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Bounded commutative residuated lattices with a retraction term. - - PowerPoint PPT Presentation

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Bounded commutative residuated lattices with a retraction term. Manuela Busaniche Instituto de Matem atica Aplicada del Litoral Santa Fe, Argentina BLAST 2018 Manuela Busaniche Blast 2018 Bounded residuated lattices Stonean residuated

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Bounded commutative residuated lattices with a retraction term.

Manuela Busaniche

Instituto de Matem´ atica Aplicada del Litoral Santa Fe, Argentina

BLAST 2018

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

To the memory of Roberto Cignoli

The ideas of this talk are based on joint works with Roberto Cignoli, Miguel Marcos and Sara Ugolini.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Residuated lattices

An integral and commutative residuated lattice is an algebra A = A, ∗, →, ∨, ∧, ⊤ such that

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Residuated lattices

An integral and commutative residuated lattice is an algebra A = A, ∗, →, ∨, ∧, ⊤ such that A, ∗, ⊤ is a commutative monoid,

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Residuated lattices

An integral and commutative residuated lattice is an algebra A = A, ∗, →, ∨, ∧, ⊤ such that A, ∗, ⊤ is a commutative monoid, L(A) = A, ∨, ∧, ⊤ is a lattice with greatest element ⊤,

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Residuated lattices

An integral and commutative residuated lattice is an algebra A = A, ∗, →, ∨, ∧, ⊤ such that A, ∗, ⊤ is a commutative monoid, L(A) = A, ∨, ∧, ⊤ is a lattice with greatest element ⊤, the following residuation condition holds: x ∗ y ≤ z iff x ≤ y → z (1)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Bounded residuated lattices

A bounded residuated lattice is an algebra A = A, ∗, →, ∨, ∧, ⊤, ⊥ such that A, ∗, →, ∨, ∧, ⊤ is a residuated lattice, and ⊥ is the smallest element of the lattice L(A).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Famous bounded residuated lattices

Boolean algebras Heyting algebras MV-algebras BL-algebras MTL-algebras NM-algebras Nelson residuated lattices

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Historical background

Montagna, F. and Ugolini, S., A categorical equivalence for product algebras, Studia Logica 103 (2015), 345-373.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Historical background

Montagna, F. and Ugolini, S., A categorical equivalence for product algebras, Studia Logica 103 (2015), 345-373. Chen, C. C. and Gr¨ atzer, G., Stone Lattices. I: Construction Theorems,

  • Canad. J. Math. 21 (1969), 884–994.

Katriˇ n´ ak, T., A new proof of the construction theorem for Stone algebras, Proc. Amer. Math. Soc., 40 (1973), 75–78. Maddana Swamy, U. and Rama Rao, V. V., Triple and sheaf representations of Stone lattices, Algebra Universalis 5 (1975), 104–113

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The retraction

Given a residuated lattice A a retraction is a homomorphism h : A ։ S(A)

  • nto a subalgebra S(A) of A such that h(h(a)) = h(a) for each a ∈ A.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The retraction

Given a residuated lattice A a retraction is a homomorphism h : A ։ S(A)

  • nto a subalgebra S(A) of A such that h(h(a)) = h(a) for each a ∈ A.

If we have a class of residuated lattices with a retraction onto a subalgebra, we have the following situation:

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

h h A S(A) h−1({⊤})

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

When can we use S(A) and h−1({⊤}) to recover A?

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

When can we use S(A) and h−1({⊤}) to recover A? When is it worth using them?

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

When can we use S(A) and h−1({⊤}) to recover A? When is it worth using them? Are they all the information we need to characterize A?

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Boolean skeleton

Let A be a bounded residuated lattice.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Boolean skeleton

Let A be a bounded residuated lattice. B(A) = { complemented elements of A}

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Boolean skeleton

Let A be a bounded residuated lattice. B(A) = { complemented elements of A} = {x ∈ A : there exists z ∈ A such that x ∧ z = ⊥ and x ∨ z = ⊤}

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Boolean skeleton

Let A be a bounded residuated lattice. B(A) = { complemented elements of A} = {x ∈ A : there exists z ∈ A such that x ∧ z = ⊥ and x ∨ z = ⊤} B(A) is a subalgebra of A which is a Boolean algebra.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Stonean residuated lattices

Stonean residuated lattices is the greatest subvariety S of bounded residuated lattices that satisfies that for each A ∈ S the application ¬¬ : A → B(A) is a retraction onto the boolean skeleton.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Importance of the Boolean skeleton in Stonean residuated lattices

Theorem The following are equivalent conditions for a bounded residuated lattice A:

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Importance of the Boolean skeleton in Stonean residuated lattices

Theorem The following are equivalent conditions for a bounded residuated lattice A:

(i)

A is Stonean,

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Importance of the Boolean skeleton in Stonean residuated lattices

Theorem The following are equivalent conditions for a bounded residuated lattice A:

(i)

A is Stonean,

(ii)

B(A) ⊇ ¬(A) := {¬ x : x ∈ A}.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Famous Stonean residuated lattices

Boolean algebras Pseudocomplemented BL-algebras Product algebras G¨

  • del algebras

Pseudocomplemented MTL-algebras Stonean Heyting algebras

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Dense elements

Let A be in S. Since ¬¬ : A → B(A) is a retraction, the kernel, D(A) = {x ∈ A : ¬¬x = ⊤} is a filter of A.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Dense elements

Let A be in S. Since ¬¬ : A → B(A) is a retraction, the kernel, D(A) = {x ∈ A : ¬¬x = ⊤} is a filter of A. We will consider D(A) = (D(A), ∗, →, ∨, ∧, ⊤) as an integral residuated lattice.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Building algebras in S

Let D be an integral residuated lattice and an element o ∈ D.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Building algebras in S

Let D be an integral residuated lattice and an element o ∈ D. Adjoining the element o as bottom element, then S(D) = ({o} ∪ D, ∗, →, ∨, ∧, ⊤, o) is in S

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

D

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

S(D) D

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Theorem Given a residuated lattice D, then

1

S(D) is a Stonean residuated lattice,

2

B(S(D)) = {⊥, ⊤}.

3

D(S(D)) = D

4

Each homomorphism h : D1 → D2 can be extended to a homomorphism S(h) : S(D1) → S(D2) by the prescription S(h)(x) =

  • h(x)

if x ∈ D1,

  • S(D2)

if x = oS(D1).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Equivalence of categories

The functor S : RL → diS given by D → S(D) and h → S(h) establish a categorical equivalence.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Isomorphic Boolean skeletons and Isomorphic dense elements

Take D an integral residuated lattice.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Isomorphic Boolean skeletons and Isomorphic dense elements

Take D an integral residuated lattice. Consider A1 ∼ =

  • N

S(D)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Isomorphic Boolean skeletons and Isomorphic dense elements

Take D an integral residuated lattice. Consider A1 ∼ =

  • N

S(D) · · · B(A1) ∼ =

  • N{⊥, ⊤} and D(A1) ∼

=

  • N D

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Isomorphic Boolean skeletons and Isomorphic dense elements

Take D an integral residuated lattice. Consider A1 ∼ =

  • N

S(D) · · · B(A1) ∼ =

  • N{⊥, ⊤} and D(A1) ∼

=

  • N D

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Isomorphic Boolean skeletons and Isomorphic dense elements

With the same D, consider C =

N D.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Isomorphic Boolean skeletons and Isomorphic dense elements

With the same D, consider C =

N D. Let

A2 ∼ = S(C) ×

  • N

{⊥, ⊤}

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Isomorphic Boolean skeletons and Isomorphic dense elements

With the same D, consider C =

N D. Let

A2 ∼ = S(C) ×

  • N

{⊥, ⊤} · · · · · · B(A2) ∼ =

  • N{⊥, ⊤} and D(A2) ∼

=

  • N D

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

A1 ≇ A2. How can we distinguish these two algebras?

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let b ∈ B(A1). Then [¬b) ∩ D(A1) ∼ =

N D.

· · ·

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let b ∈ B(A2). Then [¬b) ∩ D(A2) ∼ = {⊤} · · · · · ·

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

φA : B(A) → Fi(D(A)) b → [¬b) ∩ D(A).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

φA : B(A) → Fi(D(A)) b → [¬b) ∩ D(A). φA(b) = {x ∈ D(A) : x ≥ ¬b}.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Representation of elements

Let A be in S. For each x ∈ A x = ¬¬x ∗ (¬¬x → x)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Representation of elements

Let A be in S. For each x ∈ A x = ¬¬x

  • B(A)

∗ (¬¬x → x)

  • D(A)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Consider D a residuated lattice and A ∼ = S(D) × S(D).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Representation of elements

For d ∈ D take x = (⊥, d) ∈ A.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Representation of elements

For d ∈ D take x = (⊥, d) ∈ A. Thus (⊥, d) = (⊥, ⊤) ∗ (⊤, d)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Representation of elements

For d ∈ D take x = (⊥, d) ∈ A. Thus (⊥, d) = (⊥, ⊤) ∗ (⊤, d) But for any d′ ∈ D we also have (⊥, d) = (⊥, ⊤) ∗ (d′, d)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Representation of elements

For d ∈ D take x = (⊥, d) ∈ A. Thus (⊥, d) = (⊥, ⊤) ∗ (⊤, d) But for any d′ ∈ D we also have (⊥, d) = (⊥, ⊤) ∗ (d′, d) (⊤, d) is the unique dense satisfying the representation that belongs to φ(⊥, ⊤) = {(d1, d2) ∈ D2 : (d1, d2) ≥ ¬(⊥, ⊤)} = {(d1, d2) ∈ D2 : (d1, d2) ≥ (⊤, ⊥)}

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The category T

Objects: Triples (B, D, φ) such that: B is a Boolean algebra, D is a residuated lattice and φ is bounded lattice-homomorphism, φ : B → Fi(D).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The category T

Morphisms: Given triples (Bi, Di, φi), i = 1, 2, a morphism is a pair (h, k) : (B1, D1, φ1) → (B2, D2, φ2) is a pair such that:

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The category T

Morphisms: Given triples (Bi, Di, φi), i = 1, 2, a morphism is a pair (h, k) : (B1, D1, φ1) → (B2, D2, φ2) is a pair such that:

1

h : B1 → B2 is a Boolean algebra homomorphism,

2

k : D1 → D2 is a residuated lattice homomorphism, and

3

For all a ∈ B1, k(φ1(a)) ⊆ φ2(h(a)).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The functor T

T : S → T A → (B(A), D(A), φA) h : B(A1) → B(A2) f : A1 → A2 → k : D(A1) → D(A2)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

We need to prove that T is:

1

faithful

2

full

3

essentially surjective

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Some ideas

That T is faithful and full follows immediate from the representation of each element x in A ∈ S by x = ¬¬x ∗ (¬¬x → x).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Some ideas

To see that T is essentially surjective (dense), for each triple (B, D, φ) we need to find an algebra A such that T(A) = (B(A), D(A), φA) ∼ = (B, D, φ)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let (B, D, φ).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let (B, D, φ). Take X to be the Stone space of the Boolean algebra B, and α : C(X) → Fi(D) given by α(a) = φ(¬a).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let (B, D, φ). Take X to be the Stone space of the Boolean algebra B, and α : C(X) → Fi(D) given by α(a) = φ(¬a). Then α is a dual lattice homomorphism, i.e., for each a ⊆ b we have α(b) ⊆ α(a).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let (B, D, φ). Take X to be the Stone space of the Boolean algebra B, and α : C(X) → Fi(D) given by α(a) = φ(¬a). Then α is a dual lattice homomorphism, i.e., for each a ⊆ b we have α(b) ⊆ α(a). Moreover, we have ρa,b : D/α(b) → D/α(a) the natural projection

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Thus the system R = {D/α(a)}a∈C(X), {ρab}a⊆b is a presheaf of residuated lattices.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Thus the system R = {D/α(a)}a∈C(X), {ρab}a⊆b is a presheaf of residuated lattices. Because of the categorical equivalence, the system S = {S(D/α(a))}a∈C(X), {S(ρab)}a⊆b is a presheaf of directly indecomposable Stonean residuated lattices.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

For each x ∈ X, Fx =

  • a∈C(x)

α(a).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

For each x ∈ X, Fx =

  • a∈C(x)

α(a). Then D/Fx is the inductive limit of the system R and S(D)/Fx is the inductive limit of the system S.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

For each x ∈ X, Fx =

  • a∈C(x)

α(a). Then D/Fx is the inductive limit of the system R and S(D)/Fx is the inductive limit of the system S. Since

x∈X Fx = {⊤} the algebra D is a subdirect product of the family

{D/Fx}x∈X.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Now let S =

  • x∈X

({x} × S(D/Fx)), and for each x ∈ X, d ∈ D and a ∈ C(X) let ˆ d(x) = x, d/Fx ˆ a(x) =

  • x, oS(D/Fx)

if x ∈ a, x, ⊤ if x ∈ X \ a.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Now let S =

  • x∈X

({x} × S(D/Fx)), and for each x ∈ X, d ∈ D and a ∈ C(X) let ˆ d(x) = x, d/Fx ˆ a(x) =

  • x, oS(D/Fx)

if x ∈ a, x, ⊤ if x ∈ X \ a. Equipping S with the topology having as basis the sets {ˆ d(x) : x ∈ a} and {ˆ a(x) : x ∈ a} and defining π : S → X as the projection in the first coordinate.

Manuela Busaniche Blast 2018

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The algebra of global sections

Theorem S, π, X is the sheaf of directly indecomposable Stonean residuated lattices associated with the presheaf {S(D/α(a))}a∈C(X), {S(ρab)}a⊆b. The continuous global sections of S, π, X, with the operations defined pointwise, form a Stonean residuated lattice A = A(B, D, φ).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The algebra of global sections

Theorem S, π, X is the sheaf of directly indecomposable Stonean residuated lattices associated with the presheaf {S(D/α(a))}a∈C(X), {S(ρab)}a⊆b. The continuous global sections of S, π, X, with the operations defined pointwise, form a Stonean residuated lattice A = A(B, D, φ). T(A) ∼ = (B, D, φ).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let V be a subvariety of bounded commutative residuated lattices and A ∈ V:

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let V be a subvariety of bounded commutative residuated lattices and A ∈ V:

1

A retraction h such that h−1({⊤}) ֒ → A ։ h(A).

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Let V be a subvariety of bounded commutative residuated lattices and A ∈ V:

1

A retraction h such that h−1({⊤}) ֒ → A ։ h(A).

2

Each element x ∈ A can be written as a polinomial depending on h(x) and an element of h−1({⊤}).

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Retraction term onto MVn algebras

Let MVn be the subvariety of MV-algebras generated by the n-elements MV-chain Łn. Each M ∈ MVn is hyperarchimedean. Therefore Filters are Stonean: they are generated by a filter in B(M) Prime filters and ultrafilters coincide.

Manuela Busaniche Blast 2018

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kernel rDL-algebras

D → (D, δ) where δ : D → D

(δ1) x → δ(x) = ⊤ (δ2) δ(δ(x)) = δ(x) (δ3) δ(x ∗ y) = δ(δ(x) ∗ δ(y)) (δ4) δ(x ∧ y) = δ(x) ∧ δ(y) (δ5) δ(x ∨ y) = δ(x) ∨ δ(y)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Given: n ≥ 2 a krDL-algebra ˜ D = (D, δ)

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Given: n ≥ 2 a krDL-algebra ˜ D = (D, δ) We define the generalized δ-rotation as a bounded residuated lattice Rδ Łn(D) = ({0} × δ[D]) ∪ {{s} × {1}}s∈Ln\{0,1} ∪ ({1} × D)

Manuela Busaniche Blast 2018

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We will have algebras of the form: D ⊤

. . . n − 2

δ(D)∂ ⊥

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

n = 2, the Boolean retraction

D ⊤ δ(D)∂ ⊥

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Examples n = 2

n = 2, δ(x) = ⊤, then Rδ Ł2(D) = S(D) is a Stonean residuated lattice. n = 2, δ(x) = x, then Rδ Ł2(D) is the disconnected rotation of D. n = 2, arbitrary δ, D totally ordered, then Rδ Ł2(D) is a directly indecomposable MTL-algebra.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Examples n > 2

δ(x) = ⊤, D a basic hoop, then Rδ Łn(D) = Łn ⊕ D is a BL-algebra. n = 3, δ(x) = x and D a generalized Heyting algebra, then Rδ Ł3(D) is a Nelson residuated lattice. n = 3, δ(x) = x and D a Godel hoop, then Rδ Ł3(D) is a NM-algebra with a negation fixpoint.

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

For each fixed n, we define the variety MVRn as the subvariety of bounded residuated lattices generated by all n, δ-rotations.

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

For each fixed n, we define the variety MVRn as the subvariety of bounded residuated lattices generated by all n, δ-rotations. For each A in MVRn there are a term γ and a term β such that:

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

For each fixed n, we define the variety MVRn as the subvariety of bounded residuated lattices generated by all n, δ-rotations. For each A in MVRn there are a term γ and a term β such that: γ(A) = MA is an algebra in MVn which is a subalgebra of A. γ : A → MA is a retraction. (γ−1({⊤}), ¬¬) = DA is a krDL-algebra. B(A) ∼ = B(MA). For each x ∈ A, β(x) ∈ DA. Each x ∈ A can be written as a polynomial of γ(x) and β(x).

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The category Tn

Objects: Triples (M, D, φ) such that: M is in MVn, D = (D, δ) is krDL-algebra φ is bounded lattice-homomorphism, φ : B(M) → Fi(D).

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The category T

Morphisms: Given triples (Mi, Di, φi), i = 1, 2, a morphism is a pair (h, k) : (M1, D1, φ1) → (M2, D2, φ2) is a pair such that:

Manuela Busaniche Blast 2018

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

The category T

Morphisms: Given triples (Mi, Di, φi), i = 1, 2, a morphism is a pair (h, k) : (M1, D1, φ1) → (M2, D2, φ2) is a pair such that:

1

h : M1 → M2 is a MVn-algebra homomorphism,

2

k : D1 → D2 is krDL-homomorphism, and

3

For all a ∈ B(M1), k(φ1(a)) ⊆ φ2(h(a)).

Manuela Busaniche Blast 2018

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The functor Tn

Tn : MVRn → Tn A → (MA, DA, φA) where φA : B(MA) → Fi(DA) is given by φ(a) = [¬a) ∩ DA.

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Tn defines a categorical equivalence between MVRn and the category of triples Tn.

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Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence

Aguzzoli, S., Flaminio, T. and Ugolini, S, Equivalences between the subcategories of MTL-algebras via boolean algebras and prelinear semihoops, Journal of Logic and Computation, (2017). Busaniche, M., Cignoli, R. and Marcos, M., A categorial equivalence for Stonean residuated lattices, Studia Logica (2018). Busaniche, M., Marcos, M. and Ugolini, S., Representation by triples of algebras with an MV-retract, submitted. Chen, C. C. and Gr¨ atzer, G., Stone Lattices. I: Construction Theorems,

  • Canad. J. Math. 21 (1969).

Maddana Swamy, U. and Rama Rao, V. V., Triple and sheaf representations of Stone lattices, Algebra Universalis 5 (1975). Montagna, F. and Ugolini, S., A categorical equivalence for product algebras, Studia Logica 103 (2015).

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Thanks for your attention

Manuela Busaniche Blast 2018