A Fuzzy Gleason Algebra∗ Presentado en el X Encuentro de Geometr´ıa y sus Aplicaciones en 1999

Abstract For a complete quasi-monoidal lattice L, with some additional properties, we consider the category of Gleason L-algebras.

1

Introduction.

ˇ In their fundamental paper [3], Ulrich H¨ohle and Alexander P. Sostak have constructed the category L − TOP of L-topological spaces together with an interior operator, Manuel Surez in [5] combines the topological operators obtaining a classification in the environment of the set topology and Charles Neville in [4] studies the bitopological spaces in order to obtaining an Gleason algebra, in it is natural to investigate analogous ideas in the categories of Ltopological spaces.

2

Preliminaries.

The basic facts needed in the sequel are presented in this section. We are mainly interested in the basic ideas about cqm-lattices and L-fuzzy topo∗

Coautor: profesor Carlos O. Ochoa Castillo de la Universidad Distrital

Huellas en los Encuentros de Geometr´ıa y Aritm´etica: Joaqun Luna Torres

logical spaces. For the notions and results not discussed here, the reader is referred to reference [2].

2.1

From Lattice Theoretic Fundations.

A complete quasi-monoidal lattice or cqm-lattice is a triple (L, 6, ⊗) which satisfies: I. (L, 6) is a complete lattice where > denotes the universal upper bound and ⊥ denotes the universal lower bound II. (L, 6, ⊗) is a partially ordered groupoid, i. e. ⊗ is a binary operation on L satisfying the isotonocity axiom a 6 b and c 6 d implies a ⊗ c 6 b ⊗ d. III. α 6 α ⊗ >, α 6 > ⊗ α, for all α ∈ L. 2.1.1

The Category CQML

A morphism φ between (L1 , 61 , ⊗1 ) and (L2 , 62 , ⊗2 ) is a map φ : L1 → L2 provided with the properties: m1. φ commutes with arbitrary joins. m2. φ(α ⊗1 β) = φ(α) ⊗2 φ(β). m3. φ preserves universal upper bounds i. e. φ(>) = >. We have the category CQML where the objects are the cqm-lattices and the morphisms are the morphisms between the cqm-lattices. A partially ordered groupoid (L, 6, ⊗) is a cl-groupoid iff ⊗ is distributive over non empty joins, i. e. IV. for J 6= ∅, ! _ _ αj ⊗ β = (αj ⊗ β) j∈J

168

j∈J

and

β⊗

_ j∈J

αj =

_ j∈J

(β ⊗ αj ) .

A Fuzzy Gleason Algebra

2.1.2

Quantales and Cqm-Lattices

A triple (L, 6, ∗), where (L, 6) is a complete lattice, is a quantale if V. (L, ∗) is a semigroup, VI. ∗ is distributive over arbitrary joins, i.e. ! _ _ _ _ αi ∗ β = (αi ∗ β) and β ∗ αi = (β ∗ αi ) . i∈I

i∈I

i∈I

i∈I

A quadruple (L, 6, ⊗, ∗), where (L, 6, ⊗) is a cqm-lattice, (L, 6, ∗) is a quantale and moreover VII. ∗ is dominated by ⊗ i.e. for arbitrary α1 , α2 , β1 , β2 ∈ L we have: (α1 ⊗ β1 ) ∗ (α2 ⊗ β2 ) 6 (α1 ∗ α2 ) ⊗ (β1 ∗ β2 ), is an enriched cqm-lattice. Every quantale is left- and right-residuated i.e. there exist binary operations →r and →l on L satisfying the following axioms: α ∗ β 6 γ ⇔ β ∗ α →r γ

,

β ∗ α 6 γ ⇔ β ∗ α →l γ,

where α, β, γ ∈ L. In particular, →r and →l are determined by: _ _ α →r γ = {λ ∈ L | α ∗ λ 6 γ} and α →l γ = {λ ∈ L | λ ∗ α 6 γ}.

2.2

The Category L − FTOP.

Given (L, 6, ⊗) a cqm-lattice and X a non empty set; an L-fuzzy topology on X is a map T : LX → L satisfying the following axioms: o1. T(1X ) = >, o2. For f, g ∈ LX , T(f ) ⊗ T(g) 6 T(f ⊗ g), 169

Huellas en los Encuentros de Geometr´ıa y Aritm´etica: Joaqun Luna Torres

o3. For every subset {fλ }λ∈Λ of LX the inequality ^ _ T(fλ ) 6 T( fλ ). λ∈Λ

λ∈Λ

holds. If T is an L-fuzzy topology on X, the pair (X, T) is an L-fuzzy topological space. Since the set X is non empty, LX consists at least of two elements. In particular, the universal lower bound in (LX , 6) is given by 1∅ . When we take the empty subset of LX and we apply the axiom o3, we obtain o1’. T(1∅ ) = >. Given T1 and T2 L-fuzzy topologies on X, we say that T1 is weaker than T2 (or T2 is stronger than T1 ) if T1 (f ) ≤ T2 (f ) ∀f ∈ LX , when T1 is weaker than T2 , we denote it with T1  T2 . The relation  is a partial ordering on the set LF − T opX of all L-fuzzy topologies on X; moreover, (L − FTopX , ) is a complete lattice (cf. [3]). Given (X, X) and (Y, Y) L-fuzzy topological spaces; a map φ : X → Y is LF-continuous iff for all g ∈ LY , φ satisfies Y(g) 6 X(g ◦ φ). Thus, we have the category

L − FTOP where the objects are the LF-

topological spaces and the morphisms are the LF-continuous maps.

2.3

The Category L − TOP.

In the following considerations we always assume that (L, 6, ⊗) is an object of CQML. If we need more structure, we will state these additional requeriments explicitly. Given X a non empty set and L an object of CQML; 170

A Fuzzy Gleason Algebra

a subset τ of LX is an L-topology in X iff τ is provided with the following properties: T0. 1X , 1∅ ∈ τ , T1. If f, g ∈ τ , then f ⊗ g ∈ τ , T2. If {fi | i ∈ I} ⊆ τ , then

W

i∈I

fi ∈ τ ,

The pair (X, τ ) is called an L-topological space. Let us consider two Ltopological spaces (X1 , τ1 ) and (X2 , τ2 ); a function ψ : X1 → X2 is Lcontinuous iff for all

g ∈ τ2 we have that g ◦ ψ ∈ τ1 . It is clear that

L-topological spaces and L-continuous functions form a category that we ˇ denote with L − TOP. H¨ohle and Sostak in [3] show that every L-fuzzy topology T in X induces an L-topology τT by τT = {f ∈ LX | T (f ) = >}, and establish the functor F : L − FTOP → L − TOP defined by F(X, T ) = (X, τT ) and F(φ) = φ where φ is a LF-continuous map.

3

Important L-Operators.

Let X be a set, a map I : LX → LX is an L-interior operator (cf. [3]) iff I satisfies the following conditions: I0. I(1X ) = 1X , I1. If f ≤ g, then I(f ) ≤ I(g), I2. (I(f )) ⊗ (I(g)) ≤ I(f ⊗ g), I3. I(f ) ≤ f , 171

Huellas en los Encuentros de Geometr´ıa y Aritm´etica: Joaqun Luna Torres

I4. I(f ) ≤ I(I(f )), it is easy to see that every L-toplogy τ in X induces an L-interior operator Iτ by Iτ (f ) =

_

{g ∈ τ | g ≤ f }

and vice versa, every L-interior operator I produces an L-topology τI defined by τI = {f ∈ LX | f ≤ I(f )}, We add that each LF-topology T in X induces an L-interior operator IT defined by: IT (f ) =

_ {h ∈ LX | h ≤ f and T (h) = >}.

Let X be a set, a map A : LX → LX is an L-adherence map (cf. [5] and [2]) iff A satisfies the following conditions: A0. A(1∅ ) = 1∅ , A1. If f ≤ g, then A(f ) ≤ A(g), V W A2. (A(f )) (A(g)) ≤ A(f g), A3. f ≤ A(f ), A4. A(A(f )) ≤ A(f ), it is easy to see that every LF-toplogy T in X induces an L-adherence map Aτ by AT (f ) =

^

{h ∈ LX | f ≤ h and τ (h → 1∅ ) ∈ L0 }.

This way, given a LF-topology, we have the interior IT and adherence AT operators. We say that (X, S, T) is a comparable LF-bitopological space (cf. [4]) if X is a set and S and T are LF-topologies on X, with the condition S  172

A Fuzzy Gleason Algebra

T. Given (X, S, T) a comparable LF-bitopological space, an element f ∈ S! (L0 ) is called regular if IS AT (f ) = f . Denoting with GST the collection of the regular elements, we are interested in determining the properties of GST . Theorem 3.1. For each comparable LF-bitopological space (X, S, T), the collection GST is a Gleason algebra. Proof. First, let us see that 1X ∈ GST , from A3. we have that AT (1X ) = 1X and for I0. I(1X ) = 1X , i. e. IS AT (1X ) = 1X . To see that 0X inGST , we use A0. and I3.: IS AT (0X ) = IS (0X ) = 0X . Given f, g ∈ GST we want to prove that f ⊗ g ∈ GST ; from f ⊗ g ≤ f and f ⊗ g ≤ g and applying the properties A1. and I1. we obtain that IS AT (f ⊗ g) ≤ IS AT (f ) and IS AT (f ⊗ g) ≤ IS AT (g), therefore, IS AT (f ⊗ g) ≤ IS AT (f ) ⊗ IS AT (g) =f ⊗g

(1)

As each f ∈ GST satisfies that IS (f ) = f and f ⊗ g ≤ AT (f ⊗ g), then f ⊗ g = IS (f ) ⊗ IS (g) ≤ IS AT (f ⊗ g) from (1) and (2) we obtain that f ⊗ g ∈ GST .

(2) 

References [1] Jiri Adamek, Horst Herrlich, George Strecker Abstract and Concrete Categories, John Wiley & Sons (New York, 1990). [2] M. Garc´ıa Marrero et Al, Topolog´ıa, Vol I, Alhambra (Madrid, 1975). 173

Huellas en los Encuentros de Geometr´ıa y Aritm´etica: Joaqun Luna Torres

ˇ [3] Ulrich H¨ohle and Alexander P. Sostak, Fixed-Basis Fuzzy Topologies, In: Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publisher (Massachusetts, 1999). [4] Charles W. Neville A Loomis-Sikorski Theorem for Locales, Annals New York Academy of Sciences ( New York 1992). [5] Manuel Suarez M. Topolog´ıa Conjuntista - Una Presentaci´on BuleanaAdjunta, Universidad Pedag´ogica y Tecnol´ogica de Colombia ( Tunja, 1992).

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