PRODUCTS OF LF-TOPOLOGIES AND SEPARATION IN LF-TOP

For a GL-monoid L provided with an uniform structure, we build an LF -topology on the cartesian product of a family of LF -topological spaces. We also show that the product of an arbitrary family of Kolmogoroff (Hausdorff) LF -topological spaces is again a Kolmogoroff (Hausdorff) LF -topological space. Subjclass[2000]: 54A40, 54B10, 06D72


Introduction
Over the years, a number of descriptions of products of fuzzy topological spaces have appeared.As mathematicians have attempted to develop and extend topics of general topology in various ways using the concept of fuzzy subsets of an ordinary set, it is not surprising that searches for such products were obtined with different degrees of success, depending on the structure of the underlying lattice L. The aim of this paper is to give a characterization of arbitrary products of LF -topological spaces, when the underlying lattice L is a GL -Monoid with some additional structures.
The paper is organized as follows: After some lattice-theoretical prerequisites, where we briefly recall the concept of a GL-monoid, we present the concept of uniform structures on GL-monoids in order to get conditions for the existence of arbitrary products of elements of a GL-monoid (section 2).Then, in section 3, we shall build the LF -topology product of a given family of LF -topological spaces.Finally in sections 4, 5 and 6 we define Kolmogoroff and Hausdorff LF -topological spaces and we show that these properties are inherited by the product LF -topology from their factors, together with the separation concepts in this context.

GL -Monoids
The basic facts needed are presented in this section.We are mainly interested in the basic ideas about GL-monoids.Let (L, ) be a complete infinitely distributive lattice, i. e. (L, ) is a partially ordered set such that for every subset A ⊂ L the join W A and the meet V A are defined, and for every α ∈ L we have: In particular, > := W L and ⊥ := V L are respectively the universal upper and the universal lower bounds in L. We also assume that ⊥ 6 = >, i.e.L has at least two elements.A GL−monoid (cf.[9]) is a complete lattice enriched with a further binary operation ⊗, i.e. a triple (L, , ⊗) such that: (1) ⊗ satisfies the isotonicity axiom i.e. for all α, β, γ ∈ L, αβ implies α ⊗ γβ ⊗ γ; ( ⊗ is associative, that is to say, α⊗(β ⊗γ) = (α⊗β)⊗γ, ∀α, β, γ ∈ L; (4) (L, , ⊗) is integral, i.e. the universal upper bound > is the unit with respect to ⊗: α ⊗ > = α, ∀α ∈ L; (5) ⊥ is the zero element in (L, , ⊗), that is to say, α ⊗ ⊥ = ⊥, ∀α ∈ L; (6) ⊗ is distributive over arbitrary joins, this means that α ⊗ ( On the other hand, every GL− monoid is residuated, i.e. there exists an additional binary operation "7 −→" in L satisfying the condition: The Heyting algebras and the MV -algebras are important examples of GL-monoids.A Heyting algebra (cf.[5]), is a GL-monoid of the kind (L, , ∧, ∨, ∧) (i.e. in a Heyting algebra ∧ = ⊗).A GL-monoid is a MValgebra if (α 7 −→ ⊥) 7 −→ ⊥ = α ∀α ∈ L (cf. [9]).Thus in an MV -algebra an order reversing involution c : L → L can be naturally defined by setting If X is a set and L is a GL-monoid, then the fuzzy powerset L X in an obvious way can be pointwise endowed with a structure of a GL-monoid.In particular the L-sets 1 X and 0 X defined by 1 X (x) := > and 0 X (x) := ⊥, ∀x ∈ X, are respectively the universal upper and lower bounds in L X .
In the sequel L denotes an arbitrary GL-monoid.

Infinite Products in GL-monoids
In order to get arbitrary products of elements of a GL-monoid, let us begin with recalling the notion of infinite sums in commutative groups given by Bourbaki in [3].We briefly sketch his construction below.

Infinite Sums in Topological Groups
Let us begin with the following data: 1.A Hausdorff commutative group (G, +, τ), 2. an index set I, 3. a family (x λ ) λ∈I of points of G, indexed by I.
If P f (I) denotes the set of finite subsets of I, and with each J ∈ P f (I) we associate the element s J := P i∈J x i of G, which we call the finite partial sum of the family (x λ ) λ∈I corresponding to the set J, we have thus a mapping Now, P f (I) is a directed set (with respect to the inclusion relationship).Let Φ be the section filter of the directed set P f (I): For each J ∈ P f (I), the section of P f (I) relative to the element J is the set Then the set S = {S(J)|J ∈ P f (I)} is a filter base.The filter Φ of sections of P f (I) is the filter generated by S.
The family (x λ ) λ∈I of points of (G, +, τ) is said to be summable if the mapping P : P f (I) −→ G J 7 −→ s J has a limit with respect to the section filter Φ.When such limit exists, it is denoted by

Infinite Products in GL-Monoids
Employing the method introduced in the previous section, and proceeding in the same way, we discuss the closely related notion of infinite products in GL-Monoids.Now, we need the following data: 2. an index set I, 3. a family (x λ ) λ∈I of elements of L.
If P f (I) again denotes the set of finite subsets of I, and with each J ∈ P f (I) we associate the element N i∈J x i , of L, which we call the finite partial product of the family (x λ ) λ∈I corresponding to the set J, we have thus a mapping We would like to define the "tensorial" product of the family (x λ ) λ∈I of points of L as the limit of the mapping with respect to the section filter Φ and some convergence structure on L. A fundamental approach to constructing such convergence is the uniform structure.Uniform spaces are the carriers of uniform convergence, uniform continuity and the like.

Uniform structures on GL-Monoids
In order to get conditions for the existence of "tensorial" product on GL-Monoids, we will now introduce a uniform structure on a GL-Monoids L, paraphrasing W. Kotzé in [6], Bourbaki in [3], and Willard in [10]: Definition 2.1.A mapping f : L → L is expansive if for each a ∈ L we have that af (a), i. e. ∆ L f , where ∆ L : L → L is the identity map of L.
On the other hand we say that f : L → L commutes with arbitrary joins if for every family (x λ ) λ∈I of points of L.

We denote by f
L L the set of all expansive mappings f : L → L that commute with arbitrary joins.Now we define for each The statement bf (a 7 −→ ⊥) 7 −→ ⊥ is equivalent to hence we have that ba, and therefore Now, we wish to show that the mapping f commutes with arbitrary joins.Let {x λ } λ∈Λ be a collection of elements of L and put: Then y ∈ B if and only if In other words, for each λ ∈ Λ we have that On the other hand, f ( This concludes the proof. 2 Following W. Kotzé's paper (c.f.[6]) Now we note the set {x ∈ L | x 6 = ⊥} with L 0 ; and the foregoing definition is reworded from [3] and [4]: We distinguish the following cases: Case 1: ´.
Case 3: There exists g commutes with arbitrary joins: (lv3).Since the elements of f L L are expansive mappings, the conclusion is obvious.
(lv4).Suppose B p (a) ∈ L 0 and, as in (lv2), let f 0 = W S a .In virtue of (lu4) of definition 2.3, there exists g ∈ f L L such that g • gf 0 and U(g) ∈ L 0 .Since the elements of f L L are expansive mappings and preserve arbitrary joins, we get xg(x)g(g(x))f 0 (x), ∀x ∈ L.
Let b = g(p).It remains to show that B q (a) ∈ L 0 for all qb.Take h : L → L defined by h(x) = g(x) ∨ a, as in the proof of (lv1) (case 3), and note that qb ⇒ g(q)g(b)a, and so a = g(q) ∨ a = h(q).
proving (lv4). 2 Now, we return to the existence of arbitrary product of elements of a GL-monoid (L, , ⊗): Definition 2.6.Let (x λ ) λ∈I be an arbitrary family of points of L, let Φ be the section filter of the directed set P f (I), and let U : f L L → L an L-uniformity.A point p ∈ L is said to be a limit of the mapping with respect to the section filter Φ and with respect to the L-uniformity U if Q −1 (a) ∈ Φ for each a ∈ L such that B p (a) ∈ L 0 .When such limit exists, it is denoted by

Some examples
Example 2.7.Let (G, , +, τ) be a conditionally complete Hausdorff commutative topological l-group (i.e.(G, , +) is a patrially ordered commutative group in which every bounded subset has a supremum and infimum (c.f.[2]).Further let u be an element of the positive cone is a complete lattice.On L we consider the binary operation ⊗ defined by Then (L, , ⊗) is a complete MV -algebra (c.f.[4]).
Let I be an index set and let (x i ) i∈I be a family of point of L indexed by I. Then N i∈I x i exists whenever the family (x λ ) λ∈I of points of (L, +, τ) is summable in (G, , +, τ), (c.f.[3]) and P i∈I x i 2u.Example 2.8.Let (I, , P rod, τ) be the real unit interval provided with the usual order, the usual multiplication, and the (uniform) topology of subspace of the real line (R, τ u ).It is easy to see that (I, , P rod) is a GL-monoid (c.f.[2]).
Let Λ be an index set and let (x λ ) λ∈Λ be a family of point of I indexed by Λ.Then N λ∈Λ x λ exists whenever the family (x λ ) λ∈Λ of points of (I, P rod, τ ) is multipliable (c.f.[3]).
Example 2.9.In a GL-monoid (L, , ⊗) the product of the collection {x i } i∈I , where x i = > for each i ∈ I, is >, since for each J ∈ P f (I) one has that (This example is useful for working with products of L-Topological Spaces).

Product of LF -Topological Spaces
In this section we build the product of an arbitrary collection of LFtopological spaces, for a GL-Monoid (L, , ⊗) (which we always assume to be equipped with arbitrary "tensorial" products).Given a family {(X λ , τ λ ) | λ ∈ Λ} of LF -topological spaces, we want to build the LF -topology product on the cartesian product associate with the LF -topologies τ λ , λ ∈ Λ. ( φ λ denotes the λth component of φ, i.e. φ(λ)).

Preliminary discussion
Each projection ).Now we need to build a map such that: 2. is a LF -topology.
3. is universal in the following sense: In order to get such a mapping, we proceed as follows: For each f ∈ L Λ let us consider for all but finitely many indices α, and

Proof.
Let : L Λ → L be an LF -topology for which each projection For each β ∈ Λ, and for each

Products of Kolmogoroff and Hausdorff LF -topological Spaces
Kolmogoroff L-topological Spaces have been considered by U. Höhle, A.
Šostak in [4].In this section we shall define the notion of Kolmogoroff LF -topological space.We shall show then that the Kolmogoroff property is inherited by the product LF -topological Space from the coordenate LFtopological Spaces.

Hausdorff LF -topological Spaces
Hausdorff L-topological Spaces were considered in [4].In this section we seek generalizations of their results to LF -topological Spaces.Finally, we shall show that the Hausdorff property is inherited by the product LFtopological Space from their factors.Let (X, τ ) be an LF -topological space.For each g ∈ L X , such that τ (g) ∈ L 0 , define g * ∈ L X by ) is a Hausdorff LF -topological Space (i.e.fulfills the T 2 axiom) iff whenever p and q are distint points of X, there exists g ∈ L X satisfying: • τ (g * ) ∈ L 0 , and, • g(q) ⊗ g * (p) 6 = ⊥.Theorem 4.4.Let (X λ , τ λ ) λ∈Λ be a nonempty family of Hausdorff LFtopological spaces.Then ( Λ , ) is also a Hausdorff LF -topological space.

Proof.
If x = (x λ ) λ∈Λ and y = (y λ ) λ∈Λ are distint points of Λ , then there exists i ∈ Λ such that x i 6 = y i .Since (X i , τ i ) is a Hausdorff LFtopological space, there exists g i ∈ L X i satisfying: Consider the element It follows that Hence ( Λ , ) is a Hausdorff LF -topological space. 2

From the Quasi-coincident Neigborhoods
Let x ∈ X and λ ∈ L be, the L-point x λ is the L-set x λ : X → L defined as We note the set of L-points of X with pt(L X ).We say that x λ quasi-coincides with f ∈ L X or say that x λ is quasicoincident with f (cf [7], [11]) when if x λ quasi-coincides with f , we denote this x λ qf ; relation x λ does not quasi-coincide with f or x λ is not quasi-coincident with f is denoted by x λ ¬qf .
Let (X, τ ) be an LF -topological space and x λ ∈ pt(L X ), and define Q 5. For each x λ ∈ pt(L X ) and for all f ∈ L X ,

Separation Degrees
In contrast with the classical topology, we shall introduce a kind of separation where the topological spaces have separation degrees; these topics are due to how many or how much two L-points are separated, this question is naturally extended to the LF-topological space ambience.These ideas are inspired in [11] where the development of theoretical elements is applied on the lattice I, the unitary interval.
Let (X, τ ) be a LF-topological space, 1.Given x λ , x µ ∈ pt(L X ), i. e. L-points with the same support; the degree in which the points x λ , x µ are quasi-T 0 is The degree to which (X, τ ) is quasi-T 0 is q − T 0 (X, τ ) = ^{q − T 0 (x λ , x µ ) | x ∈ X, λ 6 = µ} We emphasize that the degree quasi-T 0 is defined on L-points with the same support.

Concluding Remarks
One of the most pervasive and widely applicable constructions in mathematics is that of products.We hope that the results outlined in this paper have exhibited the main properties of products of LF -topological spaces.Clearly, there is much work remaining to be done in this area.Here are some things that might deserve further attention: 1. Describe the relation between products of LF -topological spaces and compact of LF -topological spaces (Tychonoff Theorem).
2. Describe the products of variable-basis fuzzy topological spaces.
3. Examine the relation between products of LF -topological spaces and further separation axioms.

Definition 2 . 4 .
Let B : L → L L be a map; then for each p ∈ L the image of p under B is denoted by B p : L → L. B is an L-neighborhood system on L iff B satisfies the following axioms (lv0) B p (>) = >.(lv1) ab implies B p (a)B p (b).
(lv2) For all a, b ∈ L, B p (a) ⊗ B p (b)B p (a ⊗ b).(lv3) B p (a) ∈ L 0 implies pa, (lv4) If B p (a) ∈ L 0 then there exists b ∈ L such that B p (a)B p (b), and B q (a) ∈ L 0 , for all qb.Theorem 2.5.Let U : f L L → L Be an L-uniformity and let p ∈ L. Then B p : L → L given by