Hyper-residuated frames Nick Galatos University of Denver - - PowerPoint PPT Presentation

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 1 / 37

Hyper-residuated frames

Nick Galatos University of Denver ngalatos@du.edu

April 19, 2013

Residuated lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 2 / 37

A residuated lattice, or residuated lattice-ordered monoid, [Blount and Tsinakis] is an algebra A = (A, ∧, ∨, ·, \, /, 1) such that

■

(A, ∧, ∨) is a lattice,

■

(A, ·, 1) is a monoid and

■

for all a, b, c ∈ A, ab ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b.

Residuated lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 2 / 37

A residuated lattice, or residuated lattice-ordered monoid, [Blount and Tsinakis] is an algebra A = (A, ∧, ∨, ·, \, /, 1) such that

■

(A, ∧, ∨) is a lattice,

■

(A, ·, 1) is a monoid and

■

for all a, b, c ∈ A, ab ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b.

• Fact. The last condition is equivalent to either one of:

■

Multiplication distributes over existing ’s and, for all a, c ∈ A, both {b : ab ≤ c} (=: a\c) and {b : ba ≤ c} (=: c/a) exist.

Residuated lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 2 / 37

A residuated lattice, or residuated lattice-ordered monoid, [Blount and Tsinakis] is an algebra A = (A, ∧, ∨, ·, \, /, 1) such that

■

(A, ∧, ∨) is a lattice,

■

(A, ·, 1) is a monoid and

■

for all a, b, c ∈ A, ab ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b.

• Fact. The last condition is equivalent to either one of:

■

Multiplication distributes over existing ’s and, for all a, c ∈ A, both {b : ab ≤ c} (=: a\c) and {b : ba ≤ c} (=: c/a) exist.

■

(For complete lattices) · distributes over . [Quantales]

Residuated lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 2 / 37

A residuated lattice, or residuated lattice-ordered monoid, [Blount and Tsinakis] is an algebra A = (A, ∧, ∨, ·, \, /, 1) such that

■

(A, ∧, ∨) is a lattice,

■

(A, ·, 1) is a monoid and

■

for all a, b, c ∈ A, ab ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b.

• Fact. The last condition is equivalent to either one of:

■

Multiplication distributes over existing ’s and, for all a, c ∈ A, both {b : ab ≤ c} (=: a\c) and {b : ba ≤ c} (=: c/a) exist.

■

(For complete lattices) · distributes over . [Quantales]

■

For all a, b, c ∈ A, b ≤ a\(ab ∨ c) a ≤ (c ∨ ab)/b a(a\c ∧ b) ≤ c (a ∧ c/b)b ≤ c

Residuated lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 2 / 37

A residuated lattice, or residuated lattice-ordered monoid, [Blount and Tsinakis] is an algebra A = (A, ∧, ∨, ·, \, /, 1) such that

■

(A, ∧, ∨) is a lattice,

■

(A, ·, 1) is a monoid and

■

for all a, b, c ∈ A, ab ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b.

• Fact. The last condition is equivalent to either one of:

■

Multiplication distributes over existing ’s and, for all a, c ∈ A, both {b : ab ≤ c} (=: a\c) and {b : ba ≤ c} (=: c/a) exist.

■

(For complete lattices) · distributes over . [Quantales]

■

For all a, b, c ∈ A, b ≤ a\(ab ∨ c) a ≤ (c ∨ ab)/b a(a\c ∧ b) ≤ c (a ∧ c/b)b ≤ c So, residuated lattices form an equational class/variety.

Examples

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 3 / 37

■

Boolean algebras. x/y = y\x = y → x = ¬y ∨ x and x · y = x ∧ y.

■

Lattice-ordered groups. For x\y = x−1y, y/x = yx−1; ¬x = x−1.

■

(Reducts of) relation algebras. For x · y = x; y, x\y = (x∪; yc)c, y/x = (yc; x∪)c, 1 = id.

■

Ideals of a ring (with 1), where IJ = {

fin ij | i ∈ I, j ∈ J}

I/J = {k | kJ ⊆ I}, J\I = {k | Jk ⊆ I}, 1 = R.

■

Quantales are (definitionally equivalent) complete residuated lattices.

■

MV-algebras. For x · y = x ⊙ y and x\y = y/x = ¬(¬x ⊙ y).

■

The powerset (P(M), ∩, ∪, ·, \, /, {e}) of a monoid M = (M, ·, e), where X · Y = {x · y | x ∈ X, y ∈ Y }, X/Y = {z ∈ M | {z} · Y ⊆ X}, Y \X = {z ∈ M | Y · {z} ⊆ X}.

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c a ≤ c a ∧ b ≤ c b ≤ c a ∧ b ≤ c c ≤ a c ≤ b c ≤ a ∧ b

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c a ≤ c a ∧ b ≤ c b ≤ c a ∧ b ≤ c c ≤ a c ≤ b c ≤ a ∧ b c ≤ a c ≤ a ∨ b c ≤ b c ≤ a ∨ b a ≤ c b ≤ c a ∨ b ≤ c

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c a ≤ c a ∧ b ≤ c b ≤ c a ∧ b ≤ c c ≤ a c ≤ b c ≤ a ∧ b c ≤ a c ≤ a ∨ b c ≤ b c ≤ a ∨ b a ≤ c b ≤ c a ∨ b ≤ c Cut Eimination Theorem. (Whitman’s conditions, also Skolem) Transitivity (aka cut) is not needed.

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c a ≤ c a ∧ b ≤ c b ≤ c a ∧ b ≤ c c ≤ a c ≤ b c ≤ a ∧ b c ≤ a c ≤ a ∨ b c ≤ b c ≤ a ∨ b a ≤ c b ≤ c 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.

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c a ≤ c a ∧ b ≤ c b ≤ c a ∧ b ≤ c c ≤ a c ≤ b c ≤ a ∧ b c ≤ a c ≤ a ∨ b c ≤ b c ≤ a ∨ b a ≤ c b ≤ c 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).

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c a ≤ c a ∧ b ≤ c b ≤ c a ∧ b ≤ c c ≤ a c ≤ b c ≤ a ∧ b c ≤ a c ≤ a ∨ b c ≤ b c ≤ a ∨ b a ≤ c b ≤ c 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).

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c a ≤ c a ∧ b ≤ c b ≤ c a ∧ b ≤ c c ≤ a c ≤ b c ≤ a ∧ b c ≤ a c ≤ a ∨ b c ≤ b c ≤ a ∨ b a ≤ c b ≤ c 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.

Decidability for lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 4 / 37

a ≤ a a ≤ b b ≤ a a = b a ≤ b b ≤ c a ≤ c a ≤ c a ∧ b ≤ c b ≤ c a ∧ b ≤ c c ≤ a c ≤ b c ≤ a ∧ b c ≤ a c ≤ a ∨ b c ≤ b c ≤ a ∨ b a ≤ c b ≤ c 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.

Boolean algebras

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 5 / 37

{a, b, c} {a, b} {a, c} {b, c} {a} {b} {c} ∅ c b a c a b c b a

Every finite distributive lattice L can be recovered from its poset J(L) of join irreducibless; D ∼ = D(J(L)).

Boolean algebras

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 5 / 37

{a, b, c} {a, b} {a, c} {b, c} {a} {b} {c} ∅ c b a c a b c b a

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).

Boolean algebras

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 5 / 37

{a, b, c} {a, b} {a, c} {b, c} {a} {b} {c} ∅ c b a c a b c b a

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.

Lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 6 / 37

a b c 1 a b c c b a

For general (non-distributive) lattices, the poset of join irreducibles is not enough to recover the lattice.

Lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 6 / 37

a b c 1 a b c c b a

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).

Lattices

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 6 / 37

a b c 1 a b c c b a

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).

d c a b c b a d c a

Contexts

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 7 / 37

c′ b′ a′ a b c c′ b′ a′ a b c ≤ a′ b′ c′ a × × b × × c × × 1 a b c a b c a b c ≤ a′ b′ c′ a × b × c ×

Dedekind-McNeille

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37

d c a b a d c a b c

≤ a d c a × × b × × c × We obtain an oriented bipartite graph; an algebraic rendering of sequents!

Dedekind-McNeille

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37

d c a b a d c a b c

≤ 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?

Dedekind-McNeille

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37

d c a b a d c a b c

≤ 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.

Dedekind-McNeille

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37

d c a b a d c a b c

≤ 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.

Dedekind-McNeille

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 8 / 37

d c a b a d c a b c

≤ 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).

Lattice frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37

A lattice frame is a structure F = (L, R, N) where L and R are sets and N is a binary relation from L to R.

Lattice frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37

A lattice frame is a structure F = (L, R, N) where L and R are sets and N is a binary relation from L to R. For X ⊆ L and Y ⊆ R we define X⊲ = {b ∈ R : x N b, for all x ∈ X} Y ⊳ = {a ∈ L : a N y, for all y ∈ Y }

Lattice frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37

A lattice frame is a structure F = (L, R, N) where L and R are sets and N is a binary relation from L to R. For X ⊆ L and Y ⊆ R we define X⊲ = {b ∈ R : x N b, for all x ∈ X} Y ⊳ = {a ∈ L : a N y, for all y ∈ Y } The maps ⊲ : P(L) → P(R) and ⊳ : P(R) → P(L) form a Galois

• connection. The map γN : P(L) → P(L), where γN(X) = X⊲⊳, is

a closure operator.

Lattice frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37

A lattice frame is a structure F = (L, R, N) where L and R are sets and N is a binary relation from L to R. For X ⊆ L and Y ⊆ R we define X⊲ = {b ∈ R : x N b, for all x ∈ X} Y ⊳ = {a ∈ L : a N y, for all y ∈ Y } The maps ⊲ : P(L) → P(R) and ⊳ : P(R) → P(L) form a Galois

• connection. The map γN : P(L) → P(L), where γN(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).]

Lattice frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37

A lattice frame is a structure F = (L, R, N) where L and R are sets and N is a binary relation from L to R. For X ⊆ L and Y ⊆ R we define X⊲ = {b ∈ R : x N b, for all x ∈ X} Y ⊳ = {a ∈ L : a N y, for all y ∈ Y } The maps ⊲ : P(L) → P(R) and ⊳ : P(R) → P(L) form a Galois

• connection. The map γN : P(L) → P(L), where γN(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+ = (γN[P(L)], ∩, ∪γN) is a complete lattice.

Lattice frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 9 / 37

A lattice frame is a structure F = (L, R, N) where L and R are sets and N is a binary relation from L to R. For X ⊆ L and Y ⊆ R we define X⊲ = {b ∈ R : x N b, for all x ∈ X} Y ⊳ = {a ∈ L : a N y, for all y ∈ Y } The maps ⊲ : P(L) → P(R) and ⊳ : P(R) → P(L) form a Galois

• connection. The map γN : P(L) → P(L), where γN(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+ = (γN[P(L)], ∩, ∪γN) is a complete lattice. If A is a lattice, FA = (A, A, ≤) is a lattice frame. Also, F+

A is the

Dedekind-MacNeille completion of A and x → {x}⊳ is an embedding.

Why does it work?

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37

Enough to represent complete lattices ≡ complete-join semilattices L = (L, ) (they are also meet complete).

Why does it work?

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37

Enough to represent complete lattices ≡ complete-join semilattices L = (L, ) (they are also meet complete). The free objects in this category are (P(X), ), so for every L, there is an X and an onto

• homomorphism f : (P(X), ) → L.

Why does it work?

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37

Enough to represent complete lattices ≡ complete-join semilattices L = (L, ) (they are also meet complete). The free objects in this category are (P(X), ), so for every L, there is an X and an onto

• homomorphism f : (P(X), ) → L. Every such map is

residuated: there is f ∗ : L → P(X) such that ∀x ∈ P(X) and y ∈ L, f(x) ≤ y ⇔ x ⊆ f ∗(y).

Why does it work?

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37

Enough to represent complete lattices ≡ complete-join semilattices L = (L, ) (they are also meet complete). The free objects in this category are (P(X), ), so for every L, there is an X and an onto

• homomorphism f : (P(X), ) → L. Every such map is

residuated: there is f ∗ : L → P(X) such that ∀x ∈ P(X) and y ∈ L, f(x) ≤ y ⇔ x ⊆ f ∗(y). The composition γ = f ∗f is a closure operator on P(X), (P(X)γ,

γ) is a lattice ( γ Ai = γ( Ai)), and the map f factors

f : (P(X), )

γ

→ (P(X)γ,

γ) f|P(X)γ

→ L, where the first is surjective and the second injective.

Why does it work?

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37

Enough to represent complete lattices ≡ complete-join semilattices L = (L, ) (they are also meet complete). The free objects in this category are (P(X), ), so for every L, there is an X and an onto

• homomorphism f : (P(X), ) → L. Every such map is

residuated: there is f ∗ : L → P(X) such that ∀x ∈ P(X) and y ∈ L, f(x) ≤ y ⇔ x ⊆ f ∗(y). The composition γ = f ∗f is a closure operator on P(X), (P(X)γ,

γ) is a lattice ( γ Ai = γ( Ai)), and the map f factors

f : (P(X), )

γ

→ (P(X)γ,

γ) f|P(X)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), ).

Why does it work?

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37

Enough to represent complete lattices ≡ complete-join semilattices L = (L, ) (they are also meet complete). The free objects in this category are (P(X), ), so for every L, there is an X and an onto

• homomorphism f : (P(X), ) → L. Every such map is

residuated: there is f ∗ : L → P(X) such that ∀x ∈ P(X) and y ∈ L, f(x) ≤ y ⇔ x ⊆ f ∗(y). The composition γ = f ∗f is a closure operator on P(X), (P(X)γ,

γ) is a lattice ( γ Ai = γ( Ai)), and the map f factors

f : (P(X), )

γ

→ (P(X)γ,

γ) f|P(X)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 γN, for N given by x N y ⇔ y ∈ γ({x}); namely, it comes from a lattice frame (X, Y, N).

Why does it work?

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37

Enough to represent complete lattices ≡ complete-join semilattices L = (L, ) (they are also meet complete). The free objects in this category are (P(X), ), so for every L, there is an X and an onto

• homomorphism f : (P(X), ) → L. Every such map is

residuated: there is f ∗ : L → P(X) such that ∀x ∈ P(X) and y ∈ L, f(x) ≤ y ⇔ x ⊆ f ∗(y). The composition γ = f ∗f is a closure operator on P(X), (P(X)γ,

γ) is a lattice ( γ Ai = γ( Ai)), and the map f factors

f : (P(X), )

γ

→ (P(X)γ,

γ) f|P(X)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 γN, 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!

Why does it work?

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 10 / 37

Enough to represent complete lattices ≡ complete-join semilattices L = (L, ) (they are also meet complete). The free objects in this category are (P(X), ), so for every L, there is an X and an onto

• homomorphism f : (P(X), ) → L. Every such map is

residuated: there is f ∗ : L → P(X) such that ∀x ∈ P(X) and y ∈ L, f(x) ≤ y ⇔ x ⊆ f ∗(y). The composition γ = f ∗f is a closure operator on P(X), (P(X)γ,

γ) is a lattice ( γ Ai = γ( Ai)), and the map f factors

f : (P(X), )

γ

→ (P(X)γ,

γ) f|P(X)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 γN, 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!

Sequents

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 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 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

Let’s assume that P = N is the underlying set of a residuated lattice.

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx So, (P, ∨, ·, 1) is a semiring. [In the complete case, a quantale.]

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx So, (P, ∨, ·, 1) is a semiring. [In the complete case, a quantale.]

■

x\(y ∧ z) = (x\y) ∧ (x\z) and (y ∧ z)/x = (y/x) ∧ (z/x)

■

(y ∨ z)\x = (y\x) ∧ (z\x) and x/(y ∨ z) = (x/y) ∧ (x/z)

■

x\(y/z) = (x\y)/z

■

1\x = x = x/1

■

(yz)\x = z\(y\x) and x/(zy) = (x/y)/z

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx So, (P, ∨, ·, 1) is a semiring. [In the complete case, a quantale.]

■

x\(y ∧ z) = (x\y) ∧ (x\z) and (y ∧ z)/x = (y/x) ∧ (z/x)

■

(y ∨ z)\x = (y\x) ∧ (z\x) and x/(y ∨ z) = (x/y) ∧ (x/z)

■

x\(y/z) = (x\y)/z

■

1\x = x = x/1

■

(yz)\x = z\(y\x) and x/(zy) = (x/y)/z So, (P, ∨, ·, 1) acts on both sides on (N, ∧) by p ⋆ n = n/p and n ⋆ p = p\n. Thus, ((N, ∧), ⋆) becomes a (P, ∨, ·, 1)-bimodule.

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx So, (P, ∨, ·, 1) is a semiring. [In the complete case, a quantale.]

■

x\(y ∧ z) = (x\y) ∧ (x\z) and (y ∧ z)/x = (y/x) ∧ (z/x)

■

(y ∨ z)\x = (y\x) ∧ (z\x) and x/(y ∨ z) = (x/y) ∧ (x/z)

■

x\(y/z) = (x\y)/z

■

1\x = x = x/1

■

(yz)\x = z\(y\x) and x/(zy) = (x/y)/z So, (P, ∨, ·, 1) acts on both sides on (N, ∧) by p ⋆ n = n/p and n ⋆ p = p\n. Thus, ((N, ∧), ⋆) becomes a (P, ∨, ·, 1)-bimodule. This split matches the notion of polarity.

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx So, (P, ∨, ·, 1) is a semiring. [In the complete case, a quantale.]

■

x\(y ∧ z) = (x\y) ∧ (x\z) and (y ∧ z)/x = (y/x) ∧ (z/x)

■

(y ∨ z)\x = (y\x) ∧ (z\x) and x/(y ∨ z) = (x/y) ∧ (x/z)

■

x\(y/z) = (x\y)/z

■

1\x = x = x/1

■

(yz)\x = z\(y\x) and x/(zy) = (x/y)/z So, (P, ∨, ·, 1) acts on both sides on (N, ∧) by p ⋆ n = n/p and n ⋆ p = p\n. Thus, ((N, ∧), ⋆) becomes a (P, ∨, ·, 1)-bimodule. This split matches the notion of polarity. It also extend to , .

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx So, (P, ∨, ·, 1) is a semiring. [In the complete case, a quantale.]

■

x\(y ∧ z) = (x\y) ∧ (x\z) and (y ∧ z)/x = (y/x) ∧ (z/x)

■

(y ∨ z)\x = (y\x) ∧ (z\x) and x/(y ∨ z) = (x/y) ∧ (x/z)

■

x\(y/z) = (x\y)/z

■

1\x = x = x/1

■

(yz)\x = z\(y\x) and x/(zy) = (x/y)/z So, (P, ∨, ·, 1) acts on both sides on (N, ∧) by p ⋆ n = n/p and 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, ∧, \, /).

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx So, (P, ∨, ·, 1) is a semiring. [In the complete case, a quantale.]

■

x\(y ∧ z) = (x\y) ∧ (x\z) and (y ∧ z)/x = (y/x) ∧ (z/x)

■

(y ∨ z)\x = (y\x) ∧ (z\x) and x/(y ∨ z) = (x/y) ∧ (x/z)

■

x\(y/z) = (x\y)/z

■

1\x = x = x/1

■

(yz)\x = z\(y\x) and x/(zy) = (x/y)/z So, (P, ∨, ·, 1) acts on both sides on (N, ∧) by p ⋆ n = n/p and 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 P0 = N0 = V ar (the set of propositional variables) gives the set of all formulas.

Bi-modules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 12 / 37

Let’s assume that P = N is the underlying set of a residuated lattice.

■

x · 1 = x = 1 · x, (xy)z = x(yz)

■

x(y ∨ z) = xy ∨ xz and (y ∨ z)x = yx ∨ zx So, (P, ∨, ·, 1) is a semiring. [In the complete case, a quantale.]

■

x\(y ∧ z) = (x\y) ∧ (x\z) and (y ∧ z)/x = (y/x) ∧ (z/x)

■

(y ∨ z)\x = (y\x) ∧ (z\x) and x/(y ∨ z) = (x/y) ∧ (x/z)

■

x\(y/z) = (x\y)/z

■

1\x = x = x/1

■

(yz)\x = z\(y\x) and x/(zy) = (x/y)/z So, (P, ∨, ·, 1) acts on both sides on (N, ∧) by p ⋆ n = n/p and 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 P0 = N0 = V ar (the set of propositional variables) gives the set of all formulas. The steps of the generation process yield the substructural hierarchy.

Formula hierarchy

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 13 / 37

P3 N3 P2 N2 P1 N1 P0 N0

♣♣♣♣♣♣♣♣♣✻ ♣♣♣♣♣♣♣♣♣✻ ✻

• ✒ ✻

❅ ❅ ❅ ❅ ■ ✻

• ✒ ✻

❅ ❅ ❅ ❅ ■ ✻

• ✒ ✻

❅ ❅ ❅ ❅ ■

■

The sets Pn, Nn of formulas are defined by: (0) P0 = N0 = the set of variables (P1) Nn ⊆ Pn+1 (P2) α, β ∈ Pn+1 ⇒ α ∨ β, α · β, 1 ∈ Pn+1 (N1) Pn ⊆ Nn+1 (N2) α, β ∈ Nn+1 ⇒ α ∧ β ∈ Nn+1 (N3) α ∈ Pn+1, β ∈ Nn+1 ⇒ α\β, β/α, 0 ∈ Nn+1

■

Pn+1 = Nn, ; Nn+1 = Pn,Pn+1\,/Pn+1

■

Pn ⊆ Pn+1, Nn ⊆ Nn+1, Pn = Nn = Fm

■

P1-reduced: pi

■

N1-reduced: (p1p2 · · · pn\r/q1q2 · · · qm) p1p2 · · · pnq1q2 · · · qm ≤ r

■

Sequent: a1, a2, . . . , an ⇒ a0 (ai ∈ Fm)

• A. Ciabattoni, NG, K. Terui. From axioms to analytic rules in

nonclassical logics, Proceedings of LICS’08, 229-240, 2008.

Submodules and nuclei

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37

Given a (P, , ·, 1)-bimodule ((N, ), \, /), each sub-bimodule is defined by a -closed subset that is also closed under the actions.

Submodules and nuclei

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37

Given a (P, , ·, 1)-bimodule ((N, ), \, /), each sub-bimodule is defined by a -closed subset that is also closed under the actions. Namely, it is defined by a nucleus: a closure operator γ on N such that p ∈ P, n ∈ N implies p\n, n/p ∈ N.

Submodules and nuclei

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37

Given a (P, , ·, 1)-bimodule ((N, ), \, /), each sub-bimodule is defined by a -closed subset that is also closed under the actions. Namely, it is defined by a nucleus: a closure operator γ on N such that p ∈ P, n ∈ N implies p\n, n/p ∈ N. If P = N is the underlying set of a residuated lattice A = (A, ∧, ∨, ·, \, /, 1), a nucleus is just a closure operator that satisfies γ(x) · γ(y) ≤ γ(x · y). (Cf. phase spaces.)

Submodules and nuclei

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37

Given a (P, , ·, 1)-bimodule ((N, ), \, /), each sub-bimodule is defined by a -closed subset that is also closed under the actions. Namely, it is defined by a nucleus: a closure operator γ on N such that p ∈ P, n ∈ N implies p\n, n/p ∈ N. If P = N is the underlying set of a residuated lattice A = (A, ∧, ∨, ·, \, /, 1), a nucleus is just a closure operator that satisfies γ(x) · γ(y) ≤ γ(x · y). (Cf. phase spaces.) If we define Aγ = {γ(x) : x ∈ A}, x ∨γ y = γ(x ∨ y) and x ·γ y = γ(x · y), Aγ = (Aγ, ∧, ∨γ, ·γ, \, /, γ(1)) is also a residuated lattice.

Submodules and nuclei

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 14 / 37

Given a (P, , ·, 1)-bimodule ((N, ), \, /), each sub-bimodule is defined by a -closed subset that is also closed under the actions. Namely, it is defined by a nucleus: a closure operator γ on N such that p ∈ P, n ∈ N implies p\n, n/p ∈ N. If P = N is the underlying set of a residuated lattice A = (A, ∧, ∨, ·, \, /, 1), a nucleus is just a closure operator that satisfies γ(x) · γ(y) ≤ γ(x · y). (Cf. phase spaces.) If we define Aγ = {γ(x) : x ∈ A}, x ∨γ y = γ(x ∨ y) and x ·γ y = γ(x · y), Aγ = (Aγ, ∧, ∨γ, ·γ, \, /, γ(1)) 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).

Residuated frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37

A residuated frame is a structure F = (L, R, N, ◦, 1, , )

Residuated frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37

A residuated frame is a structure F = (L, R, N, ◦, 1, , ) where L and R are sets N ⊆ L × R,

Residuated frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37

A residuated frame is a structure F = (L, R, N, ◦, 1, , ) where L and R are sets N ⊆ L × R, (L, ◦, 1) is a monoid and : L × R → R, : R × L → R such that (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y)

Residuated frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37

A residuated frame is a structure F = (L, R, N, ◦, 1, , ) where L and R are sets N ⊆ L × R, (L, ◦, 1) is a monoid and : L × R → R, : R × L → R such that (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y)

• Theorem. If F is a frame, then γN is a nucleus on P(L, ◦, 1).

Residuated frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37

A residuated frame is a structure F = (L, R, N, ◦, 1, , ) where L and R are sets N ⊆ L × R, (L, ◦, 1) is a monoid and : L × R → R, : R × L → R such that (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y)

• Theorem. If F is a frame, then γN is a nucleus on P(L, ◦, 1).
• Corollary. If F is a residuated frame then the Galois algebra

F+ = P(L, ◦, 1)γN is a residuated lattice. Moreover, for FA, x → {x}⊳ is an embedding.

Residuated frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37

A residuated frame is a structure F = (L, R, N, ◦, 1, , ) where L and R are sets N ⊆ L × R, (L, ◦, 1) is a monoid and : L × R → R, : R × L → R such that (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y)

• Theorem. If F is a frame, then γN is a nucleus on P(L, ◦, 1).
• Corollary. If F is a residuated frame then the Galois algebra

F+ = P(L, ◦, 1)γN is a residuated lattice. Moreover, for FA, x → {x}⊳ is an embedding. If A is a RL, FA = (A, A, ≤, ·, {1}) is a residuated frame. The underlying poset of F+

A is the Dedekind-MacNeille completion of the

underlying poset reduct of A.

Residuated frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 15 / 37

A residuated frame is a structure F = (L, R, N, ◦, 1, , ) where L and R are sets N ⊆ L × R, (L, ◦, 1) is a monoid and : L × R → R, : R × L → R such that (x ◦ y) N w ⇔ y N (x w) ⇔ x N (w y)

• Theorem. If F is a frame, then γN is a nucleus on P(L, ◦, 1).
• Corollary. If F is a residuated frame then the Galois algebra

F+ = P(L, ◦, 1)γN is a residuated lattice. Moreover, for FA, x → {x}⊳ is an embedding. If A is a RL, FA = (A, A, ≤, ·, {1}) is a residuated frame. The 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).

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

xNa aNz xNz (CUT) aNa (Id)

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

xNa aNz xNz (CUT) aNa (Id) aNz bNz a ∨ bNz (∨L) xNa xNa ∨ b (∨Rℓ) xNb xNa ∨ b (∨Rr)

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

xNa aNz xNz (CUT) aNa (Id) aNz bNz a ∨ bNz (∨L) xNa xNa ∨ b (∨Rℓ) xNb xNa ∨ b (∨Rr) aNz a ∧ bNz (∧Lℓ) bNz a ∧ bNz (∧Lr) xNa xNb xNa ∧ b (∧R)

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

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

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

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

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

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

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

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

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

xNa aNz xNz (CUT) aNa (Id) aNz bNz a ∨ bNz (∨L) xNa xNa ∨ b (∨Rℓ) xNb xNa ∨ b (∨Rr) aNz a ∧ bNz (∧Lℓ) bNz a ∧ bNz (∧Lr) xNa xNb xNa ∧ b (∧R) a ◦ bNz a · bNz (·L) xNa yNb x ◦ yNa · b (·R) εNz 1Nz (1L) εN1 (1R) xNa bNz a\bNx z (\L) xNa b xNa\b (\R) xNa bNz b/aNz x (/L) xNb a xNb/a (/R) xNa bNz x ◦ (a\b)Nz xNa bN(v c u) x ◦ (a\b)N(v c u)

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

xNa aNz xNz (CUT) aNa (Id) aNz bNz a ∨ bNz (∨L) xNa xNa ∨ b (∨Rℓ) xNb xNa ∨ b (∨Rr) aNz a ∧ bNz (∧Lℓ) bNz a ∧ bNz (∧Lr) xNa xNb xNa ∧ b (∧R) a ◦ bNz a · bNz (·L) xNa yNb x ◦ yNa · b (·R) εNz 1Nz (1L) εN1 (1R) xNa bNz a\bNx z (\L) xNa b xNa\b (\R) xNa bNz b/aNz x (/L) xNb a xNb/a (/R) xNa bNz x ◦ (a\b)Nz xNa bN(v c u) x ◦ (a\b)N(v c u) xNa v ◦ b ◦ uNc v ◦ x ◦ (a\b) ◦ uNc

GN

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 16 / 37

xNa aNz xNz (CUT) aNa (Id) aNz bNz a ∨ bNz (∨L) xNa xNa ∨ b (∨Rℓ) xNb xNa ∨ b (∨Rr) aNz a ∧ bNz (∧Lℓ) bNz a ∧ bNz (∧Lr) xNa xNb xNa ∧ b (∧R) a ◦ bNz a · bNz (·L) xNa yNb x ◦ yNa · b (·R) εNz 1Nz (1L) εN1 (1R) xNa bNz a\bNx z (\L) xNa b xNa\b (\R) xNa bNz b/aNz x (/L) xNb a xNb/a (/R) xNa bNz x ◦ (a\b)Nz xNa bN(v c u) x ◦ (a\b)N(v c u) xNa v ◦ b ◦ uNc v ◦ x ◦ (a\b) ◦ uNc So, we get the sequent calculus FL, for a, b, c ∈ Fm, x, y, u, v ∈ Fm∗, z ∈ SL × Fm.

FL

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 17 / 37

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

Gentzen frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37

Given a monoid L = (L, ◦, ε), SL denotes the sections (unary linear polynomials) of L.

Gentzen frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37

Given a monoid L = (L, ◦, ε), SL denotes the sections (unary linear polynomials) of L. We define FFL, where L = Fm∗, R = SL × Fm, and x N (u, a) iff ⊢FL u(x) ⇒ a.

Gentzen frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37

Given a monoid L = (L, ◦, ε), SL denotes the sections (unary linear polynomials) of L. We define FFL, where L = Fm∗, R = SL × Fm, and x N (u, a) iff ⊢FL u(x) ⇒ a. The following properties hold for FA, FFL (and FA,B, later): 1. F is a residuated frame 2. B is a (partial) algebra of the same type, (B = A, Fm) 3. B generates (L, ◦, ε) (as a monoid) 4. R contains a copy of B (b ↔ (id, b)) 5. N satisfies GN, for all a, b ∈ B, x, y ∈ L, z ∈ R.

Gentzen frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37

Given a monoid L = (L, ◦, ε), SL denotes the sections (unary linear polynomials) of L. We define FFL, where L = Fm∗, R = SL × Fm, and x N (u, a) iff ⊢FL u(x) ⇒ a. The following properties hold for FA, FFL (and FA,B, later): 1. F is a residuated frame 2. B is a (partial) algebra of the same type, (B = A, Fm) 3. B generates (L, ◦, ε) (as a monoid) 4. R contains a copy of B (b ↔ (id, b)) 5. N satisfies GN, for all a, b ∈ B, x, y ∈ L, z ∈ R. We call such pairs (F, B) Gentzen frames. A cut-free Gentzen frame is not assumed to satisfy the (CUT)-rule.

Gentzen frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37

Given a monoid L = (L, ◦, ε), SL denotes the sections (unary linear polynomials) of L. We define FFL, where L = Fm∗, R = SL × Fm, and x N (u, a) iff ⊢FL u(x) ⇒ a. The following properties hold for FA, FFL (and FA,B, later): 1. F is a residuated frame 2. B is a (partial) algebra of the same type, (B = A, Fm) 3. B generates (L, ◦, ε) (as a monoid) 4. R contains a copy of B (b ↔ (id, b)) 5. N satisfies GN, for all a, b ∈ B, x, y ∈ L, z ∈ R. We call such pairs (F, B) Gentzen frames. A cut-free Gentzen frame is not assumed to satisfy the (CUT)-rule.

• 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}⊳).

Gentzen frames

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 18 / 37

Given a monoid L = (L, ◦, ε), SL denotes the sections (unary linear polynomials) of L. We define FFL, where L = Fm∗, R = SL × Fm, and x N (u, a) iff ⊢FL u(x) ⇒ a. The following properties hold for FA, FFL (and FA,B, later): 1. F is a residuated frame 2. B is a (partial) algebra of the same type, (B = A, Fm) 3. B generates (L, ◦, ε) (as a monoid) 4. R contains a copy of B (b ↔ (id, b)) 5. N satisfies GN, for all a, b ∈ B, x, y ∈ L, z ∈ R. We call such pairs (F, B) Gentzen frames. A cut-free Gentzen frame is not assumed to satisfy the (CUT)-rule.

• 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}⊳.

Frame applications

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 19 / 37

■

DM-completion

■

Completeness of the calculus

■

Cut elimination

■

Finite model property

■

Finite embeddability property

■

(Generalized super-)amalgamation property (Transferable injections, Congruence extension property)

■

(Craig) Interpolation property

■

Disjunction property

■

Strong separation

■

Stability under linear structural rules/equations over {∨, ·, 1}. NG and H. Ono, APAL. NG and P. Jipsen, TAMS. NG and P. Jipsen, manuscript.

• A. Ciabattoni, NG and K. Terui, APAL.

NG and K. Terui, manuscript.

Completeness - Cut elimination

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37

Key Lemma. Let (F, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ F+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then 1. a •B b ∈ X •F+ Y ⊆ {a •B b}⊳ (1B ∈ 1F+ ⊆ {1B}⊳ ) 2. In particular, a •B b ∈ {a}⊳ •F+ {b}⊳ ⊆ {a •B b}⊳. 3. Furthermore, because of (CUT), we have equality.

Completeness - Cut elimination

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37

Key Lemma. Let (F, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ F+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then 1. a •B b ∈ X •F+ Y ⊆ {a •B b}⊳ (1B ∈ 1F+ ⊆ {1B}⊳ ) 2. In particular, a •B b ∈ {a}⊳ •F+ {b}⊳ ⊆ {a •B b}⊳. 3. Furthermore, because of (CUT), we have equality. For every homomorphism f : Fm → B, let ¯ f : FmL → F+ be the homomorphism that extends ¯ f(p) = {f(p)}⊳ (p: variable.)

Completeness - Cut elimination

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37

Key Lemma. Let (F, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ F+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then 1. a •B b ∈ X •F+ Y ⊆ {a •B b}⊳ (1B ∈ 1F+ ⊆ {1B}⊳ ) 2. In particular, a •B b ∈ {a}⊳ •F+ {b}⊳ ⊆ {a •B b}⊳. 3. Furthermore, because of (CUT), we have equality. For every homomorphism f : Fm → B, let ¯ f : FmL → F+ be the homomorphism that extends ¯ f(p) = {f(p)}⊳ (p: variable.)

• Corollary. If (F, B) is a cf Gentzen frame, for every homomorphism

f : Fm → B, we have f(a) ∈ ¯ f(a) ⊆ {f(a)}⊳. If we have (CUT), then ¯ f(a) =↓ f(a).

Completeness - Cut elimination

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37

Key Lemma. Let (F, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ F+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then 1. a •B b ∈ X •F+ Y ⊆ {a •B b}⊳ (1B ∈ 1F+ ⊆ {1B}⊳ ) 2. In particular, a •B b ∈ {a}⊳ •F+ {b}⊳ ⊆ {a •B b}⊳. 3. Furthermore, because of (CUT), we have equality. For every homomorphism f : Fm → B, let ¯ f : FmL → F+ be the homomorphism that extends ¯ f(p) = {f(p)}⊳ (p: variable.)

• Corollary. If (F, B) is a cf Gentzen frame, for every homomorphism

f : Fm → B, we have f(a) ∈ ¯ f(a) ⊆ {f(a)}⊳. If we have (CUT), then ¯ f(a) =↓ f(a). We define F | = x ⇒ c by f(x) N f(c), for all f.

• Theorem. If F+

FL |

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

Completeness - Cut elimination

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37

Key Lemma. Let (F, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ F+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then 1. a •B b ∈ X •F+ Y ⊆ {a •B b}⊳ (1B ∈ 1F+ ⊆ {1B}⊳ ) 2. In particular, a •B b ∈ {a}⊳ •F+ {b}⊳ ⊆ {a •B b}⊳. 3. Furthermore, because of (CUT), we have equality. For every homomorphism f : Fm → B, let ¯ f : FmL → F+ be the homomorphism that extends ¯ f(p) = {f(p)}⊳ (p: variable.)

• Corollary. If (F, B) is a cf Gentzen frame, for every homomorphism

f : Fm → B, we have f(a) ∈ ¯ f(a) ⊆ {f(a)}⊳. If we have (CUT), then ¯ f(a) =↓ f(a). We define F | = x ⇒ c by f(x) N f(c), for all f.

• Theorem. If F+

FL |

= x· ≤ c, then FFL | = 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.

Completeness - Cut elimination

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 20 / 37

Key Lemma. Let (F, B) be a Gentzen frame. For all a, b ∈ B, k, l ∈ F+ and for every connective •, if a • b ∈ B, a ∈ X ⊆ {a}⊳ and b ∈ Y ⊆ {b}⊳, then 1. a •B b ∈ X •F+ Y ⊆ {a •B b}⊳ (1B ∈ 1F+ ⊆ {1B}⊳ ) 2. In particular, a •B b ∈ {a}⊳ •F+ {b}⊳ ⊆ {a •B b}⊳. 3. Furthermore, because of (CUT), we have equality. For every homomorphism f : Fm → B, let ¯ f : FmL → F+ be the homomorphism that extends ¯ f(p) = {f(p)}⊳ (p: variable.)

• Corollary. If (F, B) is a cf Gentzen frame, for every homomorphism

f : Fm → B, we have f(a) ∈ ¯ f(a) ⊆ {f(a)}⊳. If we have (CUT), then ¯ f(a) =↓ f(a). We define F | = x ⇒ c by f(x) N f(c), for all f.

• Theorem. If F+

FL |

= x· ≤ c, then FFL | = 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 FLf prove the same sequents.

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level.

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms.

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms. s1 ∨ · · · ∨ sm ≤ t1 ∨ · · · ∨ tn and t1 ∨ · · · ∨ tn ≤ s1 ∨ · · · ∨ sm.

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms. s1 ∨ · · · ∨ sm ≤ t1 ∨ · · · ∨ tn and t1 ∨ · · · ∨ tn ≤ s1 ∨ · · · ∨ sm. The first is equivalent to: &(sj ≤ t1 ∨ · · · ∨ tn).

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms. s1 ∨ · · · ∨ sm ≤ t1 ∨ · · · ∨ tn and t1 ∨ · · · ∨ tn ≤ s1 ∨ · · · ∨ sm. The first is equivalent to: &(sj ≤ t1 ∨ · · · ∨ tn). We proceed by example: x2y ≤ xy ∨ yx

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms. s1 ∨ · · · ∨ sm ≤ t1 ∨ · · · ∨ tn and t1 ∨ · · · ∨ tn ≤ s1 ∨ · · · ∨ sm. The first is equivalent to: &(sj ≤ t1 ∨ · · · ∨ tn). We proceed by example: x2y ≤ xy ∨ yx (x1 ∨ x2)2y ≤ (x1 ∨ x2)y ∨ y(x1 ∨ x2)

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms. s1 ∨ · · · ∨ sm ≤ t1 ∨ · · · ∨ tn and t1 ∨ · · · ∨ tn ≤ s1 ∨ · · · ∨ sm. The first is equivalent to: &(sj ≤ t1 ∨ · · · ∨ tn). We proceed by example: x2y ≤ xy ∨ yx (x1 ∨ x2)2y ≤ (x1 ∨ x2)y ∨ y(x1 ∨ x2) x2

1y ∨ x1x2y ∨ x2x1y ∨ x2 2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms. s1 ∨ · · · ∨ sm ≤ t1 ∨ · · · ∨ tn and t1 ∨ · · · ∨ tn ≤ s1 ∨ · · · ∨ sm. The first is equivalent to: &(sj ≤ t1 ∨ · · · ∨ tn). We proceed by example: x2y ≤ xy ∨ yx (x1 ∨ x2)2y ≤ (x1 ∨ x2)y ∨ y(x1 ∨ x2) x2

1y ∨ x1x2y ∨ x2x1y ∨ x2 2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2

x1x2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms. s1 ∨ · · · ∨ sm ≤ t1 ∨ · · · ∨ tn and t1 ∨ · · · ∨ tn ≤ s1 ∨ · · · ∨ sm. The first is equivalent to: &(sj ≤ t1 ∨ · · · ∨ tn). We proceed by example: x2y ≤ xy ∨ yx (x1 ∨ x2)2y ≤ (x1 ∨ x2)y ∨ y(x1 ∨ x2) x2

1y ∨ x1x2y ∨ x2x1y ∨ x2 2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2

x1x2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2 x1y ≤ z x2y ≤ z yx1 ≤ z yx2 ≤ z x1x2y ≤ z

Equations

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 21 / 37

Idea: Express equations over {∨, ·, 1} at the frame level. For an equation ε over {∨, ·, 1} we distribute products over joins to get s1 ∨ · · · ∨ sm = t1 ∨ · · · ∨ tn. si, tj: monoid terms. s1 ∨ · · · ∨ sm ≤ t1 ∨ · · · ∨ tn and t1 ∨ · · · ∨ tn ≤ s1 ∨ · · · ∨ sm. The first is equivalent to: &(sj ≤ t1 ∨ · · · ∨ tn). We proceed by example: x2y ≤ xy ∨ yx (x1 ∨ x2)2y ≤ (x1 ∨ x2)y ∨ y(x1 ∨ x2) x2

1y ∨ x1x2y ∨ x2x1y ∨ x2 2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2

x1x2y ≤ x1y ∨ x2y ∨ yx1 ∨ yx2 x1y ≤ z x2y ≤ z yx1 ≤ z yx2 ≤ z x1x2y ≤ z x1 ◦ y N z x2 ◦ y N z y ◦ x1 N z y ◦ x2 N z x1 ◦ x2 ◦ y N z R(ε)

Simple rules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 22 / 37

In the context of (FFL, Fm), R(ε) takes the form x ◦ t1 ◦ y ⇒ a · · · x ◦ tn ◦ y ⇒ a x ◦ t0 ◦ y ⇒ a (R(ε)) We call such equations and rules simple.

Simple rules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 22 / 37

In the context of (FFL, Fm), R(ε) takes the form x ◦ t1 ◦ y ⇒ a · · · x ◦ tn ◦ y ⇒ a x ◦ t0 ◦ y ⇒ a (R(ε)) We call such equations and rules simple.

• Theorem. Let (F, B) be a cf Gentzen frame and let ε be a

{∨, ·, 1}-equation. Then (F, B) satisfies R(ε) iff F+ satisfies ε.

Simple rules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 22 / 37

In the context of (FFL, Fm), R(ε) takes the form x ◦ t1 ◦ y ⇒ a · · · x ◦ tn ◦ y ⇒ a x ◦ t0 ◦ y ⇒ a (R(ε)) We call such equations and rules simple.

• Theorem. Let (F, B) be a cf Gentzen frame and let ε be a

{∨, ·, 1}-equation. Then (F, B) satisfies R(ε) iff F+ satisfies ε.

• Theorem. All extensions of FL by simple rules enjoy cut elimination.

Simple rules

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 FL Gentzen frames Frame applications Compl - CE Equations Simple rules Hypersequents Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 22 / 37

In the context of (FFL, Fm), R(ε) takes the form x ◦ t1 ◦ y ⇒ a · · · x ◦ tn ◦ y ⇒ a x ◦ t0 ◦ y ⇒ a (R(ε)) We call such equations and rules simple.

• Theorem. Let (F, B) be a cf Gentzen frame and let ε be a

{∨, ·, 1}-equation. Then (F, B) satisfies R(ε) iff F+ satisfies ε.

• Theorem. All extensions of FL by simple rules enjoy cut elimination.
• K. Terui. Which structural rules admit cut elimination? An algebraic
• criterion. J. Symbolic Logic 72 (2007), no. 3, 738-754.
• N. Galatos and H. Ono. Cut elimination and strong separation for

non-associative substructural logics, APAL 161(9) (2010), 1097–1133.

• N. Galatos and P. Jipsen. Residuated frames and applications to

decidability, Transactions of the AMS (2013).

Hypersequents

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Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 23 / 37

Hypersequents

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Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 24 / 37

FL sequents stem from N1-normal formulas. FL supports the analysis of simple structural rules, which correspond to N2-equations.

Hypersequents

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 24 / 37

FL sequents stem from N1-normal formulas. FL supports the analysis of simple structural rules, which correspond to N2-equations. To handle P3-equations, we define hypersequents, based on P2-normal formulas: (x1 . . . xn → x0) ∨ (y1 . . . yn → y0) ∨ . . . .

Hypersequents

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 24 / 37

FL sequents stem from N1-normal formulas. FL supports the analysis of simple structural rules, which correspond to N2-equations. To handle P3-equations, we define hypersequents, based on P2-normal formulas: (x1 . . . xn → x0) ∨ (y1 . . . yn → y0) ∨ . . . . A hypersequent is a multiset s1 | · · · | sm of sequents si.

Hypersequents

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 24 / 37

FL sequents stem from N1-normal formulas. FL supports the analysis of simple structural rules, which correspond to N2-equations. To handle P3-equations, we define hypersequents, based on P2-normal formulas: (x1 . . . xn → x0) ∨ (y1 . . . yn → y0) ∨ . . . . A hypersequent is a multiset s1 | · · · | sm of sequents si. For every rule s1 s2 s

• f FL, the system HFL is defined to contain the rule

H | s1 H | s2 H | s where H is a (meta)variable for hyprsequents.

Hypersequents

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 24 / 37

FL sequents stem from N1-normal formulas. FL supports the analysis of simple structural rules, which correspond to N2-equations. To handle P3-equations, we define hypersequents, based on P2-normal formulas: (x1 . . . xn → x0) ∨ (y1 . . . yn → y0) ∨ . . . . A hypersequent is a multiset s1 | · · · | sm of sequents si. For every rule s1 s2 s

• f FL, the system HFL is defined to contain the rule

H | s1 H | s2 H | s where H is a (meta)variable for hyprsequents. A hyperstructural rule is of the form H | s′

1

H | s′

2

. . . H | s′

n

H | s1 | · · · | sm

P3 and PUFs

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 25 / 37

■

Recall that prelinearity (α → β)∧1 ∨ (β → α)∧1 axiomatizes the variety generated by chains, i.e., algebras that satisfy (∀x, y)(x ≤ y or y ≤ x).

P3 and PUFs

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 25 / 37

■

Recall that prelinearity (α → β)∧1 ∨ (β → α)∧1 axiomatizes the variety generated by chains, i.e., algebras that satisfy (∀x, y)(x ≤ y or y ≤ x).

■

Weak nilpotent minimum (¬(α · β))∧1 ∨ (α ∧ β → α · β)∧1 axiomatizes the variety generated by algebras in which (∀x, y)(xy = ⊥ or xy = x ∧ y).

P3 and PUFs

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 25 / 37

■

Recall that prelinearity (α → β)∧1 ∨ (β → α)∧1 axiomatizes the variety generated by chains, i.e., algebras that satisfy (∀x, y)(x ≤ y or y ≤ x).

■

Weak nilpotent minimum (¬(α · β))∧1 ∨ (α ∧ β → α · β)∧1 axiomatizes the variety generated by algebras in which (∀x, y)(xy = ⊥ or xy = x ∧ y).

P3 and PUFs

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 25 / 37

■

Recall that prelinearity (α → β)∧1 ∨ (β → α)∧1 axiomatizes the variety generated by chains, i.e., algebras that satisfy (∀x, y)(x ≤ y or y ≤ x).

■

Weak nilpotent minimum (¬(α · β))∧1 ∨ (α ∧ β → α · β)∧1 axiomatizes the variety generated by algebras in which (∀x, y)(xy = ⊥ or xy = x ∧ y).

■

(∀x, y)(x ≤ y or 1 ≤ x) gives (x → y)∧1 ∨ (x)∧1.

• Theorem. [G. 2004] The FLe-formula

(φ1)∧1 ∨ · · · ∨ (φn)∧1 axiomatizes the variety generated by classes defined by the positive universal formula (PUFs) (∀¯ x)(1 ≤ φ1(¯ x) or · · · or 1 ≤ φ1(¯ x)).

Hyper and PUFs

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 26 / 37

Hypersequents over HFLew interpret naturally as P3-normal formulas (comma → multiplication, ⇒ → →, | → ∨).

Hyper and PUFs

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 26 / 37

Hypersequents over HFLew interpret naturally as P3-normal formulas (comma → multiplication, ⇒ → →, | → ∨). Hypersequents over HFL interpret similarly, but when | becomes ∨, the joinants inherit arbitrary iterated conjugates.

Hyper and PUFs

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 26 / 37

Hypersequents over HFLew interpret naturally as P3-normal formulas (comma → multiplication, ⇒ → →, | → ∨). Hypersequents over HFL interpret similarly, but when | becomes ∨, the joinants inherit arbitrary iterated conjugates. Being a subdirect product of chains (PUF: x ≤ y or y ≤ x) is captured by

■

(α → β) ∨ (β → α), in FLew.

■

(α → β)∧1 ∨ (β → α)∧1, in FLe.

■

γ1(α → β) ∨ γ2(β → α), in FL. All these correspond to the hypersequent (α ⇒ β)|(β ⇒ α).

Hyper-frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 27 / 37

A hyperresiduated frame is a structure of the form H = (L, R, ⊢, ◦, ε, , , ǫ), where

■

⊢⊆ H = (L × R)∗. We write ⊢ h instead of h ∈⊢.

■

(L, ◦, ε) is a monoid and ǫ ∈ R.

■

For all x, y ∈ L, z ∈ R, h ∈ H, ⊢ (x ◦ y, z) | h ⇔ ⊢ (y, x z) | h ⇔ ⊢ (x, z y) | h.

■

⊢ h implies ⊢ (x, y) | h for any (x, y) ∈ L × R.

■

⊢ (x, y) | (x, y) | h implies ⊢ (x, y) | h for any (x, y) ∈ L × R.

Hyper-frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 27 / 37

A hyperresiduated frame is a structure of the form H = (L, R, ⊢, ◦, ε, , , ǫ), where

■

⊢⊆ H = (L × R)∗. We write ⊢ h instead of h ∈⊢.

■

(L, ◦, ε) is a monoid and ǫ ∈ R.

■

For all x, y ∈ L, z ∈ R, h ∈ H, ⊢ (x ◦ y, z) | h ⇔ ⊢ (y, x z) | h ⇔ ⊢ (x, z y) | h.

■

⊢ h implies ⊢ (x, y) | h for any (x, y) ∈ L × R.

■

⊢ (x, y) | (x, y) | h implies ⊢ (x, y) | h for any (x, y) ∈ L × R.

• A. Ciabattoni, NG, K. Terui. Algebraic proof theory for substructural

logics: hypersequents and hyper-completions.

Hyper-frames

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 27 / 37

A hyperresiduated frame is a structure of the form H = (L, R, ⊢, ◦, ε, , , ǫ), where

■

⊢⊆ H = (L × R)∗. We write ⊢ h instead of h ∈⊢.

■

(L, ◦, ε) is a monoid and ǫ ∈ R.

■

For all x, y ∈ L, z ∈ R, h ∈ H, ⊢ (x ◦ y, z) | h ⇔ ⊢ (y, x z) | h ⇔ ⊢ (x, z y) | h.

■

⊢ h implies ⊢ (x, y) | h for any (x, y) ∈ L × R.

■

⊢ (x, y) | (x, y) | h implies ⊢ (x, y) | h for any (x, y) ∈ L × R.

• A. Ciabattoni, NG, K. Terui. Algebraic proof theory for substructural

logics: hypersequents and hyper-completions.

• Example. If A = (A, ∧, ∨, ·, \, /, 1, 0) is an FL-algebra, then

HA = (A, A, ⊢, ·, 1, 0) is a hyperresiduated frame, where ⊢ is defined as follows: ⊢ (x1, y1)| . . . |(xn, yn) ⇐ ⇒ 1 ≤ γ1(x1\y1) ∨ · · · ∨ γn(xn\yn).

Examples

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 28 / 37

• Example. Given a residuated frame F = (L, R, N, ◦, ε, , , ǫ), we
• btain a hyperresiduated frame h(F) = (L, R, ⊢, ◦, ε, , , ǫ) by

defining ⊢ (x1, y1) | . . . | (xn, yn) ⇐ ⇒ x1 N y1 or · · · or xn N yn.

Examples

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 28 / 37

• Example. Given a residuated frame F = (L, R, N, ◦, ε, , , ǫ), we
• btain a hyperresiduated frame h(F) = (L, R, ⊢, ◦, ε, , , ǫ) by

defining ⊢ (x1, y1) | . . . | (xn, yn) ⇐ ⇒ x1 N y1 or · · · or xn N yn.

• Example. Based on HFL we define a hyperresiduated frame

HHFL = (L, R, ⊢, ◦, ε, ǫ), where ⊢ s1 | . . . | sn ⇐ ⇒ ⊢HFL s1 | · · · | sn

Examples

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 28 / 37

• Example. Given a residuated frame F = (L, R, N, ◦, ε, , , ǫ), we
• btain a hyperresiduated frame h(F) = (L, R, ⊢, ◦, ε, , , ǫ) by

defining ⊢ (x1, y1) | . . . | (xn, yn) ⇐ ⇒ x1 N y1 or · · · or xn N yn.

• Example. Based on HFL we define a hyperresiduated frame

HHFL = (L, R, ⊢, ◦, ε, ǫ), where ⊢ s1 | . . . | sn ⇐ ⇒ ⊢HFL s1 | · · · | sn Using the cut-free version of this frame, we can prove cut elimination for HFL.

System of relations

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 29 / 37

Equivalently, a hyper-residuated frame H = (L, R, ⊢, ◦, ε, ǫ) can be identified with a set of relations Nh ⊆ L × R, where h is in the commutative monoid (H, |, ∅), such that for all h ∈ H, s ∈ L × R,

■

Nh is nuclear

■

Nh ⊆ Nh|s

■

Nh|s|s ⊆ Nh|s

■

s ∈ Nh|s′ iff s′ ∈ Nh|s (localization)

System of relations

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 29 / 37

Equivalently, a hyper-residuated frame H = (L, R, ⊢, ◦, ε, ǫ) can be identified with a set of relations Nh ⊆ L × R, where h is in the commutative monoid (H, |, ∅), such that for all h ∈ H, s ∈ L × R,

■

Nh is nuclear

■

Nh ⊆ Nh|s

■

Nh|s|s ⊆ Nh|s

■

s ∈ Nh|s′ iff s′ ∈ Nh|s (localization) Given a hyperresiduated frame H = (L, R, ⊢, ◦, ε, ǫ) we obtain a residuated frame r(H) = (L × H, R × H, N, •, (ε; ∅), (ǫ; ∅)), where H = (L × R)∗, (x; h1) • (y; h2) = (x ◦ y; h1 | h2) (x; h1) (z; h2) = (x z; h1 | h2) (z; h2) (x; h1) = (z x; h1 | h2) (x; h1) N (z; h2) ⇔ ⊢ (x, z) | h1 | h2.

System of relations

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 29 / 37

Equivalently, a hyper-residuated frame H = (L, R, ⊢, ◦, ε, ǫ) can be identified with a set of relations Nh ⊆ L × R, where h is in the commutative monoid (H, |, ∅), such that for all h ∈ H, s ∈ L × R,

■

Nh is nuclear

■

Nh ⊆ Nh|s

■

Nh|s|s ⊆ Nh|s

■

s ∈ Nh|s′ iff s′ ∈ Nh|s (localization) Given a hyperresiduated frame H = (L, R, ⊢, ◦, ε, ǫ) we obtain a residuated frame r(H) = (L × H, R × H, N, •, (ε; ∅), (ǫ; ∅)), where H = (L × R)∗, (x; h1) • (y; h2) = (x ◦ y; h1 | h2) (x; h1) (z; h2) = (x z; h1 | h2) (z; h2) (x; h1) = (z x; h1 | h2) (x; h1) N (z; h2) ⇔ ⊢ (x, z) | h1 | h2. We define H+ = r(H)+.

System of relations

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 29 / 37

Equivalently, a hyper-residuated frame H = (L, R, ⊢, ◦, ε, ǫ) can be identified with a set of relations Nh ⊆ L × R, where h is in the commutative monoid (H, |, ∅), such that for all h ∈ H, s ∈ L × R,

■

Nh is nuclear

■

Nh ⊆ Nh|s

■

Nh|s|s ⊆ Nh|s

■

s ∈ Nh|s′ iff s′ ∈ Nh|s (localization) Given a hyperresiduated frame H = (L, R, ⊢, ◦, ε, ǫ) we obtain a residuated frame r(H) = (L × H, R × H, N, •, (ε; ∅), (ǫ; ∅)), where H = (L × R)∗, (x; h1) • (y; h2) = (x ◦ y; h1 | h2) (x; h1) (z; h2) = (x z; h1 | h2) (z; h2) (x; h1) = (z x; h1 | h2) (x; h1) N (z; h2) ⇔ ⊢ (x, z) | h1 | h2. We define H+ = r(H)+. The hyper-MacNeille completion of a residuated lattice A is H+

A.

Extra structure

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 30 / 37

Given X, Y ⊆ L × H and H1, H2 ⊆ H, we define: X ⇀ Y = {(x, z) | h1 | h2 : (x; h1) ∈ X, (z; h2) ∈ Y ⊲} H1 | H2 = {h1 | h2 : h1 ∈ H1, h2 ∈ H2} ⊢ H1 ⇐ ⇒ ⊢ h for every h ∈ H1.

Extra structure

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 30 / 37

Given X, Y ⊆ L × H and H1, H2 ⊆ H, we define: X ⇀ Y = {(x, z) | h1 | h2 : (x; h1) ∈ X, (z; h2) ∈ Y ⊲} H1 | H2 = {h1 | h2 : h1 ∈ H1, h2 ∈ H2} ⊢ H1 ⇐ ⇒ ⊢ h for every h ∈ H1. Key Lemma. For every hyperresiduated frame H and every X, Y ∈ γ[P(L × H)] and H0 ⊆ H, ⊢ X ⇀ Y | H0 ⇐ ⇒ (ǫ; H0) ⊆ X\Y, where (ǫ; H0) = {(ǫ; h) : h ∈ H0}.

Extra structure

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 30 / 37

Given X, Y ⊆ L × H and H1, H2 ⊆ H, we define: X ⇀ Y = {(x, z) | h1 | h2 : (x; h1) ∈ X, (z; h2) ∈ Y ⊲} H1 | H2 = {h1 | h2 : h1 ∈ H1, h2 ∈ H2} ⊢ H1 ⇐ ⇒ ⊢ h for every h ∈ H1. Key Lemma. For every hyperresiduated frame H and every X, Y ∈ γ[P(L × H)] and H0 ⊆ H, ⊢ X ⇀ Y | H0 ⇐ ⇒ (ǫ; H0) ⊆ X\Y, where (ǫ; H0) = {(ǫ; h) : h ∈ H0}. A Genzen hyper-residuated frame is such that Nh is a Gentzen frame for all h ∈ H. (local behavior)

Extra structure

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 30 / 37

Given X, Y ⊆ L × H and H1, H2 ⊆ H, we define: X ⇀ Y = {(x, z) | h1 | h2 : (x; h1) ∈ X, (z; h2) ∈ Y ⊲} H1 | H2 = {h1 | h2 : h1 ∈ H1, h2 ∈ H2} ⊢ H1 ⇐ ⇒ ⊢ h for every h ∈ H1. Key Lemma. For every hyperresiduated frame H and every X, Y ∈ γ[P(L × H)] and H0 ⊆ H, ⊢ X ⇀ Y | H0 ⇐ ⇒ (ǫ; H0) ⊆ X\Y, where (ǫ; H0) = {(ǫ; h) : h ∈ H0}. A Genzen hyper-residuated frame is such that Nh is a Gentzen frame for all h ∈ H. (local behavior) Then we obtain the quasiembedding lemma.

Extensions

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 31 / 37

The system InFL has cut elimination, FMP (and is decidable). Its simple extension all have cut elimination.

• N. Galatos and P. Jipsen. Residuated frames and applications to

decidability, Transactions of the AMS.

Extensions

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 31 / 37

The system InFL has cut elimination, FMP (and is decidable). Its simple extension all have cut elimination.

• N. Galatos and P. Jipsen. Residuated frames and applications to

decidability, Transactions of the AMS. HInFLe has cut elimination (via a syntactic argument, for now).

• A. Ciabattoni, L. Strassburger and K. Terui. Expanding the realm of

systematic proof theory.

Extensions

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 31 / 37

The system InFL has cut elimination, FMP (and is decidable). Its simple extension all have cut elimination.

• N. Galatos and P. Jipsen. Residuated frames and applications to

decidability, Transactions of the AMS. HInFLe has cut elimination (via a syntactic argument, for now).

• A. Ciabattoni, L. Strassburger and K. Terui. Expanding the realm of

systematic proof theory. (G.-Jipsen) DFL has cut elimination (also, all of its extensions with {∧, ∨, ·, 1}-equations/rules). It also has the FMP. [G.] Every subvariety of DIRL axiomatized over {∨, ∧, ·, 1} has the FEP.

Extensions

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 31 / 37

The system InFL has cut elimination, FMP (and is decidable). Its simple extension all have cut elimination.

• N. Galatos and P. Jipsen. Residuated frames and applications to

decidability, Transactions of the AMS. HInFLe has cut elimination (via a syntactic argument, for now).

• A. Ciabattoni, L. Strassburger and K. Terui. Expanding the realm of

systematic proof theory. (G.-Jipsen) DFL has cut elimination (also, all of its extensions with {∧, ∨, ·, 1}-equations/rules). It also has the FMP. [G.] Every subvariety of DIRL axiomatized over {∨, ∧, ·, 1} has the FEP.

• Theorem. (Ciabbatoni-G.-Terui) The system HDFL has cut
• elimination. The same holds for all extensions by simple distributive

hyper-ryles corresponding to P3-equations on the distributive hierarchy.

Extensions

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 31 / 37

The system InFL has cut elimination, FMP (and is decidable). Its simple extension all have cut elimination.

• N. Galatos and P. Jipsen. Residuated frames and applications to

decidability, Transactions of the AMS. HInFLe has cut elimination (via a syntactic argument, for now).

• A. Ciabattoni, L. Strassburger and K. Terui. Expanding the realm of

systematic proof theory. (G.-Jipsen) DFL has cut elimination (also, all of its extensions with {∧, ∨, ·, 1}-equations/rules). It also has the FMP. [G.] Every subvariety of DIRL axiomatized over {∨, ∧, ·, 1} has the FEP.

• Theorem. (Ciabbatoni-G.-Terui) The system HDFL has cut
• elimination. The same holds for all extensions by simple distributive

hyper-ryles corresponding to P3-equations on the distributive hierarchy. Not done yet: HInFL, InDFL, HInDFL.

Extensions

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Hypersequents P3 and PUFs Hyper and PUFs Hyper-frames Examples System of relations Extra structure Extensions Beyond

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 31 / 37

The system InFL has cut elimination, FMP (and is decidable). Its simple extension all have cut elimination.

• N. Galatos and P. Jipsen. Residuated frames and applications to

decidability, Transactions of the AMS. HInFLe has cut elimination (via a syntactic argument, for now).

• A. Ciabattoni, L. Strassburger and K. Terui. Expanding the realm of

systematic proof theory. (G.-Jipsen) DFL has cut elimination (also, all of its extensions with {∧, ∨, ·, 1}-equations/rules). It also has the FMP. [G.] Every subvariety of DIRL axiomatized over {∨, ∧, ·, 1} has the FEP.

• Theorem. (Ciabbatoni-G.-Terui) The system HDFL has cut
• elimination. The same holds for all extensions by simple distributive

hyper-ryles corresponding to P3-equations on the distributive hierarchy. Not done yet: HInFL, InDFL, HInDFL. Beyond P3?

Beyond

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 32 / 37

Embedding theorems

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 33 / 37

Recall that given a poset C, a map f : C → C is residuated iff there exists f ∗ : C → C such that for all x, y ∈ C f(x) ≤ y ⇔ x ≤ f ∗(y).

Embedding theorems

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 33 / 37

Recall that given a poset C, a map f : C → C is residuated iff there exists f ∗ : C → C such that for all x, y ∈ C f(x) ≤ y ⇔ x ≤ f ∗(y). Recal that if C is a complete join semilattice, then f is residuated iff it preserves all joins.

Embedding theorems

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 33 / 37

Recall that given a poset C, a map f : C → C is residuated iff there exists f ∗ : C → C such that for all x, y ∈ C f(x) ≤ y ⇔ x ≤ f ∗(y). Recal that if C is a complete join semilattice, then f is residuated iff it preserves all joins. We denote the set of all residuated maps on C by Res(C).

Embedding theorems

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 33 / 37

Recall that given a poset C, a map f : C → C is residuated iff there exists f ∗ : C → C such that for all x, y ∈ C f(x) ≤ y ⇔ x ≤ f ∗(y). Recal that if C is a complete join semilattice, then f is residuated iff it preserves all joins. We denote the set of all residuated maps on C by Res(C). If C is a complete join semilattice, then Res(C) is a residuated lattice, under composition and pointwise join and meet.

Embedding theorems

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 33 / 37

Recall that given a poset C, a map f : C → C is residuated iff there exists f ∗ : C → C such that for all x, y ∈ C f(x) ≤ y ⇔ x ≤ f ∗(y). Recal that if C is a complete join semilattice, then f is residuated iff it preserves all joins. We denote the set of all residuated maps on C by Res(C). If C is a complete join semilattice, then Res(C) is a residuated lattice, under composition and pointwise join and meet. A conucleus σ on a residuated L is an interior operator on L, such that its image is a submonoid: for all x, y ∈ L, σ(1) = 1 and σ(xy) = σ(σ(x)σ(y)).

Embedding theorems

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 33 / 37

Recall that given a poset C, a map f : C → C is residuated iff there exists f ∗ : C → C such that for all x, y ∈ C f(x) ≤ y ⇔ x ≤ f ∗(y). Recal that if C is a complete join semilattice, then f is residuated iff it preserves all joins. We denote the set of all residuated maps on C by Res(C). If C is a complete join semilattice, then Res(C) is a residuated lattice, under composition and pointwise join and meet. A conucleus σ on a residuated L is an interior operator on L, such that its image is a submonoid: for all x, y ∈ L, σ(1) = 1 and σ(xy) = σ(σ(x)σ(y)). Given a residuated lattice L and a conucleus σ on it, the image σ[L] supports a residuated lattice Lσ, where ∨, · and 1 are the restrictions from L, while\, / and ∧ are the σ-images of the ones of L.

Embedding theorems

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 33 / 37

Recall that given a poset C, a map f : C → C is residuated iff there exists f ∗ : C → C such that for all x, y ∈ C f(x) ≤ y ⇔ x ≤ f ∗(y). Recal that if C is a complete join semilattice, then f is residuated iff it preserves all joins. We denote the set of all residuated maps on C by Res(C). If C is a complete join semilattice, then Res(C) is a residuated lattice, under composition and pointwise join and meet. A conucleus σ on a residuated L is an interior operator on L, such that its image is a submonoid: for all x, y ∈ L, σ(1) = 1 and σ(xy) = σ(σ(x)σ(y)). Given a residuated lattice L and a conucleus σ on it, the image σ[L] supports a residuated lattice Lσ, where ∨, · and 1 are the restrictions from L, while\, / and ∧ are the σ-images of the ones of L. Cayley’s representation for RL [G.-Horˇ c´ ık] Every residuated lattice can be embedded into the conucleus image of Res(C), for some complete join semilattice C.

Maps on a chain

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 34 / 37

Holland’s representation for RL [G.-Horˇ c´ ık] If a residuated lattice satisfies (h ∨ ca) ∧ (h ∨ db) ≤ h ∨ cb ∨ da then C can be taken to be a chain. The proof can be presented in a way that involves action hyper-residuated frames.

Maps on a chain

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 34 / 37

Holland’s representation for RL [G.-Horˇ c´ ık] If a residuated lattice satisfies (h ∨ ca) ∧ (h ∨ db) ≤ h ∨ cb ∨ da then C can be taken to be a chain. The proof can be presented in a way that involves action hyper-residuated frames. If C is a chain, the order-preserving bijections on C form an ℓ-group Aut(C). We can obtain the following celebrated theorem as a corollary.

Maps on a chain

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 34 / 37

Holland’s representation for RL [G.-Horˇ c´ ık] If a residuated lattice satisfies (h ∨ ca) ∧ (h ∨ db) ≤ h ∨ cb ∨ da then C can be taken to be a chain. The proof can be presented in a way that involves action hyper-residuated frames. If C is a chain, the order-preserving bijections on C form an ℓ-group Aut(C). We can obtain the following celebrated theorem as a corollary. Holland’s theorem for ℓ-groups [G.-Horˇ c´ ık] Every ℓ-group can be embedded in Aut(C) for some chain C. [G.-Horˇ c´ ık] Cayley’s and Holland’s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices, Semigroup Forum.

Diagrams

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 35 / 37

Holland-McCleary theorem for ℓ-groups The variety of ℓ-groups is generated by Aut(R). In other words, to test if an equation holds in the class of ℓ-groups it is enough to test if it holds in Aut(R).

Diagrams

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 35 / 37

Holland-McCleary theorem for ℓ-groups The variety of ℓ-groups is generated by Aut(R). In other words, to test if an equation holds in the class of ℓ-groups it is enough to test if it holds in Aut(R). Unfortunately, Aut(R) is infinite, but checking equations can still be done efficiently by studying diagrams.

Diagrams

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 35 / 37

Holland-McCleary theorem for ℓ-groups The variety of ℓ-groups is generated by Aut(R). In other words, to test if an equation holds in the class of ℓ-groups it is enough to test if it holds in Aut(R). Unfortunately, Aut(R) is infinite, but checking equations can still be done efficiently by studying diagrams. These provide a refutation system for checking identities [Holland-McCleary].

Diagrams

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 35 / 37

Holland-McCleary theorem for ℓ-groups The variety of ℓ-groups is generated by Aut(R). In other words, to test if an equation holds in the class of ℓ-groups it is enough to test if it holds in Aut(R). Unfortunately, Aut(R) is infinite, but checking equations can still be done efficiently by studying diagrams. These provide a refutation system for checking identities [Holland-McCleary]. In analogy to semantical tableux, one can turn this into a form of (hypersequent) calculus. [G.-Metcalfe] h|x h|x−1 h (Gp | = x = 1)

Diagrams

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 35 / 37

Holland-McCleary theorem for ℓ-groups The variety of ℓ-groups is generated by Aut(R). In other words, to test if an equation holds in the class of ℓ-groups it is enough to test if it holds in Aut(R). Unfortunately, Aut(R) is infinite, but checking equations can still be done efficiently by studying diagrams. These provide a refutation system for checking identities [Holland-McCleary]. In analogy to semantical tableux, one can turn this into a form of (hypersequent) calculus. [G.-Metcalfe] h|x h|x−1 h (Gp | = x = 1) Theorem [G.-Metcalfe] The variety of ℓ-groups is generated by Aut(R). (Syntactic argument.)

ALG and MV

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 36 / 37

The study of abelian ℓ-groups/MV-algebras is quite different in flavor than that of ℓ-groups;

ALG and MV

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 36 / 37

The study of abelian ℓ-groups/MV-algebras is quite different in flavor than that of ℓ-groups; the latter is combinatorial/group-theoretic the former is more geometeric.

ALG and MV

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 36 / 37

The study of abelian ℓ-groups/MV-algebras is quite different in flavor than that of ℓ-groups; the latter is combinatorial/group-theoretic the former is more geometeric. The variety is generated by Z, the totally odered group of the

• integers. [Weinberg]

ALG and MV

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 36 / 37

The study of abelian ℓ-groups/MV-algebras is quite different in flavor than that of ℓ-groups; the latter is combinatorial/group-theoretic the former is more geometeric. The variety is generated by Z, the totally odered group of the

• integers. [Weinberg]

Thus the decidability of the equational theory of abelian ℓ-groups can be proved using geometric/linear-programming tools.

ALG and MV

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 36 / 37

The study of abelian ℓ-groups/MV-algebras is quite different in flavor than that of ℓ-groups; the latter is combinatorial/group-theoretic the former is more geometeric. The variety is generated by Z, the totally odered group of the

• integers. [Weinberg]

Thus the decidability of the equational theory of abelian ℓ-groups can be proved using geometric/linear-programming tools. Nevertheless in [G.-Jipsen-Marra] we show that one can also use a diagram refutation system (by implementing Fourier-Motzkin into diagrams).

ℓ-pregroups

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 37 / 37

Pregroups are ordered monoids (A, ·, 1, ≤) with two additional unary

• perations l, r that satisfy the inequations

xlx ≤ 1 ≤ xxl and xxr ≤ 1 ≤ xrx. (InRL’s with x · y = x + y.) Introduced in mathematical linguistics, and studied from algebraic and proof-theoretic points of view (W. Buskowski). ℓ-pregroups are lattice-based.

ℓ-pregroups

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 37 / 37

Pregroups are ordered monoids (A, ·, 1, ≤) with two additional unary

• perations l, r that satisfy the inequations

xlx ≤ 1 ≤ xxl and xxr ≤ 1 ≤ xrx. (InRL’s with x · y = x + y.) Introduced in mathematical linguistics, and studied from algebraic and proof-theoretic points of view (W. Buskowski). ℓ-pregroups are lattice-based. ℓ-groups are exactly the ℓ-pregroups that satisfy xl = xr.

ℓ-pregroups

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 37 / 37

Pregroups are ordered monoids (A, ·, 1, ≤) with two additional unary

• perations l, r that satisfy the inequations

xlx ≤ 1 ≤ xxl and xxr ≤ 1 ≤ xrx. (InRL’s with x · y = x + y.) Introduced in mathematical linguistics, and studied from algebraic and proof-theoretic points of view (W. Buskowski). ℓ-pregroups are lattice-based. ℓ-groups are exactly the ℓ-pregroups that satisfy xl = xr. Given a chain C, the collection of all maps on C that have arbitrary residuals and arbitrary dual residuals form an ℓ-pregroup F(C).

ℓ-pregroups

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 37 / 37

Pregroups are ordered monoids (A, ·, 1, ≤) with two additional unary

• perations l, r that satisfy the inequations

xlx ≤ 1 ≤ xxl and xxr ≤ 1 ≤ xrx. (InRL’s with x · y = x + y.) Introduced in mathematical linguistics, and studied from algebraic and proof-theoretic points of view (W. Buskowski). ℓ-pregroups are lattice-based. ℓ-groups are exactly the ℓ-pregroups that satisfy xl = xr. Given a chain C, the collection of all maps on C that have arbitrary residuals and arbitrary dual residuals form an ℓ-pregroup F(C). Theorem [G.-Jipsen] Every periodic/distributive ℓ-pregroup can be embedded in F(C) for some chain C.

ℓ-pregroups

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 37 / 37

Pregroups are ordered monoids (A, ·, 1, ≤) with two additional unary

• perations l, r that satisfy the inequations

xlx ≤ 1 ≤ xxl and xxr ≤ 1 ≤ xrx. (InRL’s with x · y = x + y.) Introduced in mathematical linguistics, and studied from algebraic and proof-theoretic points of view (W. Buskowski). ℓ-pregroups are lattice-based. ℓ-groups are exactly the ℓ-pregroups that satisfy xl = xr. Given a chain C, the collection of all maps on C that have arbitrary residuals and arbitrary dual residuals form an ℓ-pregroup F(C). Theorem [G.-Jipsen] Every periodic/distributive ℓ-pregroup can be embedded in F(C) for some chain C. Goal: a diagram refuation system for distributive ℓ-pregroups.

ℓ-pregroups

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 37 / 37

Pregroups are ordered monoids (A, ·, 1, ≤) with two additional unary

• perations l, r that satisfy the inequations

xlx ≤ 1 ≤ xxl and xxr ≤ 1 ≤ xrx. (InRL’s with x · y = x + y.) Introduced in mathematical linguistics, and studied from algebraic and proof-theoretic points of view (W. Buskowski). ℓ-pregroups are lattice-based. ℓ-groups are exactly the ℓ-pregroups that satisfy xl = xr. Given a chain C, the collection of all maps on C that have arbitrary residuals and arbitrary dual residuals form an ℓ-pregroup F(C). Theorem [G.-Jipsen] Every periodic/distributive ℓ-pregroup can be embedded in F(C) for some chain C. Goal: a diagram refuation system for distributive ℓ-pregroups. The only group elements in F(Z) are the translations (isomorphic to Z). However, we obtain the following surprising result.

ℓ-pregroups

Residuated lattices Examples Decidability for lattices Boolean algebras Lattices Contexts Dedekind-McNeille Lattice frames Why does it work? Sequents Hypersequents Beyond Embedding theorems Maps on a chain Diagrams ALG and MV ℓ-pregroups

Nick Galatos, ALCOP, April 2013 Hyper-residuated frames – 37 / 37

Pregroups are ordered monoids (A, ·, 1, ≤) with two additional unary

• perations l, r that satisfy the inequations

xlx ≤ 1 ≤ xxl and xxr ≤ 1 ≤ xrx. (InRL’s with x · y = x + y.) Introduced in mathematical linguistics, and studied from algebraic and proof-theoretic points of view (W. Buskowski). ℓ-pregroups are lattice-based. ℓ-groups are exactly the ℓ-pregroups that satisfy xl = xr. Given a chain C, the collection of all maps on C that have arbitrary residuals and arbitrary dual residuals form an ℓ-pregroup F(C). Theorem [G.-Jipsen] Every periodic/distributive ℓ-pregroup can be embedded in F(C) for some chain C. Goal: a diagram refuation system for distributive ℓ-pregroups. The only group elements in F(Z) are the translations (isomorphic to Z). However, we obtain the following surprising result. Theorem [G.-Jipsen-Ball] The variety of ℓ-groups is contained in the variety generated by F(Z).