Topological spaces of monadic MV-algebras R. Grigolia (Tbilisi), A. - - PowerPoint PPT Presentation

▶
topological spaces of monadic mv algebras
SMART_READER_LITE
LIVE PREVIEW

Topological spaces of monadic MV-algebras R. Grigolia (Tbilisi), A. - - PowerPoint PPT Presentation

Sep 23, 2023 22 likes •1.24k views

Topological spaces of monadic MV-algebras R. Grigolia (Tbilisi), A. Di Nola and G. Lenzi (Salerno) TACL 2017 Prague, 26-30 June 2017 Topological spaces of monadic MV-algebras G. Lenzi Summary We generalize Belluce functor from MV-algebras

slide-1
SLIDE 1

Topological spaces of monadic MV-algebras

  • R. Grigolia (Tbilisi), A. Di Nola and G. Lenzi (Salerno)

TACL 2017 Prague, 26-30 June 2017

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-2
SLIDE 2

Summary

We generalize Belluce β functor from MV-algebras to distributive lattices we obtain a functor γ from monadic MV-algebras to Q-distributive lattices (hence to Q-spaces via Cignoli duality) and we introduce a subcategory of Q-spaces related to γ (monadic Q-spaces). γ is a tool for the spectrum problem for monadic MV-algebras.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-3
SLIDE 3

Summary

We generalize Belluce β functor from MV-algebras to distributive lattices we obtain a functor γ from monadic MV-algebras to Q-distributive lattices (hence to Q-spaces via Cignoli duality) and we introduce a subcategory of Q-spaces related to γ (monadic Q-spaces). γ is a tool for the spectrum problem for monadic MV-algebras.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-4
SLIDE 4

Summary

We generalize Belluce β functor from MV-algebras to distributive lattices we obtain a functor γ from monadic MV-algebras to Q-distributive lattices (hence to Q-spaces via Cignoli duality) and we introduce a subcategory of Q-spaces related to γ (monadic Q-spaces). γ is a tool for the spectrum problem for monadic MV-algebras.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-5
SLIDE 5

Introduction

Motivation: many valued logic Idea: truth values change from {0, 1} to [0, 1] ({0, 1}, ∧, ∨, ¬, 0, 1) is a Boolean algebra what structure can we put on [0, 1]? There are many possibilities, but the MV-algebra structure is particularly appealing (e.g. all logical connectives are continuous)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-6
SLIDE 6

Introduction

Motivation: many valued logic Idea: truth values change from {0, 1} to [0, 1] ({0, 1}, ∧, ∨, ¬, 0, 1) is a Boolean algebra what structure can we put on [0, 1]? There are many possibilities, but the MV-algebra structure is particularly appealing (e.g. all logical connectives are continuous)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-7
SLIDE 7

Introduction

Motivation: many valued logic Idea: truth values change from {0, 1} to [0, 1] ({0, 1}, ∧, ∨, ¬, 0, 1) is a Boolean algebra what structure can we put on [0, 1]? There are many possibilities, but the MV-algebra structure is particularly appealing (e.g. all logical connectives are continuous)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-8
SLIDE 8

Introduction

Motivation: many valued logic Idea: truth values change from {0, 1} to [0, 1] ({0, 1}, ∧, ∨, ¬, 0, 1) is a Boolean algebra what structure can we put on [0, 1]? There are many possibilities, but the MV-algebra structure is particularly appealing (e.g. all logical connectives are continuous)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-9
SLIDE 9

Introduction

Motivation: many valued logic Idea: truth values change from {0, 1} to [0, 1] ({0, 1}, ∧, ∨, ¬, 0, 1) is a Boolean algebra what structure can we put on [0, 1]? There are many possibilities, but the MV-algebra structure is particularly appealing (e.g. all logical connectives are continuous)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-10
SLIDE 10

The MV-algebra [0, 1]

As negation we have ¬x = 1 − x (the Liar tells half the truth) As disjunction we have x ⊕ y = min(1, x + y) (not idempotent, cfr. Ulam games with lies) Conjunction x ⊙ y is defined via De Morgan duality.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-11
SLIDE 11

The MV-algebra [0, 1]

As negation we have ¬x = 1 − x (the Liar tells half the truth) As disjunction we have x ⊕ y = min(1, x + y) (not idempotent, cfr. Ulam games with lies) Conjunction x ⊙ y is defined via De Morgan duality.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-12
SLIDE 12

The MV-algebra [0, 1]

As negation we have ¬x = 1 − x (the Liar tells half the truth) As disjunction we have x ⊕ y = min(1, x + y) (not idempotent, cfr. Ulam games with lies) Conjunction x ⊙ y is defined via De Morgan duality.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-13
SLIDE 13

The MV-algebra [0, 1]

As negation we have ¬x = 1 − x (the Liar tells half the truth) As disjunction we have x ⊕ y = min(1, x + y) (not idempotent, cfr. Ulam games with lies) Conjunction x ⊙ y is defined via De Morgan duality.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-14
SLIDE 14

Logic

Infinite valued Łukasiewicz logic is a first order logic interpreted on the MV-algebra [0, 1]. MV-algebras (introduced by Chang) are the algebraic counterpart of propositional infinite valued Łukasiewicz logic. Now full first order Łukasiewicz logic is not axiomatizable (Scarpellini) However, monadic first order Łukasiewicz logic is axiomatizable (Rutledge) This motivates monadic MV-algebras (introduced by Rutledge himself, apparently before the Scarpellini result)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-15
SLIDE 15

Logic

Infinite valued Łukasiewicz logic is a first order logic interpreted on the MV-algebra [0, 1]. MV-algebras (introduced by Chang) are the algebraic counterpart of propositional infinite valued Łukasiewicz logic. Now full first order Łukasiewicz logic is not axiomatizable (Scarpellini) However, monadic first order Łukasiewicz logic is axiomatizable (Rutledge) This motivates monadic MV-algebras (introduced by Rutledge himself, apparently before the Scarpellini result)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-16
SLIDE 16

Logic

Infinite valued Łukasiewicz logic is a first order logic interpreted on the MV-algebra [0, 1]. MV-algebras (introduced by Chang) are the algebraic counterpart of propositional infinite valued Łukasiewicz logic. Now full first order Łukasiewicz logic is not axiomatizable (Scarpellini) However, monadic first order Łukasiewicz logic is axiomatizable (Rutledge) This motivates monadic MV-algebras (introduced by Rutledge himself, apparently before the Scarpellini result)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-17
SLIDE 17

Logic

Infinite valued Łukasiewicz logic is a first order logic interpreted on the MV-algebra [0, 1]. MV-algebras (introduced by Chang) are the algebraic counterpart of propositional infinite valued Łukasiewicz logic. Now full first order Łukasiewicz logic is not axiomatizable (Scarpellini) However, monadic first order Łukasiewicz logic is axiomatizable (Rutledge) This motivates monadic MV-algebras (introduced by Rutledge himself, apparently before the Scarpellini result)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-18
SLIDE 18

Logic

Infinite valued Łukasiewicz logic is a first order logic interpreted on the MV-algebra [0, 1]. MV-algebras (introduced by Chang) are the algebraic counterpart of propositional infinite valued Łukasiewicz logic. Now full first order Łukasiewicz logic is not axiomatizable (Scarpellini) However, monadic first order Łukasiewicz logic is axiomatizable (Rutledge) This motivates monadic MV-algebras (introduced by Rutledge himself, apparently before the Scarpellini result)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-19
SLIDE 19

MV-algebras

They are structures A = (X, ⊕, ¬, 0, 1), where (X, ⊕, 0) is an abelian monoid; ¬¬x = x; ¬0 = 1; x ⊕ 1 = 1; ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x (Mangani’s axiom).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-20
SLIDE 20

MV-algebras

They are structures A = (X, ⊕, ¬, 0, 1), where (X, ⊕, 0) is an abelian monoid; ¬¬x = x; ¬0 = 1; x ⊕ 1 = 1; ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x (Mangani’s axiom).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-21
SLIDE 21

MV-algebras

They are structures A = (X, ⊕, ¬, 0, 1), where (X, ⊕, 0) is an abelian monoid; ¬¬x = x; ¬0 = 1; x ⊕ 1 = 1; ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x (Mangani’s axiom).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-22
SLIDE 22

MV-algebras

They are structures A = (X, ⊕, ¬, 0, 1), where (X, ⊕, 0) is an abelian monoid; ¬¬x = x; ¬0 = 1; x ⊕ 1 = 1; ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x (Mangani’s axiom).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-23
SLIDE 23

MV-algebras

They are structures A = (X, ⊕, ¬, 0, 1), where (X, ⊕, 0) is an abelian monoid; ¬¬x = x; ¬0 = 1; x ⊕ 1 = 1; ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x (Mangani’s axiom).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-24
SLIDE 24

MV-algebras

They are structures A = (X, ⊕, ¬, 0, 1), where (X, ⊕, 0) is an abelian monoid; ¬¬x = x; ¬0 = 1; x ⊕ 1 = 1; ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x (Mangani’s axiom).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-25
SLIDE 25

MV-algebras

They are structures A = (X, ⊕, ¬, 0, 1), where (X, ⊕, 0) is an abelian monoid; ¬¬x = x; ¬0 = 1; x ⊕ 1 = 1; ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x (Mangani’s axiom).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-26
SLIDE 26

The Łukasiewicz product

The derived operation x ⊙ y = ¬(¬x ⊕ ¬y) is the De Morgan dual of the sum ⊕.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-27
SLIDE 27

The partial order of an MV-algebra

We let x ≤ y if y = x ⊕ z for some z. This is a lattice with x ∨ y = ¬(¬x ⊕ y) ⊕ y and x ∧ y = ¬(¬x ∨ ¬y). A linearly ordered MV-algebra is called an MV-chain.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-28
SLIDE 28

Ideals

An ideal of an MV-algebra A is a set I ⊆ A such that x, y ∈ I implies x ⊕ y ∈ I x ∈ I and y ≤ x imply y ∈ I. An ideal I is prime if I = A and whenever x ∧ y ∈ I we have x ∈ I or y ∈ I. Ideals and prime ideals can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-29
SLIDE 29

Ideals

An ideal of an MV-algebra A is a set I ⊆ A such that x, y ∈ I implies x ⊕ y ∈ I x ∈ I and y ≤ x imply y ∈ I. An ideal I is prime if I = A and whenever x ∧ y ∈ I we have x ∈ I or y ∈ I. Ideals and prime ideals can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-30
SLIDE 30

Ideals

An ideal of an MV-algebra A is a set I ⊆ A such that x, y ∈ I implies x ⊕ y ∈ I x ∈ I and y ≤ x imply y ∈ I. An ideal I is prime if I = A and whenever x ∧ y ∈ I we have x ∈ I or y ∈ I. Ideals and prime ideals can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-31
SLIDE 31

Ideals

An ideal of an MV-algebra A is a set I ⊆ A such that x, y ∈ I implies x ⊕ y ∈ I x ∈ I and y ≤ x imply y ∈ I. An ideal I is prime if I = A and whenever x ∧ y ∈ I we have x ∈ I or y ∈ I. Ideals and prime ideals can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-32
SLIDE 32

Filters

A filter of an MV-algebra A is a set F ⊆ A such that x, y ∈ F implies x ⊙ y ∈ F x ∈ F and y ≥ x imply y ∈ F. A filter F is prime if F = A and whenever x ∨ y ∈ F we have x ∈ F or y ∈ F. Filters and prime filters can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-33
SLIDE 33

Filters

A filter of an MV-algebra A is a set F ⊆ A such that x, y ∈ F implies x ⊙ y ∈ F x ∈ F and y ≥ x imply y ∈ F. A filter F is prime if F = A and whenever x ∨ y ∈ F we have x ∈ F or y ∈ F. Filters and prime filters can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-34
SLIDE 34

Filters

A filter of an MV-algebra A is a set F ⊆ A such that x, y ∈ F implies x ⊙ y ∈ F x ∈ F and y ≥ x imply y ∈ F. A filter F is prime if F = A and whenever x ∨ y ∈ F we have x ∈ F or y ∈ F. Filters and prime filters can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-35
SLIDE 35

Filters

A filter of an MV-algebra A is a set F ⊆ A such that x, y ∈ F implies x ⊙ y ∈ F x ∈ F and y ≥ x imply y ∈ F. A filter F is prime if F = A and whenever x ∨ y ∈ F we have x ∈ F or y ∈ F. Filters and prime filters can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-36
SLIDE 36

Filters

A filter of an MV-algebra A is a set F ⊆ A such that x, y ∈ F implies x ⊙ y ∈ F x ∈ F and y ≥ x imply y ∈ F. A filter F is prime if F = A and whenever x ∨ y ∈ F we have x ∈ F or y ∈ F. Filters and prime filters can be also defined in lattices.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-37
SLIDE 37

The prime spectrum of a lattice

Given a lattice L, Spec(L) is the set of all prime filters of L whose topology (Zariski topology) is generated by the opens Ua = {F ∈ Spec(L)|a ∈ F}.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-38
SLIDE 38

The importance of [0, 1]

Theorem (Di Nola embedding)

Every MV algebra embeds in a power of an ultrapower of [0, 1].

Corollary

[0, 1] generates the variety of MV algebras.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-39
SLIDE 39

The importance of [0, 1]

Theorem (Di Nola embedding)

Every MV algebra embeds in a power of an ultrapower of [0, 1].

Corollary

[0, 1] generates the variety of MV algebras.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-40
SLIDE 40

The importance of [0, 1]

Theorem (Di Nola embedding)

Every MV algebra embeds in a power of an ultrapower of [0, 1].

Corollary

[0, 1] generates the variety of MV algebras.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-41
SLIDE 41

Finite MV-algebras

The finite chains are Sn = {0, 1/n, 2/n, . . . , n − 1/n, 1}. Every finite MV-algebra is a finite product of chains.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-42
SLIDE 42

The spectrum problem for MV-algebras

We do not have a good topological characterization of spectra of MV-algebras (we have it as ordered sets thanks to Cignoli-Torrens, and we have it for countable MV-algebras thanks to Wehrung). One of the tools devised for this problem is Belluce functor, which replaces the MV-algebras with “simpler” objects.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-43
SLIDE 43

The spectrum problem for MV-algebras

We do not have a good topological characterization of spectra of MV-algebras (we have it as ordered sets thanks to Cignoli-Torrens, and we have it for countable MV-algebras thanks to Wehrung). One of the tools devised for this problem is Belluce functor, which replaces the MV-algebras with “simpler” objects.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-44
SLIDE 44

The spectrum problem for MV-algebras

We do not have a good topological characterization of spectra of MV-algebras (we have it as ordered sets thanks to Cignoli-Torrens, and we have it for countable MV-algebras thanks to Wehrung). One of the tools devised for this problem is Belluce functor, which replaces the MV-algebras with “simpler” objects.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-45
SLIDE 45

The Belluce β functor

Given MV-algebra A we define the equivalence x ≡ y if x and y belong to the same prime ideals. Define β(A) = A/ ≡, which has a natural structure of a lattice. Moreover, the prime spectra of A and β(A) are homeomorphic. β can be extended to a functor from MV-algebras to bounded distributive lattices by letting β(f)(β(x)) = β(f(x)). If we consider filters rather than ideals, we obtain a dual functor β∗.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-46
SLIDE 46

The Belluce β functor

Given MV-algebra A we define the equivalence x ≡ y if x and y belong to the same prime ideals. Define β(A) = A/ ≡, which has a natural structure of a lattice. Moreover, the prime spectra of A and β(A) are homeomorphic. β can be extended to a functor from MV-algebras to bounded distributive lattices by letting β(f)(β(x)) = β(f(x)). If we consider filters rather than ideals, we obtain a dual functor β∗.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-47
SLIDE 47

The Belluce β functor

Given MV-algebra A we define the equivalence x ≡ y if x and y belong to the same prime ideals. Define β(A) = A/ ≡, which has a natural structure of a lattice. Moreover, the prime spectra of A and β(A) are homeomorphic. β can be extended to a functor from MV-algebras to bounded distributive lattices by letting β(f)(β(x)) = β(f(x)). If we consider filters rather than ideals, we obtain a dual functor β∗.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-48
SLIDE 48

The Belluce β functor

Given MV-algebra A we define the equivalence x ≡ y if x and y belong to the same prime ideals. Define β(A) = A/ ≡, which has a natural structure of a lattice. Moreover, the prime spectra of A and β(A) are homeomorphic. β can be extended to a functor from MV-algebras to bounded distributive lattices by letting β(f)(β(x)) = β(f(x)). If we consider filters rather than ideals, we obtain a dual functor β∗.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-49
SLIDE 49

The Belluce β functor

Given MV-algebra A we define the equivalence x ≡ y if x and y belong to the same prime ideals. Define β(A) = A/ ≡, which has a natural structure of a lattice. Moreover, the prime spectra of A and β(A) are homeomorphic. β can be extended to a functor from MV-algebras to bounded distributive lattices by letting β(f)(β(x)) = β(f(x)). If we consider filters rather than ideals, we obtain a dual functor β∗.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-50
SLIDE 50

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-51
SLIDE 51

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-52
SLIDE 52

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-53
SLIDE 53

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-54
SLIDE 54

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-55
SLIDE 55

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-56
SLIDE 56

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-57
SLIDE 57

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-58
SLIDE 58

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-59
SLIDE 59

The category MMV of Monadic MV-algebras

Monadic MV-algebras are structures (A, ∃), where A is an MV-algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃¬(∃x) = ¬∃x ∃(∃x ⊕ ∃y) = ∃x ⊕ ∃y ∃(x ⊙ x) = ∃x ⊙ ∃x ∃(x ⊕ x) = ∃x ⊕ ∃x. Note that the axioms imply ∃∃x = ∃x and the range of ∃ is an MV-subalgebra. ∀ : A → A is defined as ∀a = ¬(∃¬a).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-60
SLIDE 60

Monadic Boolean algebras (Halmos)

Recall that Boolean algebras are idempotent MV-algebras (x ⊕ x = x) In the same vein, Monadic Boolean algebras are structures (A, ∃), where A is a Boolean algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃x ∧ ∃y = ∃(x ∧ ∃y). Monadic Boolean algebras are dual to Boolean spaces with Boolean equivalence relations. We do not have, instead, a duality for the full category of monadic MV-algebras (not even of MV-algebras).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-61
SLIDE 61

Monadic Boolean algebras (Halmos)

Recall that Boolean algebras are idempotent MV-algebras (x ⊕ x = x) In the same vein, Monadic Boolean algebras are structures (A, ∃), where A is a Boolean algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃x ∧ ∃y = ∃(x ∧ ∃y). Monadic Boolean algebras are dual to Boolean spaces with Boolean equivalence relations. We do not have, instead, a duality for the full category of monadic MV-algebras (not even of MV-algebras).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-62
SLIDE 62

Monadic Boolean algebras (Halmos)

Recall that Boolean algebras are idempotent MV-algebras (x ⊕ x = x) In the same vein, Monadic Boolean algebras are structures (A, ∃), where A is a Boolean algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃x ∧ ∃y = ∃(x ∧ ∃y). Monadic Boolean algebras are dual to Boolean spaces with Boolean equivalence relations. We do not have, instead, a duality for the full category of monadic MV-algebras (not even of MV-algebras).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-63
SLIDE 63

Monadic Boolean algebras (Halmos)

Recall that Boolean algebras are idempotent MV-algebras (x ⊕ x = x) In the same vein, Monadic Boolean algebras are structures (A, ∃), where A is a Boolean algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃x ∧ ∃y = ∃(x ∧ ∃y). Monadic Boolean algebras are dual to Boolean spaces with Boolean equivalence relations. We do not have, instead, a duality for the full category of monadic MV-algebras (not even of MV-algebras).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-64
SLIDE 64

Monadic Boolean algebras (Halmos)

Recall that Boolean algebras are idempotent MV-algebras (x ⊕ x = x) In the same vein, Monadic Boolean algebras are structures (A, ∃), where A is a Boolean algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃x ∧ ∃y = ∃(x ∧ ∃y). Monadic Boolean algebras are dual to Boolean spaces with Boolean equivalence relations. We do not have, instead, a duality for the full category of monadic MV-algebras (not even of MV-algebras).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-65
SLIDE 65

Monadic Boolean algebras (Halmos)

Recall that Boolean algebras are idempotent MV-algebras (x ⊕ x = x) In the same vein, Monadic Boolean algebras are structures (A, ∃), where A is a Boolean algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃x ∧ ∃y = ∃(x ∧ ∃y). Monadic Boolean algebras are dual to Boolean spaces with Boolean equivalence relations. We do not have, instead, a duality for the full category of monadic MV-algebras (not even of MV-algebras).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-66
SLIDE 66

Monadic Boolean algebras (Halmos)

Recall that Boolean algebras are idempotent MV-algebras (x ⊕ x = x) In the same vein, Monadic Boolean algebras are structures (A, ∃), where A is a Boolean algebra, ∃ : A → A and x ≤ ∃x ∃(x ∨ y) = ∃x ∨ ∃y ∃x ∧ ∃y = ∃(x ∧ ∃y). Monadic Boolean algebras are dual to Boolean spaces with Boolean equivalence relations. We do not have, instead, a duality for the full category of monadic MV-algebras (not even of MV-algebras).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-67
SLIDE 67

Structure of monadic MV-algebras

If (A, ∃) is a monadic MV-algebra then A0 = {x|∃x = x} is an MV-subalgebra of A which is m-relatively complete, that is, for every a ∈ A the infimum inf{b ∈ A0|b ≥ a} exists in A0 if a ∈ A, x ∈ A0, x ≥ a ⊙ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊙ v if a ∈ A, x ∈ A0, x ≥ a ⊕ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊕ v. Conversely, every mrc-subalgebra A0 of A gives a quantifier by letting ∃a = inf{b ∈ A0|b ≥ a}.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-68
SLIDE 68

Structure of monadic MV-algebras

If (A, ∃) is a monadic MV-algebra then A0 = {x|∃x = x} is an MV-subalgebra of A which is m-relatively complete, that is, for every a ∈ A the infimum inf{b ∈ A0|b ≥ a} exists in A0 if a ∈ A, x ∈ A0, x ≥ a ⊙ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊙ v if a ∈ A, x ∈ A0, x ≥ a ⊕ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊕ v. Conversely, every mrc-subalgebra A0 of A gives a quantifier by letting ∃a = inf{b ∈ A0|b ≥ a}.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-69
SLIDE 69

Structure of monadic MV-algebras

If (A, ∃) is a monadic MV-algebra then A0 = {x|∃x = x} is an MV-subalgebra of A which is m-relatively complete, that is, for every a ∈ A the infimum inf{b ∈ A0|b ≥ a} exists in A0 if a ∈ A, x ∈ A0, x ≥ a ⊙ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊙ v if a ∈ A, x ∈ A0, x ≥ a ⊕ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊕ v. Conversely, every mrc-subalgebra A0 of A gives a quantifier by letting ∃a = inf{b ∈ A0|b ≥ a}.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-70
SLIDE 70

Structure of monadic MV-algebras

If (A, ∃) is a monadic MV-algebra then A0 = {x|∃x = x} is an MV-subalgebra of A which is m-relatively complete, that is, for every a ∈ A the infimum inf{b ∈ A0|b ≥ a} exists in A0 if a ∈ A, x ∈ A0, x ≥ a ⊙ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊙ v if a ∈ A, x ∈ A0, x ≥ a ⊕ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊕ v. Conversely, every mrc-subalgebra A0 of A gives a quantifier by letting ∃a = inf{b ∈ A0|b ≥ a}.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-71
SLIDE 71

Structure of monadic MV-algebras

If (A, ∃) is a monadic MV-algebra then A0 = {x|∃x = x} is an MV-subalgebra of A which is m-relatively complete, that is, for every a ∈ A the infimum inf{b ∈ A0|b ≥ a} exists in A0 if a ∈ A, x ∈ A0, x ≥ a ⊙ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊙ v if a ∈ A, x ∈ A0, x ≥ a ⊕ a then there is v ∈ A0 with v ≥ a and x ≥ v ⊕ v. Conversely, every mrc-subalgebra A0 of A gives a quantifier by letting ∃a = inf{b ∈ A0|b ≥ a}.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-72
SLIDE 72

Monadic ideals

A monadic ideal of (A, ∃) is an MV-algebra ideal closed under ∃. There is an isomorphism between: the lattice of monadic ideals of (A, ∃); the lattice of congruences of (A, ∃); the lattice of ideals of ∃A.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-73
SLIDE 73

Monadic ideals

A monadic ideal of (A, ∃) is an MV-algebra ideal closed under ∃. There is an isomorphism between: the lattice of monadic ideals of (A, ∃); the lattice of congruences of (A, ∃); the lattice of ideals of ∃A.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-74
SLIDE 74

Monadic ideals

A monadic ideal of (A, ∃) is an MV-algebra ideal closed under ∃. There is an isomorphism between: the lattice of monadic ideals of (A, ∃); the lattice of congruences of (A, ∃); the lattice of ideals of ∃A.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-75
SLIDE 75

Monadic ideals

A monadic ideal of (A, ∃) is an MV-algebra ideal closed under ∃. There is an isomorphism between: the lattice of monadic ideals of (A, ∃); the lattice of congruences of (A, ∃); the lattice of ideals of ∃A.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-76
SLIDE 76

Birkhoff subdirect representation

Theorem

(Rutledge) Every monadic MV-algebra (A, ∃) is a subdirect product of monadic MV-algebras (Ai, ∃i) where ∃iAi is totally ordered.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-77
SLIDE 77

The totally ordered case is trivial

Lemma

If A0 is an m-relatively complete totally ordered MV-subalgebra of an MV-algebra A, then A0 is a maximal totally ordered subalgebra of A.

Corollary

If (A, ∃) is totally ordered then A = ∃A.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-78
SLIDE 78

The totally ordered case is trivial

Lemma

If A0 is an m-relatively complete totally ordered MV-subalgebra of an MV-algebra A, then A0 is a maximal totally ordered subalgebra of A.

Corollary

If (A, ∃) is totally ordered then A = ∃A.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-79
SLIDE 79

An example of monadic MV-algebra

A diagonal construction: A = [0, 1]n ∃(x1, . . . , xn) = (m, m, . . . , m) where m = max{x1, . . . , xn}.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-80
SLIDE 80

An example of monadic MV-algebra

A diagonal construction: A = [0, 1]n ∃(x1, . . . , xn) = (m, m, . . . , m) where m = max{x1, . . . , xn}.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-81
SLIDE 81

The finite case

Theorem

If (A, ∃) is a finite monadic MV-algebra with totally ordered ∃A, then A = (∃A)n and ∃(x1, . . . , xn) = (m, m, . . . , m), where m = max{x1, . . . , xn}. More generally, if A = Sn1 × . . . × Snk is finite, then monadic structures can be found by considering homogeneous partitions of {1, . . . , k}, that is partitions where two equivalent indices correspond to equal chains. On each block of the partition, one can perform the diagonal construction.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-82
SLIDE 82

The finite case

Theorem

If (A, ∃) is a finite monadic MV-algebra with totally ordered ∃A, then A = (∃A)n and ∃(x1, . . . , xn) = (m, m, . . . , m), where m = max{x1, . . . , xn}. More generally, if A = Sn1 × . . . × Snk is finite, then monadic structures can be found by considering homogeneous partitions of {1, . . . , k}, that is partitions where two equivalent indices correspond to equal chains. On each block of the partition, one can perform the diagonal construction.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-83
SLIDE 83

The finite case

Theorem

If (A, ∃) is a finite monadic MV-algebra with totally ordered ∃A, then A = (∃A)n and ∃(x1, . . . , xn) = (m, m, . . . , m), where m = max{x1, . . . , xn}. More generally, if A = Sn1 × . . . × Snk is finite, then monadic structures can be found by considering homogeneous partitions of {1, . . . , k}, that is partitions where two equivalent indices correspond to equal chains. On each block of the partition, one can perform the diagonal construction.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-84
SLIDE 84

Dualities

In duality theory, “abstract” algebraic objects are put in correspondence with “concrete” geometric or topological objects. The theory of lattices gives a huge amount of examples. Here we will

  • nly recall some of them.

Calculate! (Leibniz) Topologize! (Stone)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-85
SLIDE 85

Dualities

In duality theory, “abstract” algebraic objects are put in correspondence with “concrete” geometric or topological objects. The theory of lattices gives a huge amount of examples. Here we will

  • nly recall some of them.

Calculate! (Leibniz) Topologize! (Stone)

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-86
SLIDE 86

Priestley spaces

Priestley discovered a duality between the category of bounded distributive lattices and the category of Priestley spaces, extending Stone duality for Boolean algebras. A Priestley space is a structure (X, R), where X is a compact topological space and R is an order relation on X such that, for all x, y ∈ X, either xRy or there is a clopen up-set V with x ∈ V and y / ∈ V. We denote by P(X) the set of clopen up-sets of X. A morphism of Priestley spaces is a continuous, order preserving map.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-87
SLIDE 87

Priestley spaces

Priestley discovered a duality between the category of bounded distributive lattices and the category of Priestley spaces, extending Stone duality for Boolean algebras. A Priestley space is a structure (X, R), where X is a compact topological space and R is an order relation on X such that, for all x, y ∈ X, either xRy or there is a clopen up-set V with x ∈ V and y / ∈ V. We denote by P(X) the set of clopen up-sets of X. A morphism of Priestley spaces is a continuous, order preserving map.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-88
SLIDE 88

Priestley duality

The dual of L is (Spec(L), ⊆) where Spec(L) is the prime spectrum of L equipped with the patch topology (the one generated by {P|a ∈ P} and {P|a / ∈ P} for a ∈ L). The dual of (X, R) is P(X). In both senses, the duality on morphisms is given by the inverse image. What if quantifiers are added?

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-89
SLIDE 89

Priestley duality

The dual of L is (Spec(L), ⊆) where Spec(L) is the prime spectrum of L equipped with the patch topology (the one generated by {P|a ∈ P} and {P|a / ∈ P} for a ∈ L). The dual of (X, R) is P(X). In both senses, the duality on morphisms is given by the inverse image. What if quantifiers are added?

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-90
SLIDE 90

Priestley duality

The dual of L is (Spec(L), ⊆) where Spec(L) is the prime spectrum of L equipped with the patch topology (the one generated by {P|a ∈ P} and {P|a / ∈ P} for a ∈ L). The dual of (X, R) is P(X). In both senses, the duality on morphisms is given by the inverse image. What if quantifiers are added?

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-91
SLIDE 91

Priestley duality

The dual of L is (Spec(L), ⊆) where Spec(L) is the prime spectrum of L equipped with the patch topology (the one generated by {P|a ∈ P} and {P|a / ∈ P} for a ∈ L). The dual of (X, R) is P(X). In both senses, the duality on morphisms is given by the inverse image. What if quantifiers are added?

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-92
SLIDE 92

The category QD of Q-distributive lattices

Intuitively, Q-distributive lattices are negation-free monadic Boolean algebras. Q-distributive lattices (Cignoli) are structures (L, ∃) where L is a bounded distributive lattice, ∃ : L → L and ∃0 = 0 a ∧ ∃a = a ∃(a ∧ ∃b) = ∃a ∧ ∃b ∃(a ∨ b) = ∃a ∨ ∃b

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-93
SLIDE 93

The category QD of Q-distributive lattices

Intuitively, Q-distributive lattices are negation-free monadic Boolean algebras. Q-distributive lattices (Cignoli) are structures (L, ∃) where L is a bounded distributive lattice, ∃ : L → L and ∃0 = 0 a ∧ ∃a = a ∃(a ∧ ∃b) = ∃a ∧ ∃b ∃(a ∨ b) = ∃a ∨ ∃b

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-94
SLIDE 94

The category QD of Q-distributive lattices

Intuitively, Q-distributive lattices are negation-free monadic Boolean algebras. Q-distributive lattices (Cignoli) are structures (L, ∃) where L is a bounded distributive lattice, ∃ : L → L and ∃0 = 0 a ∧ ∃a = a ∃(a ∧ ∃b) = ∃a ∧ ∃b ∃(a ∨ b) = ∃a ∨ ∃b

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-95
SLIDE 95

The category QD of Q-distributive lattices

Intuitively, Q-distributive lattices are negation-free monadic Boolean algebras. Q-distributive lattices (Cignoli) are structures (L, ∃) where L is a bounded distributive lattice, ∃ : L → L and ∃0 = 0 a ∧ ∃a = a ∃(a ∧ ∃b) = ∃a ∧ ∃b ∃(a ∨ b) = ∃a ∨ ∃b

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-96
SLIDE 96

The category QD of Q-distributive lattices

Intuitively, Q-distributive lattices are negation-free monadic Boolean algebras. Q-distributive lattices (Cignoli) are structures (L, ∃) where L is a bounded distributive lattice, ∃ : L → L and ∃0 = 0 a ∧ ∃a = a ∃(a ∧ ∃b) = ∃a ∧ ∃b ∃(a ∨ b) = ∃a ∨ ∃b

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-97
SLIDE 97

The category QD of Q-distributive lattices

Intuitively, Q-distributive lattices are negation-free monadic Boolean algebras. Q-distributive lattices (Cignoli) are structures (L, ∃) where L is a bounded distributive lattice, ∃ : L → L and ∃0 = 0 a ∧ ∃a = a ∃(a ∧ ∃b) = ∃a ∧ ∃b ∃(a ∨ b) = ∃a ∨ ∃b

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-98
SLIDE 98

The category QD∗ of Q-spaces

Cignoli found a dual category to Q-distributive lattices: Q-spaces. A Q-space (Cignoli) is a structure (X, R, E) where (X, R) is a Priestley space and E is an equivalence on X such that For every U ∈ P(X) we have E(U) ∈ P(X) The equivalence classes of E are closed in X. A morphism of Q spaces (X, R, E) and (Y, S, F) is a map f : X → Y which is continuous, order preserving and such that E(f −1(V)) = f −1(F(V)) for every V ∈ P(Y).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-99
SLIDE 99

The category QD∗ of Q-spaces

Cignoli found a dual category to Q-distributive lattices: Q-spaces. A Q-space (Cignoli) is a structure (X, R, E) where (X, R) is a Priestley space and E is an equivalence on X such that For every U ∈ P(X) we have E(U) ∈ P(X) The equivalence classes of E are closed in X. A morphism of Q spaces (X, R, E) and (Y, S, F) is a map f : X → Y which is continuous, order preserving and such that E(f −1(V)) = f −1(F(V)) for every V ∈ P(Y).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-100
SLIDE 100

The category QD∗ of Q-spaces

Cignoli found a dual category to Q-distributive lattices: Q-spaces. A Q-space (Cignoli) is a structure (X, R, E) where (X, R) is a Priestley space and E is an equivalence on X such that For every U ∈ P(X) we have E(U) ∈ P(X) The equivalence classes of E are closed in X. A morphism of Q spaces (X, R, E) and (Y, S, F) is a map f : X → Y which is continuous, order preserving and such that E(f −1(V)) = f −1(F(V)) for every V ∈ P(Y).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-101
SLIDE 101

The category QD∗ of Q-spaces

Cignoli found a dual category to Q-distributive lattices: Q-spaces. A Q-space (Cignoli) is a structure (X, R, E) where (X, R) is a Priestley space and E is an equivalence on X such that For every U ∈ P(X) we have E(U) ∈ P(X) The equivalence classes of E are closed in X. A morphism of Q spaces (X, R, E) and (Y, S, F) is a map f : X → Y which is continuous, order preserving and such that E(f −1(V)) = f −1(F(V)) for every V ∈ P(Y).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-102
SLIDE 102

The functor Q∗ from QD to QD∗

We define Q∗(L, ∃) = (Spec(L), ⊆, E(L, ∃)) where: Spec(L) is the prime spectrum of L, i.e. the set of prime filters of L the topology on Spec(L) is the patch topology ⊆ is the inclusion relation in Spec(L) (so (Spec(L), ⊆) is a Priestley space) E(L, ∃) = {(F, G) ∈ Spec(L)2|F ∩ ∃L = G ∩ ∃L}. Given a morphism of Q-distributive lattices h : A → B, we define Q∗(h) by Q∗(h)(P) = h−1(P) for every P ∈ Spec(B).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-103
SLIDE 103

The functor Q∗ from QD to QD∗

We define Q∗(L, ∃) = (Spec(L), ⊆, E(L, ∃)) where: Spec(L) is the prime spectrum of L, i.e. the set of prime filters of L the topology on Spec(L) is the patch topology ⊆ is the inclusion relation in Spec(L) (so (Spec(L), ⊆) is a Priestley space) E(L, ∃) = {(F, G) ∈ Spec(L)2|F ∩ ∃L = G ∩ ∃L}. Given a morphism of Q-distributive lattices h : A → B, we define Q∗(h) by Q∗(h)(P) = h−1(P) for every P ∈ Spec(B).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-104
SLIDE 104

The functor Q∗ from QD to QD∗

We define Q∗(L, ∃) = (Spec(L), ⊆, E(L, ∃)) where: Spec(L) is the prime spectrum of L, i.e. the set of prime filters of L the topology on Spec(L) is the patch topology ⊆ is the inclusion relation in Spec(L) (so (Spec(L), ⊆) is a Priestley space) E(L, ∃) = {(F, G) ∈ Spec(L)2|F ∩ ∃L = G ∩ ∃L}. Given a morphism of Q-distributive lattices h : A → B, we define Q∗(h) by Q∗(h)(P) = h−1(P) for every P ∈ Spec(B).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-105
SLIDE 105

The functor Q∗ from QD to QD∗

We define Q∗(L, ∃) = (Spec(L), ⊆, E(L, ∃)) where: Spec(L) is the prime spectrum of L, i.e. the set of prime filters of L the topology on Spec(L) is the patch topology ⊆ is the inclusion relation in Spec(L) (so (Spec(L), ⊆) is a Priestley space) E(L, ∃) = {(F, G) ∈ Spec(L)2|F ∩ ∃L = G ∩ ∃L}. Given a morphism of Q-distributive lattices h : A → B, we define Q∗(h) by Q∗(h)(P) = h−1(P) for every P ∈ Spec(B).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-106
SLIDE 106

The functor Q from QD∗ to QD

We define Q(X, R, E) = (P(X), E). Given f : (X, R, E) → (Y, S, F), we define Q(f) by Q(f)(V) = f −1(V) for every V ∈ P(Y). The pair (Q, Q∗) is a duality between QD and QD∗ (Cignoli).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-107
SLIDE 107

The functor Q from QD∗ to QD

We define Q(X, R, E) = (P(X), E). Given f : (X, R, E) → (Y, S, F), we define Q(f) by Q(f)(V) = f −1(V) for every V ∈ P(Y). The pair (Q, Q∗) is a duality between QD and QD∗ (Cignoli).

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-108
SLIDE 108

The functor γ from MMV to QD

γ is an extension of Belluce β functor. In fact, γ(A, ∃) = (A/ ≡′, ∃′) where x ≡′ y if x and y belong to the same prime ideals of A, and ∃x and ∃y belong to the same prime ideals; moreover ∃′[a] = [∃a] where [a] is the equivalence class of a modulo ≡′. γ becomes a functor from MMV-algebras to Q-distributive lattices by γ(f)(γ(x)) = γ(f(x)). Like for β, the prime spectra of (A, ∃) and γ(A, ∃) are homeomorphic.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-109
SLIDE 109

The functor γ from MMV to QD

γ is an extension of Belluce β functor. In fact, γ(A, ∃) = (A/ ≡′, ∃′) where x ≡′ y if x and y belong to the same prime ideals of A, and ∃x and ∃y belong to the same prime ideals; moreover ∃′[a] = [∃a] where [a] is the equivalence class of a modulo ≡′. γ becomes a functor from MMV-algebras to Q-distributive lattices by γ(f)(γ(x)) = γ(f(x)). Like for β, the prime spectra of (A, ∃) and γ(A, ∃) are homeomorphic.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-110
SLIDE 110

The functor γ from MMV to QD

γ is an extension of Belluce β functor. In fact, γ(A, ∃) = (A/ ≡′, ∃′) where x ≡′ y if x and y belong to the same prime ideals of A, and ∃x and ∃y belong to the same prime ideals; moreover ∃′[a] = [∃a] where [a] is the equivalence class of a modulo ≡′. γ becomes a functor from MMV-algebras to Q-distributive lattices by γ(f)(γ(x)) = γ(f(x)). Like for β, the prime spectra of (A, ∃) and γ(A, ∃) are homeomorphic.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-111
SLIDE 111

The codomain of γ

From the theory of prime spectra of MV-algebras it follows that the co-domain of the functor γ is given by the dual completely normal distributive lattices, that is (Wehrung) for every a, b there are x, y such that a ≥ b ∧ x, b ≥ a ∧ y and x ∨ y = 1.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-112
SLIDE 112

The codomain of γ

From the theory of prime spectra of MV-algebras it follows that the co-domain of the functor γ is given by the dual completely normal distributive lattices, that is (Wehrung) for every a, b there are x, y such that a ≥ b ∧ x, b ≥ a ∧ y and x ∨ y = 1.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-113
SLIDE 113

Summing up

By composing γ with Q∗, we obtain a functor from monadic MV-algebras to Q-spaces, actually monadic Q-spaces which we define right now.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-114
SLIDE 114

The category of monadic Q-spaces

A monadic Q-space is a Q-space (X, R, E) such that R(x) is a chain for every x RE(x) = ER(x) R−1E(x) = ER−1(x) R(x) ∩ E(x) = R−1(x) ∩ E(x) = {x}. A morphism of monadic Q-spaces is a strongly isotone mapping of Q-spaces. Recall that a monotonic map f : X → Y between spaces (X, R, E) and (Y, S, F) is strongly isotone if R(f(x)) = f(S(x)) for every x ∈ X.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-115
SLIDE 115

The category of monadic Q-spaces

A monadic Q-space is a Q-space (X, R, E) such that R(x) is a chain for every x RE(x) = ER(x) R−1E(x) = ER−1(x) R(x) ∩ E(x) = R−1(x) ∩ E(x) = {x}. A morphism of monadic Q-spaces is a strongly isotone mapping of Q-spaces. Recall that a monotonic map f : X → Y between spaces (X, R, E) and (Y, S, F) is strongly isotone if R(f(x)) = f(S(x)) for every x ∈ X.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-116
SLIDE 116

The category of monadic Q-spaces

A monadic Q-space is a Q-space (X, R, E) such that R(x) is a chain for every x RE(x) = ER(x) R−1E(x) = ER−1(x) R(x) ∩ E(x) = R−1(x) ∩ E(x) = {x}. A morphism of monadic Q-spaces is a strongly isotone mapping of Q-spaces. Recall that a monotonic map f : X → Y between spaces (X, R, E) and (Y, S, F) is strongly isotone if R(f(x)) = f(S(x)) for every x ∈ X.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-117
SLIDE 117

The category of monadic Q-spaces

A monadic Q-space is a Q-space (X, R, E) such that R(x) is a chain for every x RE(x) = ER(x) R−1E(x) = ER−1(x) R(x) ∩ E(x) = R−1(x) ∩ E(x) = {x}. A morphism of monadic Q-spaces is a strongly isotone mapping of Q-spaces. Recall that a monotonic map f : X → Y between spaces (X, R, E) and (Y, S, F) is strongly isotone if R(f(x)) = f(S(x)) for every x ∈ X.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-118
SLIDE 118

The category of monadic Q-spaces

A monadic Q-space is a Q-space (X, R, E) such that R(x) is a chain for every x RE(x) = ER(x) R−1E(x) = ER−1(x) R(x) ∩ E(x) = R−1(x) ∩ E(x) = {x}. A morphism of monadic Q-spaces is a strongly isotone mapping of Q-spaces. Recall that a monotonic map f : X → Y between spaces (X, R, E) and (Y, S, F) is strongly isotone if R(f(x)) = f(S(x)) for every x ∈ X.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-119
SLIDE 119

The category of monadic Q-spaces

A monadic Q-space is a Q-space (X, R, E) such that R(x) is a chain for every x RE(x) = ER(x) R−1E(x) = ER−1(x) R(x) ∩ E(x) = R−1(x) ∩ E(x) = {x}. A morphism of monadic Q-spaces is a strongly isotone mapping of Q-spaces. Recall that a monotonic map f : X → Y between spaces (X, R, E) and (Y, S, F) is strongly isotone if R(f(x)) = f(S(x)) for every x ∈ X.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-120
SLIDE 120

The category of monadic Q-spaces

A monadic Q-space is a Q-space (X, R, E) such that R(x) is a chain for every x RE(x) = ER(x) R−1E(x) = ER−1(x) R(x) ∩ E(x) = R−1(x) ∩ E(x) = {x}. A morphism of monadic Q-spaces is a strongly isotone mapping of Q-spaces. Recall that a monotonic map f : X → Y between spaces (X, R, E) and (Y, S, F) is strongly isotone if R(f(x)) = f(S(x)) for every x ∈ X.

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-121
SLIDE 121

If you want to know more...

Antonio Di Nola: adinola@unisa.it Revaz Grigolia: revaz.grigolia@tsu.ge Giacomo Lenzi: gilenzi@unisa.it

Topological spaces of monadic MV-algebras

  • G. Lenzi
slide-122
SLIDE 122

Thank you!

Topological spaces of monadic MV-algebras

  • G. Lenzi