1. Introduction

The concept of fuzzy sets was first introduced by Zadeh[8].Subsequently, Chang defined the notion of fuzzy topology[4]. Since then various aspects of general topology were investigated and. carried out in fuzzy since by several authors of this field. The notions of fuzzy ideal and and fuzzy pairwes local function introduced and studied in[2]. Nouh[6] initiated the study of fuzzy bitopological spaces. A fuzzy set equipped with two fuzzy topologies is called a fuzzy bitopological space. Concepts of fuzzy ideals and fuzzy local function were introduced by Sarkar[7]. The purpose of this paper is to to introduce and study some new pairwise fuzzy notion in fuzzy bitopological ideals spaces. We also generalize the notion of fuzzy L-open sets due to Abd El-Monsef et al[1]. In addition to generalize the concept of L-closed sets, L-continuity and L-open functions due to Abd El-Monsef et al[1].

2. Preliminaries

Throughout this paper, by , we mean a fuzzy bitopological space (fbts in short) in Nouh's[6] sense. A fuzzy point in X with support and value is denoted by in[3]. A fuzzy point is said to be contained in a fuzzy set in I^X iff and this will be denoted by [3]. For a fuzzy set in a fbts

,,, , and will respectively denote closure, interior and complement of . The constant fuzzy sets taking values 0 and 1 on X are denoted by o_x, 1_x respectively. A fuzzy set in fts is said to be quasi-coincident[5] with a fuzzy set , denoted by , if there exists such that . A fuzzy set in a fts , is called a q-nbd[3,5] of a fuzzy point iff there exists a fuzzy open set such that, we will denoted the set of all q-nbd of in by . A nonempty collection of fuzzy sets L of a set X is called fuzzy ideal[5] on X iff i) and (heredity),(ii) and (finite additivity). The fuzzy local function[7] of a fuzzy set is the union of all fuzzy points such that if and then there is at least one for which . For a fts (X, ) with fuzzy ideal L, [7] for any fuzzy set of X and be the fuzzy topology generated by cl^*[7].

Definition.2.1.[2]. A fuzzy set in a fbts is called pairwise quasi-coincident with a fuzzy set. denoted by , if there exists such that. Obviously, for any two fuzzy sets and, will imply.

Definition.2.2.[2]. A fuzzy set in a fbts) is called Pairwise quasi-neighborhood of point if and only if there exists a fuzzy , set such that we will denote the set of all pairwise q-nbd of in by ,.

Definition 2.3.[2]. Let be a fbts with fuzzy ideal L on X, and. Then the fuzzy pairwise local function , of is the union of all fuzzy points such that for and then there is at least one for which , where is the set of all q-nbd of Therefore, any , ( for any ( any fuzzy set ) implies hereafter, is not contained in the fuzzy set , i.e. implies there is at least one such that for every for which for some ∈ L.

We will occasionally write or for We define *-fuzzy closure operator, denoted by pcl* for fuzzy topology τ* (L) finer than τ as follows: for every fuzzy set on X. When there is no ambiguity, we will simply write for and for and respectively.

Definition.2.4.[2]. Let be a fbts with fuzzy ideal L on X, a fuzzy pairwise local function , of is the union of all fuzzy points xε such that for and then there is at least one for which, whereis the set of all q-nbd of in (where is fuzzy topology generated by .

Example: 2.1.[2]. One may easily verify that

If then for any, .

, then, for any , .

Theorem.2.1.[2]. Let be a fbts with fuzzy ideal L on X, and . Then we have:

i) .

ii) If then .

iii) .

iv) .

Theorem.2.2.[2]. Let be a fbts with fuzzy ideal L on X, , If , then

i) , for every ,

ii) .

iii) Cleary as

Theorem.2.3.[2]. Let be a fbts and be two fuzzy ideals on X. Then for any fuzzy sets

i.) , .

ii.) ,

iii) , .

vi)

v) .

iv)

3. On Fuzzy Pairwise Local Function

Definition.3.1. Given be a fbts with fuzzy ideal on , . Then is said to be:

i) Fuzzy pairwise -closed, (or PF-*closed) if

ii) Fuzzy pairwise PL-dense – in – itself (or PF*-dense- in – itself ) if .

(iii) Fuzzy pairwise ^*-perfect if is PF^*-closed and PF^*-dense – in itself.

Theorem.3.1. Given (, be a fbts with fuzzy ideal on , then is

i) PF*- closed iff cl*(µ) =µ.

(ii) PF^*- dense – in – itself iff cl^*(µ)= pµ^*.

(iii) PF^*- perfect iff cl^*(µ)= pµ^*=µ.

Proof: Follows directly from the fuzzy pairwise closure operator cl^* for a fuzzy bitopological in[2] and Definition 3.1

Remark 3.1. One can deduce that

(i) Every PF^*-dense- in – itself is fuzzy pairwise dense set.

(ii) Every fuzzy pairwise closed (resp. fuzzy pairwise open) set is PF^*-closed (resp. PF

Corollary.3.1. Given (, be a fbts with fuzzy ideal on , then we have :

(i) If µ is PF^*-closed then pµ^*≤ int(µ) ≤ cl(µ).

(ii) If µ is PF^*-dense- itself then int(µ)≤ pµ^*.

(iii) If µ is PF^*- perfect then int(µ) = cl(µ) = pµ^*.

Proof: Obvious.

Theorem.3.1. Given, be a fbts with fuzzy ideal on , then we have the following:

(i.) is fuzzy pairwise - closed iff µ is PF^*- closed.

(ii.) is fuzzy pairwise – open iff µ is PF^*- dense in itself.

(iii.) is fuzzy regular pairwise closed iff is PF^*- perfect.

Proof: It ̓s clear.

Corollary 3.2. For an fbts , with fuzzy ideal on , the following holds:

(i) If then is PF^*- closed.

(ii) If then

(iii) If then

Proof: Obvious.

4. Fuzzy Pairwise L-open and Fuzzy Pairwise L- closed Sets

Definition.4.1. Given , be a fbts with fuzzy ideal on , and is called a fuzzy pairwise -open set iff there exists such that .

We will denote the family of all fuzzy pairwise

-open on and .(simplify).

When there is no chance for confusion.

Theorem.4.1. let , be a fbts with fuzzy ideal L theniff for

Proof. Assume that then Definition.4.1. there exists such that , . But put . Hence . Conversely . Then there exists . Hence.

Definition.4.2.The largest (simply) set contained in is called a of . The complement of fuzzy pairwise L-open subset of X is a fuzzy pairwise L-closed subset of X (simply FPLC(X)). We denoted by FPL-.

Theorem.4.2. Let , be a fbts with fuzzy ideal L, and J is an arbitrary set then

i) The union of fuzzy pairwise L-open subsets is fuzzy pairwise L-open.

ii) If is fuzzy pairwise open and is fuzzy pairwise L-open subset of X. Then pairwise L-open subset.

Proof.

i) Let be a family of FPLO(X). Then for each and so .

ii) Assume that is fuzzy pairwise open and is fuzzy pairwise L-open subsets of X. Then .

Definition.4.3. Let (, be a fbts with fuzzy ideal . Then is said to be

i) fuzzy - closed iff .

ii) fuzzy .

iii) fuzzy if is and -dense in itself.

Theorem.4.3. Given (, a fbts with fuzzy ideal, then the following holds:

(i) If is both fuzzy pairwise L-open and -perfect then is fuzzy pairwise open.

(ii) If is both fuzzy pairwise open and dense –in – itself then is fuzzy pairwise L-open.

Proof. Follows directly from the fuzzy closure operator for and Definition.4.1

Corollary.4.1. For a fuzzy subset of a fbts , with fuzzy ideal , we have:

If is and then

If is and set then

Theorem.4.4. If (, a fbts with fuzzy ideal then

(i) is fuzzy L-open set.

(ii) FPLiff .

Proof.

(i)Since then Thus . Hence

(ii) Let) FPL. Then, impliesand so conversely assume that then. Hence .

Theorem.4.5. If , a fbts with fuzzy ideal L on X, then.

Proof. Clear

Definition.4.4: Given, a fbts with fuzzy ideal L and, called fuzzy pairwise L-closed set if its complement is fuzzy L-open set. We will denote the family of fuzzy L-closed sets by FPLC(X).

Theorem.4.6. Given,a fbts with fuzzy ideal L and is fuzzy pairwise L-closed set, then

Proof. It’s clear.

Theorem.4.7. Given , be a fbts with fuzzy ideal L on X and such that then iff

Proof. (Necessity). Follows immedially from the above theorem. (Sufficiency). Letthen from the hypothesis. Hence

Corollary.4.2. For a fbts , with fuzzy ideal , then the union of fuzzy pairwise L-closed sets is fuzzy pairwise L-closed set.

5. Fuzzy Pairwise L-Continuous Functions

By utilizing the notion of pairwise L-open sets, we establish in this article a class of fuzzy pairwise L-continuous function. Each of fuzzy pairwise L-continuous and fuzzy pairwise continuous function are independent concepts. Many characterizations and properties of this concept are investigated.

Definition.5.1.A fuzzy pairwise function : with fuzzy ideal L on X is said to be fuzzy pairwise L-continuous if for every .

Remark.5.1: Every fuzzy pairwise L-continuity is fuzzy pairwise precontinuity but the converse is not true in general as seen by the following example.

Example.5.1: Let , fuzzy pairwise indiscrete bitopological , is fuzzy pairwise discrete bitopological and L=. The fuzzy pairwise identity function : is fuzzy pairwise precontinuous but not fuzzy pairwise L-continuous, since while .

Theorem.5.1:For a function : with fuzzy ideal L on X the following are equivalent:

(i.) is fuzzy pairwise L-continuous.

(ii.) For in X and each containing there exists containing such that .

(iii.) For each fuzzy pairwise point in X and containing, is fuzzy pairwise nbd of .

(iv.) The inverse image of each fuzzy pairwise closed set in Y is fuzzy pairwiseL-closed.

Proof: (i.) (ii.). Since containing, then by (i), by putting µ= which containing , we have (ii.) (iii.). Let containing. Then by (ii) there exists containing such that so. Hence is fuzzy npbd of .

(iii.) (i.) Let, since ( is fuzzy pairwise npbd of any point, every point is a fuzzy pairwise interior point of. Then and hence is fuzzy pairwise L-continuous.

(i.) (iv.) Let be a fuzzy pairwise closed set. Then is fuzzy pairwise open set, by Thus is fuzzy PL-closed set.

The following theorem establish the relationship between fuzzy pairwise L-continuous and fuzzy pairwise continuous by using the previous fuzzy pairwise notions.

Theorem.5.2. Given: is a function with ideal L on X then we have. If is fuzzy pairwise L-continuous of each fuzzy pairwise*-perfect set in X, then is fuzzy pairwise continuous.

Proof: Obvious.

Corollary.5.1.Given a function: and each member of X is fuzzy pairwise*-dense-in-itself.

Then we have:

(i.) Every fuzzy pairwise continuous function is fuzzy pairwise L-continuous.

(ii.) Each of fuzzy pairwise precontinuous function and fuzzy pairwise L-continuous are equivalent.

Proof: It’s clear.