A duality-theoretic approach to MTL-algebras Sara Ugolini (Joint - - PowerPoint PPT Presentation

A duality-theoretic approach to MTL-algebras

Sara Ugolini (Joint work with W. Fussner) BLAST 2018 - Denver, August 6th 2018

Introduction Duality for MTL and GMTL srDL and dualized quadruples

A commutative, integral residuated lattice, or CIRL, is a structure A = (A, ·, →, ∧, ∨, 1) where: (i) (A, ∧, ∨, 1) is a lattice with top element 1, (ii) (A, ·, 1) is a commutative monoid, (iii) (·, →) is a residuated pair, i.e. it holds for every x, y, z ∈ A: x · z ≤ y iff z ≤ x → y. CIRLs constitute a variety, RL. Examples: (Z−, +, ⊖, min, max, 0), ideals of a commutative ring...

Sara Ugolini 2/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

A bounded CIRL, or BCIRL, is a CIRL A = (A, ·, →, ∧, ∨, 0, 1) with an extra constant 0 that is the least element of the lattice. Examples: Boolean algebras, Heyting algebras... In every BRL we can define further operations and abbreviations: ¬x = x → 0, x + y = ¬(¬x · ¬y), x2 = x · x. Totally ordered structures are called chains.

Sara Ugolini 3/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

A bounded CIRL, or BCIRL, is a CIRL A = (A, ·, →, ∧, ∨, 0, 1) with an extra constant 0 that is the least element of the lattice. Examples: Boolean algebras, Heyting algebras... In every BRL we can define further operations and abbreviations: ¬x = x → 0, x + y = ¬(¬x · ¬y), x2 = x · x. Totally ordered structures are called chains. A CIRL, or BCIRL, is semilinear (or prelinear, or representable) if it is a subdirect product of chains. We call semilinear CIRLs GMTL-algebras and semilinear BCIRLs MTL-algebras. They constitute varieties that we denote with GMTL and MTL. MTL-algebras are the semantics of Esteva and Godo’s MTL, the fuzzy logic of left-continuous t-norms.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Priestley duality

MTL-algebras and GMTL-algebras have a distributive lattice reduct. [Priestley, 1970]: The category BDL of bounded distributive lattices and bounded lattice homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous isotone maps. A Priestley space is a structure (X, ≤, τ), where (X, τ) is a compact topological space, (X, ≤) is a poset, and for any x ≤ y there exists a clopen U ⊆ X with x ∈ U and y / ∈ U.

Sara Ugolini 4/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

Priestley duality

MTL-algebras and GMTL-algebras have a distributive lattice reduct. [Priestley, 1970]: The category BDL of bounded distributive lattices and bounded lattice homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous isotone maps. A Priestley space is a structure (X, ≤, τ), where (X, τ) is a compact topological space, (X, ≤) is a poset, and for any x ≤ y there exists a clopen U ⊆ X with x ∈ U and y / ∈ U. Pries BDL S A

Sara Ugolini 4/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

Priestley duality

MTL-algebras and GMTL-algebras have a distributive lattice reduct. [Priestley, 1970]: The category BDL of bounded distributive lattices and bounded lattice homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous isotone maps. A Priestley space is a structure (X, ≤, τ), where (X, τ) is a compact topological space, (X, ≤) is a poset, and for any x ≤ y there exists a clopen U ⊆ X with x ∈ U and y / ∈ U. Pries BDL S A S(D): prime filters ordered by inclusion with topology generated by {ϕ(d) : d ∈ D} ∪ {ϕ(d)c : d ∈ D} where ϕ(d) = {X prime filter of D : d ∈ X}

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Priestley duality

MTL-algebras and GMTL-algebras have a distributive lattice reduct. [Priestley, 1970]: The category BDL of bounded distributive lattices and bounded lattice homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous isotone maps. A Priestley space is a structure (X, ≤, τ), where (X, τ) is a compact topological space, (X, ≤) is a poset, and for any x ≤ y there exists a clopen U ⊆ X with x ∈ U and y / ∈ U. Pries BDL S A S(D): prime filters ordered by inclusion with topology generated by {ϕ(d) : d ∈ D} ∪ {ϕ(d)c : d ∈ D} where ϕ(d) = {X prime filter of D : d ∈ X} A(X, ≤, τ): collection of clopen upsets (Cl(X), ∪, ∩, ∅, X)

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Priestley duality

Priestley duality admits numerous modifications. E.g. if one or both of the lattice bounds are dropped we obtain a dual category of pointed, or doubly-pointed, (i.e. bounded above or bounded) Priestley spaces.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Priestley duality

Priestley duality admits numerous modifications. E.g. if one or both of the lattice bounds are dropped we obtain a dual category of pointed, or doubly-pointed, (i.e. bounded above or bounded) Priestley spaces. Moreover, Priestley duality can be extended to distributive residuated lattices. Our approach is essentially drawn from [Galatos, PhD thesis, 2003] and [Urquhart, 1996], however a similar approach to duals of MTL-algebras has been developed by Cabrer and Celani in 2006. Usually, the multiplication is dualized as a ternary relation on prime filters.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Residuated spaces

We call a structure (S, R, E) an unpointed residuated space if

• S is a Priestley space
• R is a ternary relation on S,
• E is a subset of S,
• for all x, y, z, w, x′, y′, z′ ∈ S and U, V ∈ A(S):

(i) R(x, y, u) and R(u, z, w) for some u ∈ S if and only if R(y, z, v) and R(x, v, w) for some v ∈ S. (ii) If x′ ≤ x, y′ ≤ y, and z ≤ z′, then R(x, y, z) implies R(x′, y′, z′). (iii) If R(x, y, z), then there exist U, V ∈ A(S) such that x ∈ U, y ∈ V , and z / ∈ R[U, V, −]. (iv) For all U, V ∈ A(S), each of R[U, V, −], {z ∈ S : R[z, V, −] ⊆ U}, and {z ∈ P : R[B, z, −] ⊆ U} are clopen. (v) E ∈ A(S) and for all U ∈ A(S) we have R[U, E, −] = R[E, U, −] = U. Where R[U, V, −] = {z ∈ S : (∃x ∈ U)(∃y ∈ V )(R(x, y, z))}.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Residuated spaces

If S1 = (S1, ≤1, τ1, R1, E1) and S2 = (S2, ≤2, τ2, R2, E2) are unpointed residuated spaces, a map α: S1 → S2 is a bounded morphism if: (i) α is a continuous isotone map. (ii) If R1(x, y, z), then R2(α(x), α(y), α(z)). (iii) If R2(u, v, α(z)), then there exist x, y ∈ S1 such that u ≤ α(x), v ≤ α(y), and R1(x, y, z). (iv) For all U, V ∈ A(S2) and all x ∈ S1, if R1[x, α−1[U], −] ⊆ α−1[V ], then R2[α(x), U, −] ⊆ V . (v) α−1[E2] ⊆ E1. We denote the category of unpointed residuated spaces and bounded morphisms by uRS. The following is proven in [Galatos, PhD thesis].

Theorem

The category of bounded residuated lattices with residuated lattice homomorphisms preserving the lattice bounds is dually equivalent to uRS.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Extending functors

• Given an unpointed residuated space S = (S, ≤, τ, R, E), we define

A(S) = (A(S, ≤, τ), ·, →, E), where U · V = R[U, V, −] U → V = {x ∈ S : R[x, U, −] ⊆ V } for U, V ∈ A(S, ≤, τ), where R[U, V, −] = {z ∈ S : (∃x ∈ U)(∃y ∈ V )(R(x, y, z))}.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Extending functors

• Given an unpointed residuated space S = (S, ≤, τ, R, E), we define

A(S) = (A(S, ≤, τ), ·, →, E), where U · V = R[U, V, −] U → V = {x ∈ S : R[x, U, −] ⊆ V } for U, V ∈ A(S, ≤, τ), where R[U, V, −] = {z ∈ S : (∃x ∈ U)(∃y ∈ V )(R(x, y, z))}.

• Given a BCIRL A, we define a product • on prime filters as the upset of

the complex product ·: a • b = ↑(a · b) = {z ∈ A : ∃x ∈ a, y ∈ b, xy ≤ z} S(A) = (S(D), R, E), where for a bounded residuated lattice A with bounded lattice reduct D, we define a ternary relation R on S(D) and a subset of S(D) by R(a, b, c) iff a • b ⊆ c, E = {a ∈ S(D) : 1 ∈ a}.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Duality for MTL

Let A = (A, ∧, ∨, ·, \, /, 1, ⊥, ⊤) be a bounded residuated lattice and S = (S, ≤, τ, R, E) its dual space.

• A is commutative iff for all x, y, z ∈ S, R(x, y, z) iff R(y, x, z).
• A is integral iff E = S.
• In the presence of integrality and commutativity, A is semilinear iff for all

x, y, z, v, w ∈ S, if R(x, y, z) and R(x, v, w), then y ≤ w or v ≤ z. We denote by MTLτ the full subcategory of uRS whose objects satisfy these three conditions.

Theorem

MTLτ is dually equivalent to MTL.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Duality for GMTL

Let MTLdiv be the full subcategory of MTL-algebras without zero divisors and SMTLind be the full subcategory of directly indecomposable SMTL-algebras.

Theorem

The categories GMTL, MTLdiv, and SMTLind are equivalent.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Duality for GMTL

Let MTLdiv be the full subcategory of MTL-algebras without zero divisors and SMTLind be the full subcategory of directly indecomposable SMTL-algebras.

Theorem

The categories GMTL, MTLdiv, and SMTLind are equivalent. We provide a duality for GMTL, using as a bridge the full subcategory MTLτ

div,

that is characterized by having a greatest element ⊤.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Duality for GMTL

Let MTLdiv be the full subcategory of MTL-algebras without zero divisors and SMTLind be the full subcategory of directly indecomposable SMTL-algebras.

Theorem

The categories GMTL, MTLdiv, and SMTLind are equivalent. We provide a duality for GMTL, using as a bridge the full subcategory MTLτ

div,

that is characterized by having a greatest element ⊤. Thus, let GMTLτ be the category whose objects are of the form (S, R, E, ⊤), where (S, R, E) is an object of MTLτ with a ⊤. The morphisms are bounded morphisms that preserve ⊤.

Theorem

GMTL and GMTLτ are dually equivalent.

Sara Ugolini 10/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualities for classes of distributive residuated lattices typically employ a ternary relation on dual structures. For prelinear residuated structures, the ternary relation on duals may be presented by means of a (possibly partially-defined) binary operation • on the underlying Priestley dual of the lattice reduct.

Sara Ugolini 11/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualities for classes of distributive residuated lattices typically employ a ternary relation on dual structures. For prelinear residuated structures, the ternary relation on duals may be presented by means of a (possibly partially-defined) binary operation • on the underlying Priestley dual of the lattice reduct. In a general setting, the functionality of duals is treated in [Gehrke, 2016] and [Fussner, Palmigiano 2018]. We explore the cases of MTL and GMTL.

Sara Ugolini 11/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualities for classes of distributive residuated lattices typically employ a ternary relation on dual structures. For prelinear residuated structures, the ternary relation on duals may be presented by means of a (possibly partially-defined) binary operation • on the underlying Priestley dual of the lattice reduct. In a general setting, the functionality of duals is treated in [Gehrke, 2016] and [Fussner, Palmigiano 2018]. We explore the cases of MTL and GMTL.

Lemma

Let A be a GMTL-algebra or an MTL-algebra, and let a, b be nonempty filters of

• A. Then if either one of a or b is prime, we have that either a • b is prime or

a • b = A.

Sara Ugolini 11/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualities for classes of distributive residuated lattices typically employ a ternary relation on dual structures. For prelinear residuated structures, the ternary relation on duals may be presented by means of a (possibly partially-defined) binary operation • on the underlying Priestley dual of the lattice reduct. In a general setting, the functionality of duals is treated in [Gehrke, 2016] and [Fussner, Palmigiano 2018]. We explore the cases of MTL and GMTL.

Lemma

Let A be a GMTL-algebra or an MTL-algebra, and let a, b be nonempty filters of

• A. Then if either one of a or b is prime, we have that either a • b is prime or

a • b = A.

Corollary

If A is an GMTL-algebra, then • is a binary operation on S(A). If A is an MTL-algebra, then • is gives a partial operation on S(A) and is undefined for a, b ∈ S(A) only if a • b = A.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

The following provides a mechanism for defining • on MTLτ and GMTLτ.

Lemma

Let S = (S, ≤, τ, R, E) be an object of MTLτ. If x, y, z ∈ S with R(x, y, z), then there exists a least element z′ ∈ S such that R(x, y, z′). If S is in GMTLτ, then for any x, y ∈ S there exists a least z′ ∈ S with R(x, y, z′). x • y =

• min{z ∈ S : R(x, y, z)},

if {z ∈ S : R(x, y, z)} = ∅ undefined,

• therwise

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

The following provides a mechanism for defining • on MTLτ and GMTLτ.

Lemma

Let S = (S, ≤, τ, R, E) be an object of MTLτ. If x, y, z ∈ S with R(x, y, z), then there exists a least element z′ ∈ S such that R(x, y, z′). If S is in GMTLτ, then for any x, y ∈ S there exists a least z′ ∈ S with R(x, y, z′). x • y =

• min{z ∈ S : R(x, y, z)},

if {z ∈ S : R(x, y, z)} = ∅ undefined,

• therwise

Lemma

Let S be an object of MTLτ or GMTLτ, and let x, y, z ∈ S. (i) R(x, y, z) iff x • y ≤ z. (ii) Each of the following holds (whenever • is defined).

1 x • (y • z) = (x • y) • z. 2 x • y = y • x. 3 x ≤ y implies that x • z ≤ y • z and z • x ≤ z • y.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Moreover, • possesses a partial residual.

Proposition

Let A be an MTL-algebra or GMTL-algebra, and suppose that b, c ∈ S(A) are such that there exists a ∈ S(A) with a • b ⊆ c. Then there is a greatest such a, and it is given by b ⇒ c :=

• {a ∈ S(A) : a • b ⊆ c}.

Moreover, a • b ⊆ c if and only if a ⊆ b ⇒ c.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Moreover, • possesses a partial residual.

Proposition

Let A be an MTL-algebra or GMTL-algebra, and suppose that b, c ∈ S(A) are such that there exists a ∈ S(A) with a • b ⊆ c. Then there is a greatest such a, and it is given by b ⇒ c :=

• {a ∈ S(A) : a • b ⊆ c}.

Moreover, a • b ⊆ c if and only if a ⊆ b ⇒ c.

Corollary

Let S be an object of MTLτ or GMTLτ, and suppose that y, z ∈ S are such that there exists x ∈ S with R(x, y, z). Then there is a greatest such x, which we denote by y ⇒ z. Moreover, x • y ≤ z if and only if x ≤ y ⇒ z.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

In order to treat negation, we use the Routley star ([Routley, Meyer - 1972,1973], [Urquhart, 1996]). If A is an MTL-algebra and a ∈ S(A), we define a∗ = {a ∈ A : ¬a ∈ a} It is easy to see that if a is a prime filter, then so is a∗. Moreover, ∗ is an

• rder-reversing operation on prime filters.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

In order to treat negation, we use the Routley star ([Routley, Meyer - 1972,1973], [Urquhart, 1996]). If A is an MTL-algebra and a ∈ S(A), we define a∗ = {a ∈ A : ¬a ∈ a} It is easy to see that if a is a prime filter, then so is a∗. Moreover, ∗ is an

• rder-reversing operation on prime filters.

Lemma

Let A be an MTL-algebra and let a ∈ S(A). Then a∗ is the largest prime filter

• f A such that a • a∗ = A.

Corollary

Let S be an object of MTLτ, and let x ∈ S. Then there exists a greatest y ∈ S such that R(x, y, z) for some z ∈ S. Equivalently, there exists a greatest y ∈ S such that x • y is defined. In light of the previous corollary, for an abstract object S of MTLτ we define for any x ∈ S, x∗ := max{y ∈ S : ∃z ∈ S, R(x, y, z)}.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

We now focus on a special class of MTL-algebras: srDL-algebras, that are the axiomatic extension of MTL by: (x + x)2 = x2 + x2 (DL) ¬(x2) → (¬¬x → x) = 1 (r) Relevant subvarieties are: product algebras, G¨

• del algebras, the variety generated

by perfect MV-algebras, nilpotent minimum without negation fixpoint, pseudocomplemented MTL algebras... [Cignoli, Torrens - 2006], [Aguzzoli, Flaminio, U. - 2017]: srDL is generated by δ-rotations of GMTL-algebras.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

δ-rotation

Let R = (R, ·, →, ∧, ∨, 1) be a GMTL-algebra, and let δ : R → R be a wdl-admissible

• perator:
• closure operator;
• δ(x) · δ(y) ≤ δ(x · y) (nucleus on R);
• δ(x ∨ y) = δ(x) ∨ δ(y),

δ(x ∧ y) = δ(x) ∧ δ(y).

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

δ-rotation

Let R = (R, ·, →, ∧, ∨, 1) be a GMTL-algebra, and let δ : R → R be a wdl-admissible

• perator:
• closure operator;
• δ(x) · δ(y) ≤ δ(x · y) (nucleus on R);
• δ(x ∨ y) = δ(x) ∨ δ(y),

δ(x ∧ y) = δ(x) ∧ δ(y). Examples: δD = id, δL = ¯ 1, i.e. δL(x) = 1 for every x ∈ R.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

δ-rotation

Let R = (R, ·, →, ∧, ∨, 1) be a GMTL-algebra, and let δ : R → R be a wdl-admissible

• perator:
• closure operator;
• δ(x) · δ(y) ≤ δ(x · y) (nucleus on R);
• δ(x ∨ y) = δ(x) ∨ δ(y),

δ(x ∧ y) = δ(x) ∧ δ(y). Examples: δD = id, δL = ¯ 1, i.e. δL(x) = 1 for every x ∈ R. We define the δ-rotation Rδ(R) as the structure with domain ({1} × R) ∪ ({0} × δ[R]) and suitably defined operations.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Notable subvarieties

With δ = id we get directly indecomposable sIDL-algebras (introduced as IBP0-algebras in [Noguera, Esteva, Gispert - 2005]), i.e. disconnected rotation (as defined by Jenei) of GMTL-algebras. Examples: the variety generated by perfect MV-algebras, NM− of nilpotent minimum without negation fixpoint.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Notable subvarieties

With δ = ¯ 1 we get directly indecomposable pseudocomplemented MTL-algebras: algebras A isomorphic to 2 ⊕ R for some GMTL R. Examples: G¨

• del algebras (prelinear Heyting algebras), Product algebras.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

srDL-algebras

[Aguzzoli, Flaminio, U. - 2017]: srDL-algebras are equivalent to categories whose objects are quadruples (B, R, ∨e, δ):

• B is a Boolean algebra,
• R is a GMTL-algebra,
• δ : R → R is wdl-admissible,
• ∨e : B × R → R is an external join:

(V1) For fixed b ∈ B and c ∈ R: νb(x) = b ∨e x is an endomorphism of R, γc(x) = x ∨e c is a lattice homomorphism from B to R. (V2) ν0 is the identity on R, ν1 is constantly equal to 1. (V3) For all b, b′ ∈ B and for all c, c′ ∈ R, νb(c) ∨ νb′(c′) = νb∨b′(c ∨ c′) = νb(hb′(c ∨ c′)).

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

BRL B MTL MVRn HA rDL srDL IMTL sIDL NR− NM− G SMTL SHA SRL P DLMV wIDL

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

In analogy to the algebraic side, we expect that if A is an srDL-algebra, S(A) can be reconstructed by:

• the Stone space associated to its Boolean skeleton B(A);
• the object in GMTLτ associated to Rad(A);
• the maps induced on the dual side by the (external) join;
• the dual of the wdl-admissible operator δ (¬¬ of A).

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

In analogy to the algebraic side, we expect that if A is an srDL-algebra, S(A) can be reconstructed by:

• the Stone space associated to its Boolean skeleton B(A);
• the object in GMTLτ associated to Rad(A);
• the maps induced on the dual side by the (external) join;
• the dual of the wdl-admissible operator δ (¬¬ of A).

Indeed, we can reconstruct S(A) by FA: sets of pairs (u, y) such that (i) u is an ultrafilter of B(A), (ii) y is a prime lattice filter of Rad(A), (iii) for b ∈ B(A), c ∈ Rad(A), if b ∨ c ∈ y, then b ∈ u or c ∈ y. plus the information given by ¬¬.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Let u, v, w ∈ S(B(A)). Aaa

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Below u: x ∈ S(R(A)) that respect the “external primality condition” wrt u,

• rdered by inclusion.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Same for v, w, . . . Aaa

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Rotate upwards the δ-images of the elements below u. Aaa

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

The dualized rotation construction obtained, that we denote SB(A) ⊗∆

Υ SR(A)

(where ∆ = δ−1 and Υ = {ν−1

b

}b∈B(A)), is isomorphic to S(A).

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Our aim is to find the correct definition of abstract dual quadruples, and to understand how to abstractly perform the dualized rotation construction.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Our aim is to find the correct definition of abstract dual quadruples, and to understand how to abstractly perform the dualized rotation construction. Every prime filter a of an srDL-algebra A contains a unique ultrafilter of its Boolean skeleton.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Our aim is to find the correct definition of abstract dual quadruples, and to understand how to abstractly perform the dualized rotation construction. Every prime filter a of an srDL-algebra A contains a unique ultrafilter of its Boolean skeleton. Indeed, let a ∈ S(A), then for each u ∈ B(A) we have that u ∨ ¬u = 1 ∈ a. But a is prime, thus u ∈ a or ¬u ∈ a. We denote the ultrafilter associated to a by ua.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Our aim is to find the correct definition of abstract dual quadruples, and to understand how to abstractly perform the dualized rotation construction. Every prime filter a of an srDL-algebra A contains a unique ultrafilter of its Boolean skeleton. Indeed, let a ∈ S(A), then for each u ∈ B(A) we have that u ∨ ¬u = 1 ∈ a. But a is prime, thus u ∈ a or ¬u ∈ a. We denote the ultrafilter associated to a by ua. For each ultrafilter u, set Su = {a ∈ S(A) : ua = u} and call this the site at u.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Let u be an ultrafilter of B(A), y ∈ S(Rad(A)). We say that u fixes y if there exists a ∈ Su such that y = a ∩ Rad(A).

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Let u be an ultrafilter of B(A), y ∈ S(Rad(A)). We say that u fixes y if there exists a ∈ Su such that y = a ∩ Rad(A). For every b ∈ B(A), let µb(y) = ν−1

b

[y] = {x ∈ Rad(A) : b ∨ x ∈ y}.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized construction

Let u be an ultrafilter of B(A), y ∈ S(Rad(A)). We say that u fixes y if there exists a ∈ Su such that y = a ∩ Rad(A). For every b ∈ B(A), let µb(y) = ν−1

b

[y] = {x ∈ Rad(A) : b ∨ x ∈ y}.

Lemma

Let A be an srDL-algebra, u be an ultrafilter of B(A), and y ∈ S(Rad(A)). The following are equivalent.

• For every b ∈ B(A) and c ∈ Rad(A), if b ∨ c ∈ y, then b ∈ u or c ∈ y.
• u fixes y.
• µb(y) = y for each b /

∈ u.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Example

Let us consider Chang MV-algebra C, with domain C = {0, c, . . . , nc, . . . , 1 − nc, . . . , 1 − c, 1}. C+ = {1, 1 − c, . . . , 1 − nc, . . .} is isomorphic to Z−, and C is isomorphic to the disconnected rotation of Z−. C generates the variety of perfect MV-algebras.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Example

Let us consider Chang MV-algebra C, with domain C = {0, c, . . . , nc, . . . , 1 − nc, . . . , 1 − c, 1}. C+ = {1, 1 − c, . . . , 1 − nc, . . .} is isomorphic to Z−, and C is isomorphic to the disconnected rotation of Z−. C generates the variety of perfect MV-algebras. Consider C2 = C × C: (1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

(1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

The Boolean skeleton of C2 is the Boolean algebra of 4 elements B(C2) = {(0, 0), (0, 1), (1, 0), (1, 1)}. The radical R(C2) is isomorphic to Z− × Z−, and is the upper square.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

(1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

We want pairs (u, x) such that for every b / ∈ u, {y ∈ R(C2) : b ∨ y ∈ x} = x

Sara Ugolini 31/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

(1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

We want pairs (u, x) such that for every b / ∈ u, {y ∈ R(C2) : b ∨ y ∈ x} = x Let u1 be the Boolean ultrafilter generated by (1, 0), u2 the one generated by (0, 1), Z1 = {(1, y) : y ∈ C+}, and Z2 = {(x, 1) : x ∈ C+}

Sara Ugolini 31/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

(1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

We want pairs (u, x) such that for every b / ∈ u, {y ∈ R(C2) : b ∨ y ∈ x} = x Let u1 be the Boolean ultrafilter generated by (1, 0), u2 the one generated by (0, 1), Z1 = {(1, y) : y ∈ C+}, and Z2 = {(x, 1) : x ∈ C+} We get: (u2, [rn)) and (u1, [rm)).

Sara Ugolini 31/37

Introduction Duality for MTL and GMTL srDL and dualized quadruples

(1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

We want pairs (u, x) such that for every b / ∈ u, {y ∈ R(C2) : b ∨ y ∈ x} = x Let u1 be the Boolean ultrafilter generated by (1, 0), u2 the one generated by (0, 1), Z1 = {(1, y) : y ∈ C+}, and Z2 = {(x, 1) : x ∈ C+} We get: (u2, [rn)) and (u1, [rm)).

(u1, R(C2)) (u1, Z1) (u2, R(C2)) (u2, [rn)) (u1, [rm)) (u2, Z2)

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

(1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

(u1, R(C2)) (u1, Z1) (u2, R(C2)) (u1, δ[Z1]) (u1, [rm)) (u1, δ[[rm)]) (u2, [rn)) (u2, δ[[rn)]) (u2, Z2) (u2, δ[Z2])

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

(1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

(u1, R(C2)) (u1, Z1) (u2, R(C2)) (u1, δ[Z1]) (u1, [rm)) (u1, δ[[rm)]) (u2, [rn)) (u2, δ[[rn)]) (u2, Z2) (u2, δ[Z2])

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

(1, 1) (1, 0) (0, 1) (0, 0) (0, 1 − nc) (0, nc) (1 − mc, 0) (mc, 0)

rn rm

(u1, R(C2)) (u1, Z1) (u2, R(C2)) (u1, δ[Z1]) (u1, [rm)) (u1, δ[[rm)]) (u2, [rn)) (u2, δ[[rn)]) (u2, Z2) (u2, δ[Z2])

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized quadruples

Let a dual quadruple be a structure (S, X, Υ, ∆) where (i) S is a Stone space; (ii) X is in GMTLτ; (iii) Υ = {υU}U∈A(S) is an indexed family of GMTLτ-morphisms υU : X → X such that the map ∨e : A(S) × A(X) → A(X) defined by ∨e(U, X) = υ−1

U [X]

is an external join; (iv) ∆ : X → X is a continuous closure operator such that R(x, y, z) implies R(∆x, ∆y, ∆z).

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Dualized quadruples

Given a dual quadruple (S, X, Υ, ∆), let S ⊗∆

Υ X be the structure obtained by

the dual rotation construction, with suitably defined topology and multiplication.

Theorem

Let (S, X, Υ, ∆) be a dual quadruple. Then S ⊗∆

Υ X is the extended Priestley

dual of some srDL-algebra.

Theorem

Let Y be the extended Priestley dual of an srDL-algebra. Then there exists a dual quadruple (S, X, Υ, ∆) such that Y ∼ = S ⊗∆

Υ X.

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Introduction Duality for MTL and GMTL srDL and dualized quadruples

Thank you!

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