 , 3 L 7 , 4 L 7 , 5 L 7 , 6 L 7 , 7 L 7 , 8 L 7 , 9 L 7 , 10 L 7 , 11 L 7 , 12 L 7 , 13 L 7 , 14 L 7 , 15 L 7 , 16 L 7 , 17 L 7 , 18 L 7 , 19 L 7 , 20 K 7 , 1 K 7 , 2 K 7 , 3 K 7 , 4 K 7 , 5 K 7 , 6 K 7 , 7 K 7 , 8 K 7 , 9 K 7 , 10 K 7 , 11 K 7 , 12 K 7 , 13 P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 9 / 39
FL x ⇒ a y ◦ a ◦ z ⇒ c (cut) a ⇒ a (Id) y ◦ x ◦ z ⇒ c y ◦ a ◦ z ⇒ c y ◦ b ◦ z ⇒ c x ⇒ a x ⇒ b y ◦ a ∧ b ◦ z ⇒ c ( ∧ L ℓ ) y ◦ a ∧ b ◦ z ⇒ c ( ∧ L r ) ( ∧ R) x ⇒ a ∧ b y ◦ a ◦ z ⇒ c y ◦ b ◦ z ⇒ c x ⇒ a x ⇒ b ( ∨ L) x ⇒ a ∨ b ( ∨ R ℓ ) x ⇒ a ∨ b ( ∨ R r ) y ◦ a ∨ b ◦ z ⇒ c x ⇒ a y ◦ b ◦ z ⇒ c a ◦ x ⇒ b y ◦ x ◦ ( a \ b ) ◦ z ⇒ c ( \ L) x ⇒ a \ b ( \ R) x ⇒ a y ◦ b ◦ z ⇒ c x ◦ a ⇒ b y ◦ ( b / a ) ◦ x ◦ z ⇒ c ( / L) x ⇒ b / a ( / R) y ◦ a ◦ b ◦ z ⇒ c x ⇒ a y ⇒ b y ◦ a · b ◦ z ⇒ c ( · L) ( · R) x ◦ y ⇒ a · b 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
FL with context notation x ⇒ a u [ a ] ⇒ c (cut) a ⇒ a (Id) u [ x ] ⇒ c u [ a ] ⇒ c u [ b ] ⇒ c x ⇒ a x ⇒ b u [ a ∧ b ] ⇒ c ( ∧ L ℓ ) u [ a ∧ b ] ⇒ c ( ∧ L r ) ( ∧ R) x ⇒ a ∧ b u [ a ] ⇒ c u [ b ] ⇒ c x ⇒ a x ⇒ b ( ∨ L) x ⇒ a ∨ b ( ∨ R ℓ ) x ⇒ a ∨ b ( ∨ R r ) u [ a ∨ b ] ⇒ c x ⇒ a u [ b ] ⇒ c a ◦ x ⇒ b u [ x ◦ ( a \ b )] ⇒ c ( \ L) x ⇒ a \ b ( \ R) x ⇒ a u [ b ] ⇒ c x ◦ a ⇒ b u [( b / a ) ◦ x ] ⇒ c ( / L) x ⇒ b / a ( / R) u [ a ◦ b ] ⇒ c x ⇒ a y ⇒ b u [ a · b ] ⇒ c ( · L) ( · R) x ◦ y ⇒ a · b | u | ⇒ a u [1] ⇒ a (1L) ε ⇒ 1 (1R) P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 11 / 39
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 ( c ) x ≤ x 2 u [ x ] ⇒ c (contraction) | u | ⇒ c u [ x ] ⇒ c ( i ) (integrality) x ≤ 1 We write FL ec for FL + ( e ) + ( c ). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39
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 ( c ) x ≤ x 2 u [ x ] ⇒ c (contraction) | u | ⇒ c u [ x ] ⇒ c ( i ) (integrality) x ≤ 1 We write FL ec for FL + ( e ) + ( c ). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39
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 ( c ) x ≤ x 2 u [ x ] ⇒ c (contraction) | u | ⇒ c u [ x ] ⇒ c ( i ) (integrality) x ≤ 1 We write FL ec for FL + ( e ) + ( c ). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39
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 ( c ) x ≤ x 2 u [ x ] ⇒ c (contraction) | u | ⇒ c u [ x ] ⇒ c ( i ) (integrality) x ≤ 1 We write FL ec for FL + ( e ) + ( c ). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39
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 ( c ) x ≤ x 2 u [ x ] ⇒ c (contraction) | u | ⇒ c u [ x ] ⇒ c ( i ) (integrality) x ≤ 1 We write FL ec for FL + ( e ) + ( c ). P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 12 / 39
Examples of frames (FL) Consider the Gentzen system FL (full Lambek calculus). We define the frame W FL , where ( W , ◦ , ε ) to be the free monoid over the set Fm of all formulas W ′ = S W × Fm , where S W 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
Examples of frames (FL) Consider the Gentzen system FL (full Lambek calculus). We define the frame W FL , where ( W , ◦ , ε ) to be the free monoid over the set Fm of all formulas W ′ = S W × Fm , where S W 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
Examples of frames (FL) Consider the Gentzen system FL (full Lambek calculus). We define the frame W FL , where ( W , ◦ , ε ) to be the free monoid over the set Fm of all formulas W ′ = S W × Fm , where S W 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
Examples of frames (FL) Consider the Gentzen system FL (full Lambek calculus). We define the frame W FL , where ( W , ◦ , ε ) to be the free monoid over the set Fm of all formulas W ′ = S W × Fm , where S W 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
Examples of frames (FL) Consider the Gentzen system FL (full Lambek calculus). We define the frame W FL , where ( W , ◦ , ε ) to be the free monoid over the set Fm of all formulas W ′ = S W × Fm , where S W 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
Examples of frames (FEP) Let A be a residuated lattice and B a partial subalgebra of A . We define the frame W A , B , where ( W , · , 1) to be the submonoid of A generated by B , W ′ = S B × B , where S W 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
Examples of frames (FEP) Let A be a residuated lattice and B a partial subalgebra of A . We define the frame W A , B , where ( W , · , 1) to be the submonoid of A generated by B , W ′ = S B × B , where S W 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
Examples of frames (FEP) Let A be a residuated lattice and B a partial subalgebra of A . We define the frame W A , B , where ( W , · , 1) to be the submonoid of A generated by B , W ′ = S B × B , where S W 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
Examples of frames (FEP) Let A be a residuated lattice and B a partial subalgebra of A . We define the frame W A , B , where ( W , · , 1) to be the submonoid of A generated by B , W ′ = S B × B , where S W 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
Examples of frames (FEP) Let A be a residuated lattice and B a partial subalgebra of A . We define the frame W A , B , where ( W , · , 1) to be the submonoid of A generated by B , W ′ = S B × B , where S W 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
GN xNa aNz (CUT) aNa (Id) xNz xNa bNz a ◦ xNb x ◦ ( a \ b ) Nz ( \ L) xNa \ b ( \ R) xNa bNz x ◦ aNb ( b / a ) ◦ xNz ( / L) xNb / a ( / R) xNa yNb a ◦ bNz a · bNz ( · L) x ◦ yNa · b ( · R) aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) xNa ∧ b aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) a ∨ bNz ε Nz 1 Nz (1L) ε N 1 (1R) P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 15 / 39
Gentzen frames The following properties hold for W L , W FL and W A , 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
Gentzen frames The following properties hold for W L , W FL and W A , 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
Gentzen frames The following properties hold for W L , W FL and W A , 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
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 } ⊳ (1 B ∈ 1 W + ⊆ { 1 B } ⊳ ) 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 ( ∨ R ℓ ); 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
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 } ⊳ (1 B ∈ 1 W + ⊆ { 1 B } ⊳ ) 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 ( ∨ R ℓ ); 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
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 } ⊳ (1 B ∈ 1 W + ⊆ { 1 B } ⊳ ) 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 ( ∨ R ℓ ); 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
DM-completion For a residuated lattice L , we associated the Gentzen frame ( W L , 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
DM-completion For a residuated lattice L , we associated the Gentzen frame ( W L , 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
Completeness - Cut elimination f : Fm L → W + be the For every homomorphism f : Fm → B , let ¯ f ( p ) = { f ( p ) } ⊳ for any variable p homomorphism that extends ¯ 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 W FL | = x ⇒ c if f ( x ) N f ( c ) for all f : Fm → Fm = x · ≤ c , then W FL | Theorem. If W + FL | = 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 W FL f corresponds to cut-free FL . Corollary (CE). FL and FL f 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
Completeness - Cut elimination f : Fm L → W + be the For every homomorphism f : Fm → B , let ¯ f ( p ) = { f ( p ) } ⊳ for any variable p homomorphism that extends ¯ 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 W FL | = x ⇒ c if f ( x ) N f ( c ) for all f : Fm → Fm = x · ≤ c , then W FL | Theorem. If W + FL | = 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 W FL f corresponds to cut-free FL . Corollary (CE). FL and FL f 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
Completeness - Cut elimination f : Fm L → W + be the For every homomorphism f : Fm → B , let ¯ f ( p ) = { f ( p ) } ⊳ for any variable p homomorphism that extends ¯ 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 W FL | = x ⇒ c if f ( x ) N f ( c ) for all f : Fm → Fm = x · ≤ c , then W FL | Theorem. If W + FL | = 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 W FL f corresponds to cut-free FL . Corollary (CE). FL and FL f 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
Completeness - Cut elimination f : Fm L → W + be the For every homomorphism f : Fm → B , let ¯ f ( p ) = { f ( p ) } ⊳ for any variable p homomorphism that extends ¯ 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 W FL | = x ⇒ c if f ( x ) N f ( c ) for all f : Fm → Fm = x · ≤ c , then W FL | Theorem. If W + FL | = 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 W FL f corresponds to cut-free FL . Corollary (CE). FL and FL f 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
Completeness - Cut elimination f : Fm L → W + be the For every homomorphism f : Fm → B , let ¯ f ( p ) = { f ( p ) } ⊳ for any variable p homomorphism that extends ¯ 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 W FL | = x ⇒ c if f ( x ) N f ( c ) for all f : Fm → Fm = x · ≤ c , then W FL | Theorem. If W + FL | = 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 W FL f corresponds to cut-free FL . Corollary (CE). FL and FL f 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
Finite model property For W FL , 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 FL f . 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 γ N ′ . 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
Finite model property For W FL , 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 FL f . 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 γ N ′ . 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
Finite model property For W FL , 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 FL f . 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 γ N ′ . 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
Finite model property For W FL , 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 FL f . 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 γ N ′ . 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
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 . T he 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
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 . T he 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
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 . T he 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
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 . T he 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
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. A , B via { } ⊳ : B → W + B embeds in 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
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. A , B via { } ⊳ : B → W + B embeds in 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
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. A , B via { } ⊳ : B → W + B embeds in 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
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 W A , B by one W M 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. N f − W ′ M − → W − . P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 23 / 39
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 W A , B by one W M 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. N f − W ′ M − → W − . P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 23 / 39
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 W A , B by one W M 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. N f − W ′ M − → W − . P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 23 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 1 Idea: Express equations over {∨ , · , 1 } at the frame level. For an equation ε over {∨ , · , 1 } we distribute products over joins to get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ). We proceed by example: x 2 y ≤ xy ∨ yx ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 x 1 y ≤ v x 2 y ≤ v yx 1 ≤ v yx 2 ≤ v x 1 x 2 y ≤ v x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 24 / 39
Equations 2 Theorem. If ( W , B ) is a Gentzen frame and ε an equation over {∨ , · , 1 } , then ( W , B ) satisfies R( ε ) iff W + satisfies ε . (The linearity of the denominator of R ( ε ) plays an important role in the proof.) Corollary. If an equation over {∨ , · , 1 } is valid in A , then it is also valid in W + A , B , for every partial subalgebra B of A . Consequently, W + A , B ∈ V . P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 25 / 39
Equations 2 Theorem. If ( W , B ) is a Gentzen frame and ε an equation over {∨ , · , 1 } , then ( W , B ) satisfies R( ε ) iff W + satisfies ε . (The linearity of the denominator of R ( ε ) plays an important role in the proof.) Corollary. If an equation over {∨ , · , 1 } is valid in A , then it is also valid in W + A , B , for every partial subalgebra B of A . Consequently, W + A , B ∈ V . P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 25 / 39
Equations 2 Theorem. If ( W , B ) is a Gentzen frame and ε an equation over {∨ , · , 1 } , then ( W , B ) satisfies R( ε ) iff W + satisfies ε . (The linearity of the denominator of R ( ε ) plays an important role in the proof.) Corollary. If an equation over {∨ , · , 1 } is valid in A , then it is also valid in W + A , B , for every partial subalgebra B of A . Consequently, W + A , B ∈ V . P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 25 / 39
Structural rules Given an equation ε of the form t 0 ≤ t 1 ∨ · · · ∨ t n , where t i are {· , 1 } -terms we construct the rule R ( ε ) u [ t 1 ] ⇒ a · · · u [ t n ] ⇒ a ( R ( ε )) u [ t 0 ] ⇒ a where the t i ’s are evaluated in ( W , ◦ , ε ). Such a rule is called linear if all variables in t 0 are distinct. Theorem. Every system obtained from FL by adding linear rules has the cut elimination property. A set of rules of the form R ( ε ) is called reducing if there is a complexity measure that decreases with upward applications of the rules (and the rules of FL ). Theorem. Every system obtained from FL by adding linear reducing rules is decidable. The subvariety of residuated lattices axiomatized by the corresponding equations has decidable equational theory. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 26 / 39
Structural rules Given an equation ε of the form t 0 ≤ t 1 ∨ · · · ∨ t n , where t i are {· , 1 } -terms we construct the rule R ( ε ) u [ t 1 ] ⇒ a · · · u [ t n ] ⇒ a ( R ( ε )) u [ t 0 ] ⇒ a where the t i ’s are evaluated in ( W , ◦ , ε ). Such a rule is called linear if all variables in t 0 are distinct. Theorem. Every system obtained from FL by adding linear rules has the cut elimination property. A set of rules of the form R ( ε ) is called reducing if there is a complexity measure that decreases with upward applications of the rules (and the rules of FL ). Theorem. Every system obtained from FL by adding linear reducing rules is decidable. The subvariety of residuated lattices axiomatized by the corresponding equations has decidable equational theory. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 26 / 39
Structural rules Given an equation ε of the form t 0 ≤ t 1 ∨ · · · ∨ t n , where t i are {· , 1 } -terms we construct the rule R ( ε ) u [ t 1 ] ⇒ a · · · u [ t n ] ⇒ a ( R ( ε )) u [ t 0 ] ⇒ a where the t i ’s are evaluated in ( W , ◦ , ε ). Such a rule is called linear if all variables in t 0 are distinct. Theorem. Every system obtained from FL by adding linear rules has the cut elimination property. A set of rules of the form R ( ε ) is called reducing if there is a complexity measure that decreases with upward applications of the rules (and the rules of FL ). Theorem. Every system obtained from FL by adding linear reducing rules is decidable. The subvariety of residuated lattices axiomatized by the corresponding equations has decidable equational theory. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 26 / 39
Structural rules Given an equation ε of the form t 0 ≤ t 1 ∨ · · · ∨ t n , where t i are {· , 1 } -terms we construct the rule R ( ε ) u [ t 1 ] ⇒ a · · · u [ t n ] ⇒ a ( R ( ε )) u [ t 0 ] ⇒ a where the t i ’s are evaluated in ( W , ◦ , ε ). Such a rule is called linear if all variables in t 0 are distinct. Theorem. Every system obtained from FL by adding linear rules has the cut elimination property. A set of rules of the form R ( ε ) is called reducing if there is a complexity measure that decreases with upward applications of the rules (and the rules of FL ). Theorem. Every system obtained from FL by adding linear reducing rules is decidable. The subvariety of residuated lattices axiomatized by the corresponding equations has decidable equational theory. P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 26 / 39
Applications Cut-elimination (CE) and finite model property (FMP) for FL and (cyclic) InFL . Generation by finite members for RL, InFL M. Kozak 2008 proved distributive FL has the FMP, and using our approach the same result holds for any extension of DFL with linear reducing structural rules The finite embeddability property (FEP) for integral RL with {∨ , · , 1 } -axioms The above extend to the non-associative case, also with the addition of suitable structural rules P Jipsen (Chapman), N Galatos (DU) Semantics: Residuated Frames ICLA, January, 2009 27 / 39
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