Co-Stone Residuated Lattices Claudia MURES AN c.muresan@yahoo.com, - - PowerPoint PPT Presentation

Co-Stone Residuated Lattices

Claudia MURES ¸AN c.muresan@yahoo.com, cmuresan11@gmail.com

University of Bucharest Faculty of Mathematics and Computer Science Bucharest

December 12, 2009

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Contents

1

Preliminaries

2

Co-Stone Algebras

3

The Strongly Co-Stone Hull of a Residuated Lattice

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1

Preliminaries

2

Co-Stone Algebras

3

The Strongly Co-Stone Hull of a Residuated Lattice

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Definition

Residuated lattice: (A, ∨, ∧, ⊙, →, 0, 1), with: (A, ∨, ∧, 0, 1) bounded lattice; (A, ⊙, 1) commutative monoid; the law of residuation: for all a, b, c ∈ A, a ≤ b → c iff a ⊙ b ≤ c. monoidal logic A residuated lattice a, b ∈ A n ∈ N∗ a ↔ b = (a → b) ∧ (b → a) ¬ a = a → 0 an = a ⊙ . . . ⊙ a

• n of a

; a0 = 1

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A lattice (residuated lattice) F, G filters of A F ∨ G =< F ∪ G > More generally: {Ft|t ∈ T} family of filters of A

• t∈T

Ft =<

• t∈T

Ft > (F(A), ∨, ∩, {1}, A) = the lattice of filters of A

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A bounded lattice or residuated lattice B(A) = the Boolean center of A for A a bounded distributive lattice or a residuated lattice, B(A) is a Boolean algebra with the operations induced by those of A

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A lattice (residuated lattice) Spec(A), Max(A) topological spaces with the Stone topologies RL = the category of residuated lattices D01 = the category of bounded distributive lattices

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A bounded distributive lattice or residuated lattice ∅ = X ⊆ A x ∈ A the co-annihilator of X: X ⊤ = {a ∈ A|(∀y ∈ X)a ∨ y = 1} the co-annihilator of x: x⊤ = {x}⊤ = {a ∈ A|a ∨ x = 1} X ⊤⊤ = (X ⊤)⊤ x⊤⊤ = (x⊤)⊤ the co-annihilators are filters of A

Definition

Let A be a bounded distributive lattice or a residuated lattice. Then A is said to be co-Stone (respectively strongly co-Stone) iff, for all x ∈ A (respectively all ∅ = X ⊆ A), there exists an element e ∈ B(A) such that x⊤ =< e > (respectively X ⊤ =< e >).

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A bounded distributive lattice or residuated lattice CoAnn(A) = {X ⊤|∅ = X ⊆ A} F, G ∈ CoAnn(A) F ∨⊤ G = (F ⊤ ∩ G ⊤)⊤ More generally: {Ft|t ∈ T} family of filters of A

• t∈T

⊤Ft = t∈T

F ⊤

t

⊤

Proposition

Let A be a bounded distributive lattice or a residuated lattice. Then (CoAnn(A), ∨⊤, ∩,⊤ , {1}, A) is a complete Boolean algebra.

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Definition

Let m be an infinite cardinal. An m-complete lattice is a lattice L with the property that any subset X of L with |X| ≤ m has an infimum and a supremum in L.

Theorem

Let L be a bounded distributive lattice and m an infinite cardinal. Then the following are equivalent: (i) for each nonempty subset X of L with |X| ≤ m, there exists an element e ∈ B(L) such that X ⊤ =< e >; (ii) L is a co-Stone lattice and B(L) is an m-complete Boolean algebra; (iii) L⊤⊤ = {l⊤⊤|l ∈ L} is an m-complete Boolean sublattice of F(L); (iv) for all l, p ∈ L, (l ∨ p)⊤ = l⊤ ∨ p⊤ and, for each nonempty subset X

• f L with |X| ≤ m, there exists an element x ∈ L such that X ⊤⊤ = x⊤;

(v) for each nonempty subset X of L with |X| ≤ m, X ⊤ ∨ X ⊤⊤ = L. A bounded distributive lattice will be called an m-co-Stone lattice iff the conditions of the previous theorem hold for it.

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Definition

Let A be a bounded lattice (residuated lattice) and B a subalgebra of A. We say that B is co-dense in A iff, for all a ∈ A \ {1}, there exists b ∈ B such that a ≤ b < 1 (that is a ≤ b ≤ 1 and b = 1).

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Previous definitions of the reticulation:

• H. Simmons: commutative rings (1980, [21])
• L. P. Belluce: MV-algebras (1986, [2])
• L. P. Belluce: non-commutative rings (1991, [3])
• G. Georgescu: quantales (1995, [9])
• L. Leu¸

stean: BL-algebras (2003, [15], [16]) In each of the papers cited above, although it is not explicitely defined this way, the reticulation of an algebra A is a pair (L(A), λ) consisting of a bounded distributive lattice L(A) and a surjection λ : A → L(A) such that the function given by the inverse image of λ induces (by restriction) a homeomorphism between the prime spectrum of L(A) and that of A. This construction allows many properties to be transferred between L(A) and A, and this transfer of properties between the category of bounded distributive lattices and another category (in our case that of residuated lattices) is the very purpose of the reticulation.

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An axiomatic purely algebraic definition of the reticulation: innovation in the study of the reticulation, as in previous work the reticulation of an algebra was defined by its construction very useful in practice

Definition

[17] Let A be a residuated lattice. A reticulation of A is a pair (L, λ), where L is a bounded distributive lattice and λ : A → L is a function that satisfies conditions 1)-5) below: 1) for all a, b ∈ A, λ(a ⊙ b) = λ(a) ∧ λ(b); 2) for all a, b ∈ A, λ(a ∨ b) = λ(a) ∨ λ(b); 3) λ(0) = 0; λ(1) = 1; 4) λ is surjective; 5) for all a, b ∈ A, λ(a) ≤ λ(b) iff (∃ n ∈ N∗) an ≤ b.

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In [17] and [18] we proved that this definition is in accordance with the general notion of reticulation applied to residuated lattices, more precisely that, given a residuated lattice A and a pair (L, λ) consisting of a bounded distributive lattice L and a function λ : A → L, we have: if λ satisfies conditions 1)-5) above, then its inverse image induces (by restriction) a homeomorphism between the prime spectrum of L and that of A; and conversely: if the function given by the inverse image of λ takes prime filters of L to prime filters of A and its restriction to the prime spectrum of L is a homeomorphism between the prime spectrum of L and that of A, then λ satisfies conditions 1)-5) from the definition above.

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The following theorem states the existence and uniqueness of the reticulation for any residuated lattice. usefulness of the axiomatic purely algebraic definition of the reticulation: simple algebraic proof for the uniqueness of the reticulation in previous work the argument for the uniqueness of the reticulation was of topological nature and consisted of the fact that there is at most one bounded distributive lattice whose prime spectrum is homeomorphic to a given topological space

Theorem

[17] Let A be a residuated lattice. Then there exists a reticulation of A. Let (L1, λ1), (L2, λ2) be two reticulations of A. Then there exists an isomorphism of bounded lattices f : L1 → L2 such that f ◦ λ1 = λ2.

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Definition of the reticulation functor L : RL → D01: A residuated lattice (L(A), λA) the reticulation of A L(A) = L(A) B residuated lattice (L(B), λB) the reticulation of B f : A → B a morphism of residuated lattices L(f ) : L(A) = L(A) → L(B) = L(B) for all a ∈ A, L(f )(λA(a)) = λB(f (a)) A

❄

λA λB L(A)

✲

f B L(B)

✲ ❄

L(f ) This definition makes L a covariant functor from RL to D01.

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1

Preliminaries

2

Co-Stone Algebras

3

The Strongly Co-Stone Hull of a Residuated Lattice

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structure = bounded distributive lattice or residuated lattice any strongly co-Stone structure is co-Stone and any complete co-Stone structure is strongly co-Stone the trivial structure is strongly co-Stone any chain is strongly co-Stone, because a chain A clearly has all co-annihilators equal to {1} =< 1 >, except for 1⊤, which is equal to A =< 0 >

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In the following, unless mentioned otherwise, let A be a residuated lattice.

Proposition

A is a co-Stone residuated lattice iff L(A) is a co-Stone lattice.

Proposition

A is a strongly co-Stone residuated lattice iff L(A) is a strongly co-Stone lattice.

Proposition

CoAnn(A) and CoAnn(L(A)) are isomorphic Boolean algebras.

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Theorem

Let A be a residuated lattice and m an infinite cardinal. Then the following are equivalent: (I) for each nonempty subset X of A with |X| ≤ m, there exists an element e ∈ B(A) such that X ⊤ =< e >; (II) A is a co-Stone residuated lattice and B(A) is an m-complete Boolean algebra; (III) A⊤⊤ = {a⊤⊤|a ∈ A} is an m-complete Boolean sublattice of F(A); (IV) for all a, b ∈ A, (a ∨ b)⊤ = a⊤ ∨ b⊤ and, for each nonempty subset X

• f A with |X| ≤ m, there exists an element x ∈ A such that X ⊤⊤ = x⊤;

(V) for each nonempty subset X of A with |X| ≤ m, X ⊤ ∨ X ⊤⊤ = A. A residuated lattice will be called an m-co-Stone residuated lattice iff the conditions of the previous theorem hold for it.

Proposition

A is an m-co-Stone residuated lattice iff L(A) is an m-co-Stone lattice.

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• R. Cignoli: Stone residuated lattices are those residuated lattices A

that satisfy the identity ¬ a ∨ ¬ ¬ a = 1 for all a ∈ A ([5])

Remark

There exist co-Stone residuated lattices A with elements a ∈ A that do not satisfy the identity ¬ a ∨ ¬ ¬ a = 1.

Remark

There exist residuated lattices A that satisfy the identity ¬ a ∨ ¬ ¬ a = 1 for all a ∈ A and that are not co-Stone. hence co-Stone residuated lattices do not have a characterization like the one in [1, Theorem 8.7.1, page 164] for Stone pseudocomplemented distributive lattices and thus our definition of co-Stone residuated lattices does not coincide with the one given by R. Cignoli in [5]

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Remark

There exist residuated lattices A that do not satisfy the identity ¬ a ∨ ¬ ¬ a = 1 for all a ∈ A, but whose reticulations L(A) are pseudocomplemented lattices and satisfy this identity: l∗ ∨ l∗∗ = 1 for all l ∈ L(A), where we have denoted by l∗ the pseudocomplement of l. thus the definition from [5] is not transferrable through the reticulation, which is the reason why we have chosen our definition

• ver it
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1

Preliminaries

2

Co-Stone Algebras

3

The Strongly Co-Stone Hull of a Residuated Lattice

• C. Mure¸

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In the following, let A be a residuated lattice. We shall define the strongly co-Stone hull of A, in a manner similar to the definition from [8] for the strongly Stone hull of an MV-algebra. B = CoAnn(A) the poset Π(A) = P(B) of the finite partitions of B, with the partial

• rder relation given by:

for any C, D ∈ Π(A), C ≤ D iff D is a refinement of C for any C ∈ Π(A), AC =

• C∈C

A/(C ⊤) for every C, D ∈ Π(A) with C ≤ D, PCD : AC → AD, for all (aC)C∈C, PCD((aC/(C ⊤))C∈C) = (bD/(D⊤))D∈D, where, by definition, for all D ∈ D, bD = aC, where C is the unique member of C such that D ⊆ C PCD is an injective morphism of residuated lattices ((AC)C∈Π(A), (PCD)C≤D) is an inductive system of residuated lattices ˜ A = lim − →

C∈Π(A)

AC by the uniqueness of the inductive limit, it follows that ˜ A is unique up to a residuated lattice isomorphism

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Definition

We define ˜ A to be the strongly co-Stone hull of A. a ∈ A C ∈ Π(A) aC = (a/(C ⊤))C∈C [aC] = the congruence class of aC in ˜ A ǫ(a) = [aC] the definition of ǫ does not depend on C ǫ : A → ˜ A is an injective morphism of residuated lattices

• ne can identify A and ǫ(A), hence one can consider A to be a

subalgebra of ˜ A

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Lemma

˜ A is a strongly co-Stone residuated lattice.

Proposition

A is co-dense in ˜ A.

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Conjecture

The strongly co-Stone hull of A, ˜ A, verifies the following universality property: for any strongly co-Stone residuated lattice A1 and any morphism of residuated lattices f : A → A1 with the property that, for any ∅ = K ⊆ A, we have f (K ⊤) = f (K)⊤, there exists a unique morphism of residuated lattices f : ˜ A → A1 such that f ◦ ǫ = f . A

✲ ˜

A ǫ

❄

f A1

❅ ❅ ❅ ❘

f

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A similar definition can be given for the unique strongly co-Stone hull of a bounded distributive lattice. This definition is in accordance with the one from [7].

Proposition

L preserves the strongly co-Stone hull, namely the reticulation of the strongly co-Stone hull of a residuated lattice equals the strongly co-Stone hull of the reticulation of that residuated lattice. In order to prove that L(A) and L(˜ A) are isomorphic bounded lattices, we have used the fact that L preserves inductive limits.

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Next we will determine the strongly co-Stone hull of the following residuated lattice A, as an example of computation of the strongly co-Stone hull:

r r a ❅ ❅

• r

b

r c

• ❅

❅ r

1 → a b c 1 1 1 1 1 1 a 1 1 1 1 b c 1 c 1 c b b 1 1 1 a b c 1 and ⊙ = ∧.

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Let B = CoAnn(A). 0⊤ = a⊤ = {1}, 1⊤ = A, b⊤ = {c, 1} =< c > and c⊤ = {b, 1} =< b >, hence B = {1⊤, b⊤, c⊤, 0⊤} and Π(A) = P(B) = {{1⊤}, {b⊤, c⊤}}. Let C = {1⊤} and D = {b⊤, c⊤}. C ≤ D. AC = A/A = {1/A} and AD = A/b⊤ × A/c⊤. As the table of the operation ↔ shows, 0/b⊤ = {0}, a/b⊤ = {a, b} = b/b⊤ and c/b⊤ = 1/b⊤ = b⊤, so A/b⊤ = {0/b⊤, a/b⊤, 1/b⊤}, and 0/c⊤ = {0}, a/c⊤ = {a, c} = c/c⊤ and b/c⊤ = 1/c⊤ = c⊤, so A/c⊤ = {0/c⊤, a/c⊤, 1/c⊤}.

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Therefore AD = {0/b⊤, a/b⊤, 1/b⊤} × {0/c⊤, a/c⊤, 1/c⊤} = {0, x0a, x01, xa0, xaa, xa1, x10, x1a, 1}, where we denoted: 0 = (0/b⊤, 0/c⊤), 1 = (1/b⊤, 1/c⊤) and xij = (i/b⊤, j/c⊤) for all i, j ∈ {0, a, 1} with (i, j) / ∈ {(0, 0), (1, 1)}. (AD, {PCD, idD}) is the inductive limit of the inductive system ({AC, AD}, {PCD}). Therefore ˜ A = lim − →

E∈Π(A)

AE = AD.

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A/b⊤ and A/c⊤ share the same lattice structure, namely that of the three-element chain, hence the lattice structure of AD = ˜ A is the following:

r ❅ ❅ r

x0a

• rxa0
• ❅

❅ ❅ ❅ r

xaa

r

x01

r

xa1

rx10 rx1a

• ❅

❅ r

1 The operations of AD = ˜ A are defined componentwise from those of the quotient lattices A/b⊤ and A/c⊤, hence, like in A, ⊙ = ∧ also in ˜ A.

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Here is the table of the operation → in AD = ˜ A: → x0a x01 xa0 xaa xa1 x10 x1a 1 1 1 1 1 1 1 1 1 1 x0a 1 1 x10 1 1 x10 1 1 x01 x1a 1 x10 x1a 1 x10 x1a 1 xa0 x01 x01 1 1 1 1 1 1 xaa x01 x01 x10 1 1 x10 1 1 xa1 x0a x01 x10 x1a 1 x10 x1a 1 x10 x01 x01 xa1 xa1 xa1 1 1 1 x1a x01 x01 xa0 xa1 xa1 x10 1 1 1 x0a x01 xa0 xaa xa1 x10 x1a 1

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THANK YOU FOR YOUR ATTENTION!

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