Decidability for lattices Residuated lattices Examples Decidability for lattices Boolean algebras a ≤ b b ≤ a a ≤ b b ≤ c Lattices Contexts a ≤ a a ≤ c a = b Dedekind-McNeille Lattice frames Why does it work? a ≤ c b ≤ c c ≤ a c ≤ b Sequents a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Hypersequents Beyond c ≤ a c ≤ b a ≤ c b ≤ c c ≤ a ∨ b c ≤ a ∨ b a ∨ b ≤ c Cut Eimination Theorem. (Whitman’s conditions, also Skolem) Transitivity (aka cut) is not needed. Corollary. The equational theory of lattices is decidable. Note that the system is based on quasi-inequalities ( rules ).The rules are distinguished in structural and logical (depending on whether they involve connectives). Logical rules involve each a single connective which they introduce one one side of the inequality. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37
Decidability for lattices Residuated lattices Examples Decidability for lattices Boolean algebras a ≤ b b ≤ a a ≤ b b ≤ c Lattices Contexts a ≤ a a ≤ c a = b Dedekind-McNeille Lattice frames Why does it work? a ≤ c b ≤ c c ≤ a c ≤ b Sequents a ∧ b ≤ c a ∧ b ≤ c c ≤ a ∧ b Hypersequents Beyond c ≤ a c ≤ b a ≤ c b ≤ c c ≤ a ∨ b c ≤ a ∨ b a ∨ b ≤ c Cut Eimination Theorem. (Whitman’s conditions, also Skolem) Transitivity (aka cut) is not needed. Corollary. The equational theory of lattices is decidable. Note that the system is based on quasi-inequalities ( rules ).The rules are distinguished in structural and logical (depending on whether they involve connectives). Logical rules involve each a single connective which they introduce one one side of the inequality. We write Lat for the above system and Lat for the variety of lattices. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37
Boolean algebras Residuated lattices Examples Decidability for lattices Boolean algebras Lattices { a, b, c } Contexts Dedekind-McNeille { a, c } Lattice frames { a, b } { b, c } Why does it work? a c b Sequents { a } { b } { c } Hypersequents Beyond ∅ c c a a b b Every finite distributive lattice L can be recovered from its poset J ( L ) of join irreducibless; D ∼ = D ( J ( L )) . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 5 / 37
Boolean algebras Residuated lattices Examples Decidability for lattices Boolean algebras Lattices { a, b, c } Contexts Dedekind-McNeille { a, c } Lattice frames { a, b } { b, c } Why does it work? a c b Sequents { a } { b } { c } Hypersequents Beyond ∅ c c a a b b Every finite distributive lattice L can be recovered from its poset J ( L ) of join irreducibless; D ∼ = D ( J ( L )) . For general DLs we use prime filters. Using topology we can recover the original algebra (Priestley). Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 5 / 37
Boolean algebras Residuated lattices Examples Decidability for lattices Boolean algebras Lattices { a, b, c } Contexts Dedekind-McNeille { a, c } Lattice frames { a, b } { b, c } Why does it work? a c b Sequents { a } { b } { c } Hypersequents Beyond ∅ c c a a b b Every finite distributive lattice L can be recovered from its poset J ( L ) of join irreducibless; D ∼ = D ( J ( L )) . For general DLs we use prime filters. Using topology we can recover the original algebra (Priestley). This is the basis of Kripke semantics. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 5 / 37
Lattices Residuated lattices Examples Decidability for lattices Boolean algebras Lattices 1 Contexts Dedekind-McNeille Lattice frames Why does it work? a c a c b b Sequents a c b Hypersequents 0 Beyond For general (non-distributive) lattices, the poset of join irreducibles is not enough to recover the lattice. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 6 / 37
Lattices Residuated lattices Examples Decidability for lattices Boolean algebras Lattices 1 Contexts Dedekind-McNeille Lattice frames Why does it work? a c a c b b Sequents a c b Hypersequents 0 Beyond For general (non-distributive) lattices, the poset of join irreducibles is not enough to recover the lattice. We also need the meet irreducibles ; we denote their poset by M ( L ) . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 6 / 37
Lattices Residuated lattices Examples Decidability for lattices Boolean algebras Lattices 1 Contexts Dedekind-McNeille Lattice frames Why does it work? a c a c b b Sequents a c b Hypersequents 0 Beyond For general (non-distributive) lattices, the poset of join irreducibles is not enough to recover the lattice. We also need the meet irreducibles ; we denote their poset by M ( L ) . For every distributive lattice M ( L ) is isomorphic to J ( L ) . Note ↑ a ∪ ↓ c = ↑ b ∪ ↓ a = ↑ c ∪ ↓ d = L . Splitting pairs : ( a, c ) , ( b, a ) , ( c, d ) . c c d d a a b a c b Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 6 / 37
Contexts Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames c ′ b ′ a ′ c ′ b ′ a ′ a ′ b ′ c ′ ≤ Why does it work? Sequents × × a Hypersequents a c a c b b × × b Beyond c × × 1 a c b a ′ b ′ c ′ ≤ a c b × a a c b b × c × 0 Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 7 / 37
Dedekind-McNeille Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille c a c d d Lattice frames Why does it work? Sequents a a c b b Hypersequents Beyond ≤ a d c × × a × × b c × We obtain an oriented bipartite graph; an algebraic rendering of sequents! Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37
Dedekind-McNeille Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille c a c d d Lattice frames Why does it work? Sequents a a c b b Hypersequents Beyond ≤ a d c × × a × × b c × We obtain an oriented bipartite graph; an algebraic rendering of sequents! How do we recover the lattice? Which subsets of join irreducibles should we consider? Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37
Dedekind-McNeille Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille c a c d d Lattice frames Why does it work? Sequents a a c b b Hypersequents Beyond ≤ a d c × × a × × b c × We obtain an oriented bipartite graph; an algebraic rendering of sequents! How do we recover the lattice? Which subsets of join irreducibles should we consider? Let’s go back to Dedekind’s construction of R from Q using cuts. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37
Dedekind-McNeille Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille c a c d d Lattice frames Why does it work? Sequents a a c b b Hypersequents Beyond ≤ a d c × × a × × b c × We obtain an oriented bipartite graph; an algebraic rendering of sequents! How do we recover the lattice? Which subsets of join irreducibles should we consider? Let’s go back to Dedekind’s construction of R from Q using cuts. A subset is a Dedekind cut if when we take common upper bounds and then common lower bounds of them, we get back the original subset. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37
Dedekind-McNeille Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille c a c d d Lattice frames Why does it work? Sequents a a c b b Hypersequents Beyond ≤ a d c × × a × × b c × We obtain an oriented bipartite graph; an algebraic rendering of sequents! How do we recover the lattice? Which subsets of join irreducibles should we consider? Let’s go back to Dedekind’s construction of R from Q using cuts. A subset is a Dedekind cut if when we take common upper bounds and then common lower bounds of them, we get back the original subset. McNeille extended this definition to arbitrary posets, and Birkhoff to arbitrary relations between two sets (contexts). Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37
Lattice frames Residuated lattices Examples Decidability for lattices Boolean algebras A lattice frame is a structure F = ( L, R, N ) where L and R are sets Lattices Contexts and N is a binary relation from L to R . Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37
Lattice frames Residuated lattices Examples Decidability for lattices Boolean algebras A lattice frame is a structure F = ( L, R, N ) where L and R are sets Lattices Contexts and N is a binary relation from L to R . Dedekind-McNeille Lattice frames For X ⊆ L and Y ⊆ R we define Why does it work? Sequents X ⊲ = { b ∈ R : x N b, for all x ∈ X } Hypersequents Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37
Lattice frames Residuated lattices Examples Decidability for lattices Boolean algebras A lattice frame is a structure F = ( L, R, N ) where L and R are sets Lattices Contexts and N is a binary relation from L to R . Dedekind-McNeille Lattice frames For X ⊆ L and Y ⊆ R we define Why does it work? Sequents X ⊲ = { b ∈ R : x N b, for all x ∈ X } Hypersequents Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Beyond The maps ⊲ : P ( L ) → P ( R ) and ⊳ : P ( R ) → P ( L ) form a Galois connection. The map γ N : P ( L ) → P ( L ) , where γ N ( X ) = X ⊲⊳ , is a closure operator. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37
Lattice frames Residuated lattices Examples Decidability for lattices Boolean algebras A lattice frame is a structure F = ( L, R, N ) where L and R are sets Lattices Contexts and N is a binary relation from L to R . Dedekind-McNeille Lattice frames For X ⊆ L and Y ⊆ R we define Why does it work? Sequents X ⊲ = { b ∈ R : x N b, for all x ∈ X } Hypersequents Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Beyond The maps ⊲ : P ( L ) → P ( R ) and ⊳ : P ( R ) → P ( L ) form a Galois connection. The map γ N : P ( L ) → P ( L ) , where γ N ( X ) = X ⊲⊳ , is a closure operator. Lemma. If A = ( A, ∧ , ∨ ) is a lattice and γ is a cl.op. on L , then ( γ [ A ] , ∧ , ∨ γ ) is a lattice. [ x ∨ γ y = γ ( x ∨ y ) .] Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37
Lattice frames Residuated lattices Examples Decidability for lattices Boolean algebras A lattice frame is a structure F = ( L, R, N ) where L and R are sets Lattices Contexts and N is a binary relation from L to R . Dedekind-McNeille Lattice frames For X ⊆ L and Y ⊆ R we define Why does it work? Sequents X ⊲ = { b ∈ R : x N b, for all x ∈ X } Hypersequents Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Beyond The maps ⊲ : P ( L ) → P ( R ) and ⊳ : P ( R ) → P ( L ) form a Galois connection. The map γ N : P ( L ) → P ( L ) , where γ N ( X ) = X ⊲⊳ , is a closure operator. Lemma. If A = ( A, ∧ , ∨ ) is a lattice and γ is a cl.op. on L , then ( γ [ A ] , ∧ , ∨ γ ) is a lattice. [ x ∨ γ y = γ ( x ∨ y ) .] Corollary. If F is a lattice frame then the Galois algebra F + = ( γ N [ P ( L )] , ∩ , ∪ γ N ) is a complete lattice. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37
Lattice frames Residuated lattices Examples Decidability for lattices Boolean algebras A lattice frame is a structure F = ( L, R, N ) where L and R are sets Lattices Contexts and N is a binary relation from L to R . Dedekind-McNeille Lattice frames For X ⊆ L and Y ⊆ R we define Why does it work? Sequents X ⊲ = { b ∈ R : x N b, for all x ∈ X } Hypersequents Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Beyond The maps ⊲ : P ( L ) → P ( R ) and ⊳ : P ( R ) → P ( L ) form a Galois connection. The map γ N : P ( L ) → P ( L ) , where γ N ( X ) = X ⊲⊳ , is a closure operator. Lemma. If A = ( A, ∧ , ∨ ) is a lattice and γ is a cl.op. on L , then ( γ [ A ] , ∧ , ∨ γ ) is a lattice. [ x ∨ γ y = γ ( x ∨ y ) .] Corollary. If F is a lattice frame then the Galois algebra F + = ( γ N [ P ( L )] , ∩ , ∪ γ N ) is a complete lattice. If A is a lattice, F A = ( A, A, ≤ ) is a lattice frame. Also, F + A is the Dedekind-MacNeille completion of A and x �→ { x } ⊳ is an embedding. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37
Why does it work? Residuated lattices Examples Decidability for lattices Boolean algebras Enough to represent complete lattices ≡ complete-join semilattices Lattices Contexts L = ( L, � ) (they are also meet complete). Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37
Why does it work? Residuated lattices Examples Decidability for lattices Boolean algebras Enough to represent complete lattices ≡ complete-join semilattices Lattices Contexts L = ( L, � ) (they are also meet complete). The free objects in this Dedekind-McNeille category are ( P ( X ) , � ) , so for every L , there is an X and an onto Lattice frames Why does it work? � -homomorphism f : ( P ( X ) , � ) → L . Sequents Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37
Why does it work? Residuated lattices Examples Decidability for lattices Boolean algebras Enough to represent complete lattices ≡ complete-join semilattices Lattices Contexts L = ( L, � ) (they are also meet complete). The free objects in this Dedekind-McNeille category are ( P ( X ) , � ) , so for every L , there is an X and an onto Lattice frames Why does it work? � -homomorphism f : ( P ( X ) , � ) → L . Every such map is Sequents residuated : there is f ∗ : L → P ( X ) such that ∀ x ∈ P ( X ) and y ∈ L , Hypersequents Beyond f ( x ) ≤ y ⇔ x ⊆ f ∗ ( y ) . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37
Why does it work? Residuated lattices Examples Decidability for lattices Boolean algebras Enough to represent complete lattices ≡ complete-join semilattices Lattices Contexts L = ( L, � ) (they are also meet complete). The free objects in this Dedekind-McNeille category are ( P ( X ) , � ) , so for every L , there is an X and an onto Lattice frames Why does it work? � -homomorphism f : ( P ( X ) , � ) → L . Every such map is Sequents residuated : there is f ∗ : L → P ( X ) such that ∀ x ∈ P ( X ) and y ∈ L , Hypersequents Beyond f ( x ) ≤ y ⇔ x ⊆ f ∗ ( y ) . The composition γ = f ∗ f is a closure operator on P ( X ) , γ A i = γ ( � A i )), and the map f factors ( P ( X ) γ , � γ ) is a lattice ( � f |P ( X ) γ γ f : ( P ( X ) , � ) → ( P ( X ) γ , � γ ) → L , where the first is surjective and the second injective. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37
Why does it work? Residuated lattices Examples Decidability for lattices Boolean algebras Enough to represent complete lattices ≡ complete-join semilattices Lattices Contexts L = ( L, � ) (they are also meet complete). The free objects in this Dedekind-McNeille category are ( P ( X ) , � ) , so for every L , there is an X and an onto Lattice frames Why does it work? � -homomorphism f : ( P ( X ) , � ) → L . Every such map is Sequents residuated : there is f ∗ : L → P ( X ) such that ∀ x ∈ P ( X ) and y ∈ L , Hypersequents Beyond f ( x ) ≤ y ⇔ x ⊆ f ∗ ( y ) . The composition γ = f ∗ f is a closure operator on P ( X ) , γ A i = γ ( � A i )), and the map f factors ( P ( X ) γ , � γ ) is a lattice ( � f |P ( X ) γ γ f : ( P ( X ) , � ) → ( P ( X ) γ , � γ ) → L , where the first is surjective and the second injective. L is isomorphic to ( P ( X ) γ , � γ ) , a closure-operator image of ( P ( X ) , � ) . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37
Why does it work? Residuated lattices Examples Decidability for lattices Boolean algebras Enough to represent complete lattices ≡ complete-join semilattices Lattices Contexts L = ( L, � ) (they are also meet complete). The free objects in this Dedekind-McNeille category are ( P ( X ) , � ) , so for every L , there is an X and an onto Lattice frames Why does it work? � -homomorphism f : ( P ( X ) , � ) → L . Every such map is Sequents residuated : there is f ∗ : L → P ( X ) such that ∀ x ∈ P ( X ) and y ∈ L , Hypersequents Beyond f ( x ) ≤ y ⇔ x ⊆ f ∗ ( y ) . The composition γ = f ∗ f is a closure operator on P ( X ) , γ A i = γ ( � A i )), and the map f factors ( P ( X ) γ , � γ ) is a lattice ( � f |P ( X ) γ γ f : ( P ( X ) , � ) → ( P ( X ) γ , � γ ) → L , where the first is surjective and the second injective. L is isomorphic to ( P ( X ) γ , � γ ) , a closure-operator image of ( P ( X ) , � ) . Further, a closure operator γ on a poewerset P ( X ) is always of the form γ N , for N given by x N y ⇔ y ∈ γ ( { x } ) ; namely, it comes from a lattice frame ( X, Y, N ) . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37
Why does it work? Residuated lattices Examples Decidability for lattices Boolean algebras Enough to represent complete lattices ≡ complete-join semilattices Lattices Contexts L = ( L, � ) (they are also meet complete). The free objects in this Dedekind-McNeille category are ( P ( X ) , � ) , so for every L , there is an X and an onto Lattice frames Why does it work? � -homomorphism f : ( P ( X ) , � ) → L . Every such map is Sequents residuated : there is f ∗ : L → P ( X ) such that ∀ x ∈ P ( X ) and y ∈ L , Hypersequents Beyond f ( x ) ≤ y ⇔ x ⊆ f ∗ ( y ) . The composition γ = f ∗ f is a closure operator on P ( X ) , γ A i = γ ( � A i )), and the map f factors ( P ( X ) γ , � γ ) is a lattice ( � f |P ( X ) γ γ f : ( P ( X ) , � ) → ( P ( X ) γ , � γ ) → L , where the first is surjective and the second injective. L is isomorphic to ( P ( X ) γ , � γ ) , a closure-operator image of ( P ( X ) , � ) . Further, a closure operator γ on a poewerset P ( X ) is always of the form γ N , for N given by x N y ⇔ y ∈ γ ( { x } ) ; namely, it comes from a lattice frame ( X, Y, N ) . The atomic formulas in the language of frames are of the form x N y . Hence we discover algebraically lattice sequents! Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37
Why does it work? Residuated lattices Examples Decidability for lattices Boolean algebras Enough to represent complete lattices ≡ complete-join semilattices Lattices Contexts L = ( L, � ) (they are also meet complete). The free objects in this Dedekind-McNeille category are ( P ( X ) , � ) , so for every L , there is an X and an onto Lattice frames Why does it work? � -homomorphism f : ( P ( X ) , � ) → L . Every such map is Sequents residuated : there is f ∗ : L → P ( X ) such that ∀ x ∈ P ( X ) and y ∈ L , Hypersequents Beyond f ( x ) ≤ y ⇔ x ⊆ f ∗ ( y ) . The composition γ = f ∗ f is a closure operator on P ( X ) , γ A i = γ ( � A i )), and the map f factors ( P ( X ) γ , � γ ) is a lattice ( � f |P ( X ) γ γ f : ( P ( X ) , � ) → ( P ( X ) γ , � γ ) → L , where the first is surjective and the second injective. L is isomorphic to ( P ( X ) γ , � γ ) , a closure-operator image of ( P ( X ) , � ) . Further, a closure operator γ on a poewerset P ( X ) is always of the form γ N , for N given by x N y ⇔ y ∈ γ ( { x } ) ; namely, it comes from a lattice frame ( X, Y, N ) . The atomic formulas in the language of frames are of the form x N y . Hence we discover algebraically lattice sequents! Also, by writing down the basic algebraic properties in the language of N or ≤ we discover the proof-theoretic system Lat ! Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37
Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN Sequents FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 11 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents So, ( P, ∨ , · , 1) is a semiring. [In the complete case, a quantale.] Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents So, ( P, ∨ , · , 1) is a semiring. [In the complete case, a quantale.] Bi-modules Formula hierarchy Submodules and nuclei x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Residuated frames GN ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ FL x \ ( y/z ) = ( x \ y ) /z ■ Gentzen frames Frame applications 1 \ x = x = x/ 1 ■ Compl - CE ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents So, ( P, ∨ , · , 1) is a semiring. [In the complete case, a quantale.] Bi-modules Formula hierarchy Submodules and nuclei x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Residuated frames GN ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ FL x \ ( y/z ) = ( x \ y ) /z ■ Gentzen frames Frame applications 1 \ x = x = x/ 1 ■ Compl - CE ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ Equations Simple rules So, ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by p ⋆ n = n/p and Hypersequents Beyond n ⋆ p = p \ n . Thus, (( N, ∧ ) , ⋆ ) becomes a ( P, ∨ , · , 1) -bimodule. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents So, ( P, ∨ , · , 1) is a semiring. [In the complete case, a quantale.] Bi-modules Formula hierarchy Submodules and nuclei x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Residuated frames GN ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ FL x \ ( y/z ) = ( x \ y ) /z ■ Gentzen frames Frame applications 1 \ x = x = x/ 1 ■ Compl - CE ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ Equations Simple rules So, ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by p ⋆ n = n/p and Hypersequents Beyond n ⋆ p = p \ n . Thus, (( N, ∧ ) , ⋆ ) becomes a ( P, ∨ , · , 1) -bimodule. This split matches the notion of polarity . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents So, ( P, ∨ , · , 1) is a semiring. [In the complete case, a quantale.] Bi-modules Formula hierarchy Submodules and nuclei x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Residuated frames GN ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ FL x \ ( y/z ) = ( x \ y ) /z ■ Gentzen frames Frame applications 1 \ x = x = x/ 1 ■ Compl - CE ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ Equations Simple rules So, ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by p ⋆ n = n/p and Hypersequents Beyond n ⋆ p = p \ n . Thus, (( N, ∧ ) , ⋆ ) becomes a ( P, ∨ , · , 1) -bimodule. This split matches the notion of polarity . It also extend to � , � . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents So, ( P, ∨ , · , 1) is a semiring. [In the complete case, a quantale.] Bi-modules Formula hierarchy Submodules and nuclei x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Residuated frames GN ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ FL x \ ( y/z ) = ( x \ y ) /z ■ Gentzen frames Frame applications 1 \ x = x = x/ 1 ■ Compl - CE ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ Equations Simple rules So, ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by p ⋆ n = n/p and Hypersequents Beyond n ⋆ p = p \ n . Thus, (( N, ∧ ) , ⋆ ) becomes a ( P, ∨ , · , 1) -bimodule. This split matches the notion of polarity . It also extend to � , � . The bimodule can be viewed as a two-sorted algebra ( P, ∨ , · , 1 , N, ∧ , \ , / ) . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents So, ( P, ∨ , · , 1) is a semiring. [In the complete case, a quantale.] Bi-modules Formula hierarchy Submodules and nuclei x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Residuated frames GN ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ FL x \ ( y/z ) = ( x \ y ) /z ■ Gentzen frames Frame applications 1 \ x = x = x/ 1 ■ Compl - CE ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ Equations Simple rules So, ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by p ⋆ n = n/p and Hypersequents Beyond n ⋆ p = p \ n . Thus, (( N, ∧ ) , ⋆ ) becomes a ( P, ∨ , · , 1) -bimodule. This split matches the notion of polarity . It also extend to � , � . The bimodule can be viewed as a two-sorted algebra ( P, ∨ , · , 1 , N, ∧ , \ , / ) . The absolutely free algebra for P = N generated by P 0 = N 0 = V ar (the set of propositional variables) gives the set of all formulas. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Bi-modules Residuated lattices Examples Decidability for lattices Boolean algebras Let’s assume that P = N is the underlying set of a residuated lattice. Lattices Contexts Dedekind-McNeille x · 1 = x = 1 · x , ( xy ) z = x ( yz ) ■ Lattice frames x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Why does it work? Sequents So, ( P, ∨ , · , 1) is a semiring. [In the complete case, a quantale.] Bi-modules Formula hierarchy Submodules and nuclei x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Residuated frames GN ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ FL x \ ( y/z ) = ( x \ y ) /z ■ Gentzen frames Frame applications 1 \ x = x = x/ 1 ■ Compl - CE ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ Equations Simple rules So, ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by p ⋆ n = n/p and Hypersequents Beyond n ⋆ p = p \ n . Thus, (( N, ∧ ) , ⋆ ) becomes a ( P, ∨ , · , 1) -bimodule. This split matches the notion of polarity . It also extend to � , � . The bimodule can be viewed as a two-sorted algebra ( P, ∨ , · , 1 , N, ∧ , \ , / ) . The absolutely free algebra for P = N generated by P 0 = N 0 = V ar (the set of propositional variables) gives the set of all formulas. The steps of the generation process yield the substructural hierarchy . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37
Formula hierarchy Residuated lattices Examples Decidability for lattices Boolean algebras Lattices The sets P n , N n of formulas are defined by: ♣♣♣♣♣♣♣♣♣ ✻ ♣♣♣♣♣♣♣♣♣ ✻ ■ Contexts Dedekind-McNeille (0) P 0 = N 0 = the set of variables Lattice frames Why does it work? N n ⊆ P n +1 (P1) Sequents α, β ∈ P n +1 ⇒ α ∨ β, α · β, 1 ∈ P n +1 (P2) P 3 N 3 Bi-modules P n ⊆ N n +1 (N1) Formula hierarchy ✻ ❅ ■ ✒ ✻ � ❅ � Submodules and nuclei α, β ∈ N n +1 ⇒ α ∧ β ∈ N n +1 (N2) � ❅ Residuated frames α ∈ P n +1 , β ∈ N n +1 ⇒ α \ β, β/α, 0 ∈ N n +1 (N3) � ❅ GN FL P 2 N 2 P n +1 = �N n � � , � ; N n +1 = �P n � � , P n +1 \ ,/ P n +1 Gentzen frames ■ Frame applications ✻ ❅ ■ ❅ � � ✒ ✻ P n ⊆ P n +1 , N n ⊆ N n +1 , � P n = � N n = Fm Compl - CE ■ � ❅ Equations P 1 -reduced: � � p i Simple rules � ❅ ■ Hypersequents P 1 N 1 N 1 -reduced: � ( p 1 p 2 · · · p n \ r/q 1 q 2 · · · q m ) ■ Beyond ✻ ■ ❅ � ✒ ✻ ❅ � p 1 p 2 · · · p n q 1 q 2 · · · q m ≤ r � ❅ Sequent: a 1 , a 2 , . . . , a n ⇒ a 0 ( a i ∈ Fm ) ■ � ❅ P 0 N 0 A. Ciabattoni, NG, K. Terui. From axioms to analytic rules in nonclassical logics, Proceedings of LICS’08, 229-240, 2008. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 13 / 37
Submodules and nuclei Residuated lattices Examples Decidability for lattices Boolean algebras Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Lattices Contexts defined by a � -closed subset that is also closed under the actions. Dedekind-McNeille Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37
Submodules and nuclei Residuated lattices Examples Decidability for lattices Boolean algebras Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Lattices Contexts defined by a � -closed subset that is also closed under the actions. Dedekind-McNeille Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames Why does it work? that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37
Submodules and nuclei Residuated lattices Examples Decidability for lattices Boolean algebras Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Lattices Contexts defined by a � -closed subset that is also closed under the actions. Dedekind-McNeille Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames Why does it work? that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Sequents Bi-modules Formula hierarchy If P = N is the underlying set of a residuated lattice Submodules and nuclei A = ( A, ∧ , ∨ , · , \ , /, 1) , a nucleus is just a closure operator that Residuated frames GN satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37
Submodules and nuclei Residuated lattices Examples Decidability for lattices Boolean algebras Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Lattices Contexts defined by a � -closed subset that is also closed under the actions. Dedekind-McNeille Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames Why does it work? that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Sequents Bi-modules Formula hierarchy If P = N is the underlying set of a residuated lattice Submodules and nuclei A = ( A, ∧ , ∨ , · , \ , /, 1) , a nucleus is just a closure operator that Residuated frames GN satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) FL Gentzen frames Frame applications If we define A γ = { γ ( x ) : x ∈ A } , x ∨ γ y = γ ( x ∨ y ) and Compl - CE x · γ y = γ ( x · y ) , Equations Simple rules Hypersequents A γ = ( A γ , ∧ , ∨ γ , · γ , \ , /, γ (1)) Beyond is also a residuated lattice. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37
Submodules and nuclei Residuated lattices Examples Decidability for lattices Boolean algebras Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Lattices Contexts defined by a � -closed subset that is also closed under the actions. Dedekind-McNeille Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames Why does it work? that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Sequents Bi-modules Formula hierarchy If P = N is the underlying set of a residuated lattice Submodules and nuclei A = ( A, ∧ , ∨ , · , \ , /, 1) , a nucleus is just a closure operator that Residuated frames GN satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) FL Gentzen frames Frame applications If we define A γ = { γ ( x ) : x ∈ A } , x ∨ γ y = γ ( x ∨ y ) and Compl - CE x · γ y = γ ( x · y ) , Equations Simple rules Hypersequents A γ = ( A γ , ∧ , ∨ γ , · γ , \ , /, γ (1)) Beyond is also a residuated lattice. Residuated frames arise from studying submodules of P ( M ) , where M is a monoid, namely nuclei on powersets (of monoids). Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37
Residuated frames Residuated lattices Examples Decidability for lattices Boolean algebras A residuated frame is a structure F = ( L, R, N, ◦ , 1 , � , � ) Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37
Residuated frames Residuated lattices Examples Decidability for lattices Boolean algebras A residuated frame is a structure F = ( L, R, N, ◦ , 1 , � , � ) where L Lattices Contexts and R are sets N ⊆ L × R , Dedekind-McNeille Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37
Residuated frames Residuated lattices Examples Decidability for lattices Boolean algebras A residuated frame is a structure F = ( L, R, N, ◦ , 1 , � , � ) where L Lattices Contexts and R are sets N ⊆ L × R , ( L, ◦ , 1) is a monoid and Dedekind-McNeille � : L × R → R , � : R × L → R such that Lattice frames Why does it work? Sequents ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37
Residuated frames Residuated lattices Examples Decidability for lattices Boolean algebras A residuated frame is a structure F = ( L, R, N, ◦ , 1 , � , � ) where L Lattices Contexts and R are sets N ⊆ L × R , ( L, ◦ , 1) is a monoid and Dedekind-McNeille � : L × R → R , � : R × L → R such that Lattice frames Why does it work? Sequents ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Bi-modules Formula hierarchy Submodules and nuclei Residuated frames Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37
Residuated frames Residuated lattices Examples Decidability for lattices Boolean algebras A residuated frame is a structure F = ( L, R, N, ◦ , 1 , � , � ) where L Lattices Contexts and R are sets N ⊆ L × R , ( L, ◦ , 1) is a monoid and Dedekind-McNeille � : L × R → R , � : R × L → R such that Lattice frames Why does it work? Sequents ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Bi-modules Formula hierarchy Submodules and nuclei Residuated frames Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . GN FL Gentzen frames Corollary. If F is a residuated frame then the Galois algebra Frame applications F + = P ( L, ◦ , 1) γ N is a residuated lattice. Moreover, for F A , Compl - CE x �→ { x } ⊳ is an embedding. Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37
Residuated frames Residuated lattices Examples Decidability for lattices Boolean algebras A residuated frame is a structure F = ( L, R, N, ◦ , 1 , � , � ) where L Lattices Contexts and R are sets N ⊆ L × R , ( L, ◦ , 1) is a monoid and Dedekind-McNeille � : L × R → R , � : R × L → R such that Lattice frames Why does it work? Sequents ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Bi-modules Formula hierarchy Submodules and nuclei Residuated frames Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . GN FL Gentzen frames Corollary. If F is a residuated frame then the Galois algebra Frame applications F + = P ( L, ◦ , 1) γ N is a residuated lattice. Moreover, for F A , Compl - CE x �→ { x } ⊳ is an embedding. Equations Simple rules Hypersequents If A is a RL, F A = ( A, A, ≤ , · , { 1 } ) is a residuated frame. The Beyond underlying poset of F + A is the Dedekind-MacNeille completion of the underlying poset reduct of A . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37
Residuated frames Residuated lattices Examples Decidability for lattices Boolean algebras A residuated frame is a structure F = ( L, R, N, ◦ , 1 , � , � ) where L Lattices Contexts and R are sets N ⊆ L × R , ( L, ◦ , 1) is a monoid and Dedekind-McNeille � : L × R → R , � : R × L → R such that Lattice frames Why does it work? Sequents ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Bi-modules Formula hierarchy Submodules and nuclei Residuated frames Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . GN FL Gentzen frames Corollary. If F is a residuated frame then the Galois algebra Frame applications F + = P ( L, ◦ , 1) γ N is a residuated lattice. Moreover, for F A , Compl - CE x �→ { x } ⊳ is an embedding. Equations Simple rules Hypersequents If A is a RL, F A = ( A, A, ≤ , · , { 1 } ) is a residuated frame. The Beyond underlying poset of F + A is the Dedekind-MacNeille completion of the underlying poset reduct of A . N. Galatos and P. Jipsen. Residuated frames and applications to decidability, Transactions of the AMS (2013). Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Formula hierarchy xNa ∧ b Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Formula hierarchy xNa ∧ b Submodules and nuclei Residuated frames xNa yNb a ◦ bNz εNz GN a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Formula hierarchy xNa ∧ b Submodules and nuclei Residuated frames xNa yNb a ◦ bNz εNz GN a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) FL Gentzen frames Frame applications xNa � b xNa bNz Compl - CE a \ bNx � z ( \ L) ( \ R) Equations xNa \ b Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Formula hierarchy xNa ∧ b Submodules and nuclei Residuated frames xNa yNb a ◦ bNz εNz GN a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) FL Gentzen frames Frame applications xNa � b xNa bNz Compl - CE a \ bNx � z ( \ L) ( \ R) Equations xNa \ b Simple rules Hypersequents xNb � a xNa bNz Beyond b/aNz � x ( / L) ( / R) xNb/a Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Formula hierarchy xNa ∧ b Submodules and nuclei Residuated frames xNa yNb a ◦ bNz εNz GN a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) FL Gentzen frames Frame applications xNa � b xNa bNz Compl - CE a \ bNx � z ( \ L) ( \ R) Equations xNa \ b Simple rules Hypersequents xNb � a xNa bNz Beyond b/aNz � x ( / L) ( / R) xNb/a xNa bNz x ◦ ( a \ b ) Nz Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Formula hierarchy xNa ∧ b Submodules and nuclei Residuated frames xNa yNb a ◦ bNz εNz GN a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) FL Gentzen frames Frame applications xNa � b xNa bNz Compl - CE a \ bNx � z ( \ L) ( \ R) Equations xNa \ b Simple rules Hypersequents xNb � a xNa bNz Beyond b/aNz � x ( / L) ( / R) xNb/a xNa bN ( v � c � u ) xNa bNz x ◦ ( a \ b ) Nz x ◦ ( a \ b ) N ( v � c � u ) Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Formula hierarchy xNa ∧ b Submodules and nuclei Residuated frames xNa yNb a ◦ bNz εNz GN a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) FL Gentzen frames Frame applications xNa � b xNa bNz Compl - CE a \ bNx � z ( \ L) ( \ R) Equations xNa \ b Simple rules Hypersequents xNb � a xNa bNz Beyond b/aNz � x ( / L) ( / R) xNb/a xNa bN ( v � c � u ) v ◦ b ◦ uNc xNa bNz xNa x ◦ ( a \ b ) Nz x ◦ ( a \ b ) N ( v � c � u ) v ◦ x ◦ ( a \ b ) ◦ uNc Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
GN Residuated lattices Examples Decidability for lattices Boolean algebras xNa aNz Lattices (CUT) aNa (Id) Contexts xNz Dedekind-McNeille Lattice frames aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Why does it work? a ∨ bNz Sequents Bi-modules aNz bNz xNa xNb a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Formula hierarchy xNa ∧ b Submodules and nuclei Residuated frames xNa yNb a ◦ bNz εNz GN a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) FL Gentzen frames Frame applications xNa � b xNa bNz Compl - CE a \ bNx � z ( \ L) ( \ R) Equations xNa \ b Simple rules Hypersequents xNb � a xNa bNz Beyond b/aNz � x ( / L) ( / R) xNb/a xNa bN ( v � c � u ) v ◦ b ◦ uNc xNa bNz xNa x ◦ ( a \ b ) Nz x ◦ ( a \ b ) N ( v � c � u ) v ◦ x ◦ ( a \ b ) ◦ uNc So, we get the sequent calculus FL , for a, b, c ∈ Fm , x, y, u, v ∈ Fm ∗ , z ∈ S L × Fm . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37
FL Residuated lattices Examples Decidability for lattices Boolean algebras x ⇒ a y ◦ a ◦ z ⇒ c Lattices (cut) a ⇒ a (Id) y ◦ x ◦ z ⇒ c Contexts Dedekind-McNeille y ◦ a ◦ z ⇒ c y ◦ b ◦ z ⇒ c Lattice frames x ⇒ a x ⇒ b y ◦ a ∧ b ◦ z ⇒ c ( ∧ L ℓ ) y ◦ a ∧ b ◦ z ⇒ c ( ∧ L r ) ( ∧ R) Why does it work? x ⇒ a ∧ b Sequents Bi-modules y ◦ a ◦ z ⇒ c y ◦ b ◦ z ⇒ c x ⇒ a x ⇒ b Formula hierarchy ( ∨ L) x ⇒ a ∨ b ( ∨ R ℓ ) x ⇒ a ∨ b ( ∨ R r ) Submodules and nuclei y ◦ a ∨ b ◦ z ⇒ c Residuated frames GN x ⇒ a y ◦ b ◦ z ⇒ c a ◦ x ⇒ b FL y ◦ x ◦ ( a \ b ) ◦ z ⇒ c ( \ L) x ⇒ a \ b ( \ R) Gentzen frames Frame applications Compl - CE x ⇒ a y ◦ b ◦ z ⇒ c x ◦ a ⇒ b Equations y ◦ ( b/a ) ◦ x ◦ z ⇒ c ( / L) x ⇒ b/a ( / R) Simple rules Hypersequents y ◦ a ◦ b ◦ z ⇒ c x ⇒ a y ⇒ b Beyond y ◦ a · b ◦ z ⇒ c ( · L) ( · R) x ◦ y ⇒ a · b y ◦ z ⇒ a y ◦ 1 ◦ z ⇒ a ( 1 L) ε ⇒ 1 ( 1 R) where a, b, c ∈ Fm , x, y, z ∈ Fm ∗ . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 17 / 37
Gentzen frames Residuated lattices Examples Decidability for lattices Boolean algebras Given a monoid L = ( L, ◦ , ε ) , S L denotes the sections (unary linear Lattices Contexts polynomials) of L . Dedekind-McNeille Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37
Gentzen frames Residuated lattices Examples Decidability for lattices Boolean algebras Given a monoid L = ( L, ◦ , ε ) , S L denotes the sections (unary linear Lattices Contexts polynomials) of L . We define F FL , where L = Fm ∗ , R = S L × Fm , Dedekind-McNeille and x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37
Gentzen frames Residuated lattices Examples Decidability for lattices Boolean algebras Given a monoid L = ( L, ◦ , ε ) , S L denotes the sections (unary linear Lattices Contexts polynomials) of L . We define F FL , where L = Fm ∗ , R = S L × Fm , Dedekind-McNeille and x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Lattice frames Why does it work? The following properties hold for F A , F FL (and F A , B , later): Sequents Bi-modules Formula hierarchy 1. F is a residuated frame Submodules and nuclei 2. B is a (partial) algebra of the same type, ( B = A , Fm ) Residuated frames GN B generates ( L, ◦ , ε ) (as a monoid) 3. FL Gentzen frames R contains a copy of B ( b ↔ ( id, b ) ) 4. Frame applications Compl - CE N satisfies GN , for all a, b ∈ B , x, y ∈ L , z ∈ R . 5. Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37
Gentzen frames Residuated lattices Examples Decidability for lattices Boolean algebras Given a monoid L = ( L, ◦ , ε ) , S L denotes the sections (unary linear Lattices Contexts polynomials) of L . We define F FL , where L = Fm ∗ , R = S L × Fm , Dedekind-McNeille and x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Lattice frames Why does it work? The following properties hold for F A , F FL (and F A , B , later): Sequents Bi-modules Formula hierarchy 1. F is a residuated frame Submodules and nuclei 2. B is a (partial) algebra of the same type, ( B = A , Fm ) Residuated frames GN B generates ( L, ◦ , ε ) (as a monoid) 3. FL Gentzen frames R contains a copy of B ( b ↔ ( id, b ) ) 4. Frame applications Compl - CE N satisfies GN , for all a, b ∈ B , x, y ∈ L , z ∈ R . 5. Equations Simple rules We call such pairs ( F , B ) Gentzen frames . A cut-free Gentzen frame Hypersequents is not assumed to satisfy the (CUT)-rule. Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37
Gentzen frames Residuated lattices Examples Decidability for lattices Boolean algebras Given a monoid L = ( L, ◦ , ε ) , S L denotes the sections (unary linear Lattices Contexts polynomials) of L . We define F FL , where L = Fm ∗ , R = S L × Fm , Dedekind-McNeille and x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Lattice frames Why does it work? The following properties hold for F A , F FL (and F A , B , later): Sequents Bi-modules Formula hierarchy 1. F is a residuated frame Submodules and nuclei 2. B is a (partial) algebra of the same type, ( B = A , Fm ) Residuated frames GN B generates ( L, ◦ , ε ) (as a monoid) 3. FL Gentzen frames R contains a copy of B ( b ↔ ( id, b ) ) 4. Frame applications Compl - CE N satisfies GN , for all a, b ∈ B , x, y ∈ L , z ∈ R . 5. Equations Simple rules We call such pairs ( F , B ) Gentzen frames . A cut-free Gentzen frame Hypersequents is not assumed to satisfy the (CUT)-rule. Beyond Theorem. (NG-Jipsen) Given a Gentzen frame ( F , B ) , the map {} ⊳ : B → F + , 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 } ⊳ ). Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37
Gentzen frames Residuated lattices Examples Decidability for lattices Boolean algebras Given a monoid L = ( L, ◦ , ε ) , S L denotes the sections (unary linear Lattices Contexts polynomials) of L . We define F FL , where L = Fm ∗ , R = S L × Fm , Dedekind-McNeille and x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Lattice frames Why does it work? The following properties hold for F A , F FL (and F A , B , later): Sequents Bi-modules Formula hierarchy 1. F is a residuated frame Submodules and nuclei 2. B is a (partial) algebra of the same type, ( B = A , Fm ) Residuated frames GN B generates ( L, ◦ , ε ) (as a monoid) 3. FL Gentzen frames R contains a copy of B ( b ↔ ( id, b ) ) 4. Frame applications Compl - CE N satisfies GN , for all a, b ∈ B , x, y ∈ L , z ∈ R . 5. Equations Simple rules We call such pairs ( F , B ) Gentzen frames . A cut-free Gentzen frame Hypersequents is not assumed to satisfy the (CUT)-rule. Beyond Theorem. (NG-Jipsen) Given a Gentzen frame ( F , B ) , the map {} ⊳ : B → F + , 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 } ⊳ ). For cut-free Genzten frames, we get only a quasihomomorphism . a • B b ∈ { a } ⊳ • F + { b } ⊳ ⊆ { a • B b } ⊳ . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37
Frame applications Residuated lattices Examples Decidability for lattices Boolean algebras DM-completion ■ Lattices Completeness of the calculus ■ Contexts Dedekind-McNeille Cut elimination ■ Lattice frames Finite model property ■ Why does it work? Finite embeddability property ■ Sequents Bi-modules (Generalized super-)amalgamation property (Transferable ■ Formula hierarchy injections, Congruence extension property) Submodules and nuclei Residuated frames (Craig) Interpolation property ■ GN Disjunction property FL ■ Gentzen frames Strong separation ■ Frame applications Stability under linear structural rules/equations over {∨ , · , 1 } . Compl - CE ■ Equations Simple rules NG and H. Ono, APAL. Hypersequents NG and P. Jipsen, TAMS. Beyond NG and P. Jipsen, manuscript. A. Ciabattoni, NG and K. Terui, APAL. NG and K. Terui, manuscript. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 19 / 37
Completeness - Cut elimination Residuated lattices Examples Decidability for lattices Boolean algebras Key Lemma. Let ( F , B ) be a Gentzen frame. For all a, b ∈ B , Lattices k, l ∈ F + and for every connective • , if a • b ∈ B , a ∈ X ⊆ { a } ⊳ Contexts Dedekind-McNeille and b ∈ Y ⊆ { b } ⊳ , then Lattice frames Why does it work? a • B b ∈ X • F + Y ⊆ { a • B b } ⊳ ( 1 B ∈ 1 F + ⊆ { 1 B } ⊳ ) 1. Sequents In particular, a • B b ∈ { a } ⊳ • F + { b } ⊳ ⊆ { a • B b } ⊳ . Bi-modules 2. Formula hierarchy 3. Furthermore, because of (CUT), we have equality. Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37
Completeness - Cut elimination Residuated lattices Examples Decidability for lattices Boolean algebras Key Lemma. Let ( F , B ) be a Gentzen frame. For all a, b ∈ B , Lattices k, l ∈ F + and for every connective • , if a • b ∈ B , a ∈ X ⊆ { a } ⊳ Contexts Dedekind-McNeille and b ∈ Y ⊆ { b } ⊳ , then Lattice frames Why does it work? a • B b ∈ X • F + Y ⊆ { a • B b } ⊳ ( 1 B ∈ 1 F + ⊆ { 1 B } ⊳ ) 1. Sequents In particular, a • B b ∈ { a } ⊳ • F + { b } ⊳ ⊆ { a • B b } ⊳ . Bi-modules 2. Formula hierarchy 3. Furthermore, because of (CUT), we have equality. Submodules and nuclei Residuated frames GN f : Fm L → F + be the For every homomorphism f : Fm → B , let ¯ FL Gentzen frames f ( p ) = { f ( p ) } ⊳ ( p : variable.) homomorphism that extends ¯ Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37
Completeness - Cut elimination Residuated lattices Examples Decidability for lattices Boolean algebras Key Lemma. Let ( F , B ) be a Gentzen frame. For all a, b ∈ B , Lattices k, l ∈ F + and for every connective • , if a • b ∈ B , a ∈ X ⊆ { a } ⊳ Contexts Dedekind-McNeille and b ∈ Y ⊆ { b } ⊳ , then Lattice frames Why does it work? a • B b ∈ X • F + Y ⊆ { a • B b } ⊳ ( 1 B ∈ 1 F + ⊆ { 1 B } ⊳ ) 1. Sequents In particular, a • B b ∈ { a } ⊳ • F + { b } ⊳ ⊆ { a • B b } ⊳ . Bi-modules 2. Formula hierarchy 3. Furthermore, because of (CUT), we have equality. Submodules and nuclei Residuated frames GN f : Fm L → F + be the For every homomorphism f : Fm → B , let ¯ FL Gentzen frames f ( p ) = { f ( p ) } ⊳ ( p : variable.) homomorphism that extends ¯ Frame applications Compl - CE Corollary. If ( F , B ) is a cf Gentzen frame, for every homomorphism Equations Simple rules f : Fm → B , we have f ( a ) ∈ ¯ f ( a ) ⊆ { f ( a ) } ⊳ . If we have (CUT), Hypersequents then ¯ f ( a ) = ↓ f ( a ) . Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37
Completeness - Cut elimination Residuated lattices Examples Decidability for lattices Boolean algebras Key Lemma. Let ( F , B ) be a Gentzen frame. For all a, b ∈ B , Lattices k, l ∈ F + and for every connective • , if a • b ∈ B , a ∈ X ⊆ { a } ⊳ Contexts Dedekind-McNeille and b ∈ Y ⊆ { b } ⊳ , then Lattice frames Why does it work? a • B b ∈ X • F + Y ⊆ { a • B b } ⊳ ( 1 B ∈ 1 F + ⊆ { 1 B } ⊳ ) 1. Sequents In particular, a • B b ∈ { a } ⊳ • F + { b } ⊳ ⊆ { a • B b } ⊳ . Bi-modules 2. Formula hierarchy 3. Furthermore, because of (CUT), we have equality. Submodules and nuclei Residuated frames GN f : Fm L → F + be the For every homomorphism f : Fm → B , let ¯ FL Gentzen frames f ( p ) = { f ( p ) } ⊳ ( p : variable.) homomorphism that extends ¯ Frame applications Compl - CE Corollary. If ( F , B ) is a cf Gentzen frame, for every homomorphism Equations Simple rules f : Fm → B , we have f ( a ) ∈ ¯ f ( a ) ⊆ { f ( a ) } ⊳ . If we have (CUT), Hypersequents then ¯ f ( a ) = ↓ f ( a ) . Beyond We define F | = x ⇒ c by f ( x ) N f ( c ) , for all f . = x · ≤ c , then F FL | Theorem. If F + FL | = x ⇒ c . Idea: For f : Fm → B , f ( x ) ∈ ¯ f ( x ) ⊆ ¯ f ( c ) ⊆ { f ( c ) } ⊳ , so f ( x ) N f ( c ) . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37
Completeness - Cut elimination Residuated lattices Examples Decidability for lattices Boolean algebras Key Lemma. Let ( F , B ) be a Gentzen frame. For all a, b ∈ B , Lattices k, l ∈ F + and for every connective • , if a • b ∈ B , a ∈ X ⊆ { a } ⊳ Contexts Dedekind-McNeille and b ∈ Y ⊆ { b } ⊳ , then Lattice frames Why does it work? a • B b ∈ X • F + Y ⊆ { a • B b } ⊳ ( 1 B ∈ 1 F + ⊆ { 1 B } ⊳ ) 1. Sequents In particular, a • B b ∈ { a } ⊳ • F + { b } ⊳ ⊆ { a • B b } ⊳ . Bi-modules 2. Formula hierarchy 3. Furthermore, because of (CUT), we have equality. Submodules and nuclei Residuated frames GN f : Fm L → F + be the For every homomorphism f : Fm → B , let ¯ FL Gentzen frames f ( p ) = { f ( p ) } ⊳ ( p : variable.) homomorphism that extends ¯ Frame applications Compl - CE Corollary. If ( F , B ) is a cf Gentzen frame, for every homomorphism Equations Simple rules f : Fm → B , we have f ( a ) ∈ ¯ f ( a ) ⊆ { f ( a ) } ⊳ . If we have (CUT), Hypersequents then ¯ f ( a ) = ↓ f ( a ) . Beyond We define F | = x ⇒ c by f ( x ) N f ( c ) , for all f . = x · ≤ c , then F FL | Theorem. If F + 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 F + FL . Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37
Completeness - Cut elimination Residuated lattices Examples Decidability for lattices Boolean algebras Key Lemma. Let ( F , B ) be a Gentzen frame. For all a, b ∈ B , Lattices k, l ∈ F + and for every connective • , if a • b ∈ B , a ∈ X ⊆ { a } ⊳ Contexts Dedekind-McNeille and b ∈ Y ⊆ { b } ⊳ , then Lattice frames Why does it work? a • B b ∈ X • F + Y ⊆ { a • B b } ⊳ ( 1 B ∈ 1 F + ⊆ { 1 B } ⊳ ) 1. Sequents In particular, a • B b ∈ { a } ⊳ • F + { b } ⊳ ⊆ { a • B b } ⊳ . Bi-modules 2. Formula hierarchy 3. Furthermore, because of (CUT), we have equality. Submodules and nuclei Residuated frames GN f : Fm L → F + be the For every homomorphism f : Fm → B , let ¯ FL Gentzen frames f ( p ) = { f ( p ) } ⊳ ( p : variable.) homomorphism that extends ¯ Frame applications Compl - CE Corollary. If ( F , B ) is a cf Gentzen frame, for every homomorphism Equations Simple rules f : Fm → B , we have f ( a ) ∈ ¯ f ( a ) ⊆ { f ( a ) } ⊳ . If we have (CUT), Hypersequents then ¯ f ( a ) = ↓ f ( a ) . Beyond We define F | = x ⇒ c by f ( x ) N f ( c ) , for all f . = x · ≤ c , then F FL | Theorem. If F + 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 F + FL . Corollary (CE). FL and FL f prove the same sequents. Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . Bi-modules Formula hierarchy Submodules and nuclei Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . Bi-modules Formula hierarchy Submodules and nuclei The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Residuated frames GN FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . Bi-modules Formula hierarchy Submodules and nuclei The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Residuated frames GN We proceed by example: x 2 y ≤ xy ∨ yx FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . Bi-modules Formula hierarchy Submodules and nuclei The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Residuated frames GN We proceed by example: x 2 y ≤ xy ∨ yx FL Gentzen frames Frame applications ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Compl - CE Equations Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . Bi-modules Formula hierarchy Submodules and nuclei The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Residuated frames GN We proceed by example: x 2 y ≤ xy ∨ yx FL Gentzen frames Frame applications ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Compl - CE Equations 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 Simple rules Hypersequents Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . Bi-modules Formula hierarchy Submodules and nuclei The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Residuated frames GN We proceed by example: x 2 y ≤ xy ∨ yx FL Gentzen frames Frame applications ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Compl - CE Equations 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 Simple rules Hypersequents x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 Beyond Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . Bi-modules Formula hierarchy Submodules and nuclei The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Residuated frames GN We proceed by example: x 2 y ≤ xy ∨ yx FL Gentzen frames Frame applications ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Compl - CE Equations 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 Simple rules Hypersequents x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 Beyond x 1 y ≤ z x 2 y ≤ z yx 1 ≤ z yx 2 ≤ z x 1 x 2 y ≤ z Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Equations Residuated lattices Examples Decidability for lattices Boolean algebras Idea: Express equations over {∨ , · , 1 } at the frame level. Lattices Contexts Dedekind-McNeille For an equation ε over {∨ , · , 1 } we distribute products over joins to Lattice frames get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Why does it work? Sequents s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . Bi-modules Formula hierarchy Submodules and nuclei The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Residuated frames GN We proceed by example: x 2 y ≤ xy ∨ yx FL Gentzen frames Frame applications ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Compl - CE Equations 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 Simple rules Hypersequents x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 Beyond x 1 y ≤ z x 2 y ≤ z yx 1 ≤ z yx 2 ≤ z x 1 x 2 y ≤ z 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 Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37
Recommend
More recommend
