Applied Mathematics and Computation 155 (2004) 39–54

www.elsevier.com/locate/amc

On various weak fuzzy pairwise retracts

F.S. Mahmoud *, A.A. Abd-Ellatif

Department of Mathematics, Faculty of Science, Sohag 82524, Egypt

Abstract

The notion of a fuzzy retract was introduced by Rodabaugh [J. Math. Anal. Appl. 79

(1981) 273] and this notion was extended to the fuzzy bitopological setting [J. Fuzzy

Math. 9(1) (2001) 207]. In this paper, we define the notions of fuzzy pairwise semire-

tracts, fuzzy pairwise preretracts, fuzzy pairwise strongly semiretracts, fuzzy pairwise

less strongly semiretracts and fuzzy pairwise semi-preretracts. We establish some

properties of these concepts. Moreover the relations between these new concepts are

discussed.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Fuzzy bitopological spaces; Fuzzy pairwise (neighborhood) strongly semiretracts; Fuzzy

pairwise (neighborhood) less strongly semiretracts; Fuzzy pairwise (neighborhood) semi-preretracts

1. Introduction

In 1965 Zadeh [18] introduced the fundamental concept of fuzzy sets which

formed the backbone of fuzzy mathematics. In 1986 Chang [4] introduced the

concepts of fuzzy topological spaces as a generalization of topological spaces.

Since then many topologist have contributed to the theory of fuzzy topological

spaces. In 1989 Kandil [6] introduced and studied the notion of fuzzy bi-

topological spaces as a natural generalization of fuzzy topological space. In thispaper, we define the notions of fuzzy pairwise semiretracts, fuzzy pairwise

preretracts, fuzzy pairwise strongly semiretracts, fuzzy pairwise less strongly

semiretracts, fuzzy pairwise semi-preretracts, fuzzy pairwise nbd semiretracts,

fuzzy pairwise nbd preretracts, fuzzy pairwise nbd strongly semiretracts, fuzzy

* Corresponding author.

E-mail address: fsmmel_2000@yahoo.com (F.S. Mahmoud).

0096-3003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2003.06.001

40 F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54

pairwise nbd less strongly semiretracts, fuzzy pairwise nbd semi-preretracts.

Moreover the relations between these new concepts are discussed. Throughoutthis paper, the symbol I will denote the closed unit interval. Let X be a non-

empty set. A fuzzy set in X is a mapping with domain X and values in I [18].

For definitions and results not explained in this paper, we refer to [7,9,12]

assuming them to be well known.

Definition 1.1 [8]. Let ðX ; dÞ be a F-ts and Y � X . The collection

dY ¼ fljY : l 2 dg is a F-topology on Y , we refer to ðY ; dY Þ as the F-topo-logical subspace.

Lemma 1.1 [3]. Let g : X ! X Y be the graph of a mapping f : X ! Y . If k isa F-set of X and l is a F-set of Y then,

g ðk lÞ ¼ k ^ f ðlÞ:

Definition 1.2 [11]. A family of F-ts� is said to be contained in one productivity

class if the F-closure operator is finitely productive on that family.

Lemma 1.2 [11]. If a family of F-ts’ is contained in one productivity class, thenthe F-interior operator is finitely productive on that family.

Definition 1.3 [4]. Let f : ðX ; dÞ ! ðY ; rÞ be a mapping. Then f is F-contin-

uous if 8t 2 r, f ðtÞ 2 d.

Definition 1.4 [5,14]. Let ðX ; dÞ be a F-topological space and A � X . Then the

F-subspace ðA; dAÞ of ðX ; dÞ is called a F-retract of ðX ; dÞ, briefly, FR if there

exists a F-continuous mapping r : ðX ; dÞ ! ðA; dAÞ such that rðaÞ ¼ a, 8a 2 A.

In this case r is called a F-retraction.

Definition 1.5 [3]. The product f1 f2 : X1 X2 ! Y1 Y2 of mappings

f1 : X1 ! Y1 and f2 : X2 ! Y2 is defined by

ðf1 f2Þðx1; x2Þ ¼ ðf1ðx1Þ; f2ðx2ÞÞ

8ðx1; x2Þ 2 X1 X2.

Lemma 1.3 [3]. For mappings fi : Xi ! Yi and F-sets ki of Yi, i ¼ 1; 2 we have

ðf1 f2Þ ðk1 k2Þ ¼ f1 ðk1Þ f2 ðk2Þ:

Definition 1.6 [6]. A system ðX ; d1; d2Þ consisting of a non-empty set X with

two fuzzy topologies d1 and d2 on X is called a F-bitopological space.

F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54 41

Definition 1.7. Let k be a F-set of a F-bts ðX ; d1; d2Þ. Then, k is called:

ii(i) a ðdi; djÞ-F-semiopen (briefly ðdi; djÞ-F-so) set of X if, k6 dj clðdi intkÞ[17];

i(ii) a ðdi; djÞ-F-preopen (briefly ðdi; djÞ-F-po) set of X if, k6 di intðdj clkÞ[15];

(iii) a ðdi; djÞ-F-strongly semiopen (briefly ðdi; djÞ-F-sso) set of X if, k6 di intðdj clðdi intkÞÞ [15];

(iv) a ðdi; djÞ-F-less strongly semiopen (briefly ðdi; djÞ-F-lsso) set of X if, there

exists a di-F-o set m s.t. m6 k6 di sintðdj sclmÞ, equivalently if,k6 di sintðdj sclðdi intkÞÞ [1];

i(v) a ðdi; djÞ-F-semi-preopen (briefly ðdi; djÞ-F-spo) set of X if, k6 dj clðdi intðdj clkÞÞ [13].

Remark 1.1. The concepts of F-sso set and F-lsso set in F-topological spaces

are equivalent [19], but the concepts of ðdi; djÞ-F-sso set and ðdi; djÞ-F-lsso set in

F-bitoplogical spaces are independent as shown by the following example.

Example 1.1. Let k1, k2, k3 and m be F-sets of X ¼ fa; b; cg defined as follows:

k1 ¼ a0:5 _ b0:5 _ c0:3;

k2 ¼ a0:7 _ b0:7 _ c0:4;

k3 ¼ a0:2 _ b0:2 _ c0:3;

m ¼ a0:6 _ b0:6 _ c0:4:

Let ðX ; d1; d2Þ be a F-bts where d1 ¼ f0; 1; k1; k2g and d2 ¼ f0; 1; k3g. Then mis a ðd1; d2Þ-F-sso set but not a ðd1; d2Þ-F-lsso set.

Definition 1.8 [16]. A mapping f : ðX ; d1; d2Þ ! ðY ; r1;r2Þ is said to F-pairwise

continuous (resp. F-pairwise open), briefly, F-pc (resp. F-p open) mapping if

f : ðX ; d1Þ ! ðY ; r1Þ and f : ðX ; d2Þ ! ðY ; r2Þ are F-continuous (resp. F-open).

Definition 1.9. Let f : ðX ; d1; d2Þ ! ðY ; r1; r2Þ be a mapping from a F-btsðX ; d1; d2Þ to another F-bts ðY ; r1; r2Þ. Then, f is called:

ii(i) a F-pairwise semicontinuous (briefly F-psc) mapping if, f ðmÞ is a

ðdi; djÞ-F-so set of X 8m 2 ri [16];

i(ii) a F-pairwise precontinuous (briefly F-ppc) mapping if, f ðmÞ is a ðdi; djÞ-F-po set of X 8m 2 ri [15];

(iii) a F-pairwise strongly semicontinuous (briefly F-pssc) mapping if, f ðmÞis a ðdi; djÞ-F-sso set of X 8m 2 ri [15];

(iv) a F-pairwise less strongly semicontinuous (briefly F-plssc) mapping if,

f ðmÞ is a ðdi; djÞ-F-lsso set of X 8m 2 ri [1];

42 F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54

i(v) a F-pairwise semi-precontinuous (briefly F-pspc) mapping if, f ðmÞ is a

ðdi; djÞ-F-spo set of X 8m 2 ri [13].

Definition 1.10 [2]. Let ðX ; d1; d2Þ be a F-bts, and A � X . Then, the family

diA ¼ fkjA : k 2 dig ði ¼ 1; 2Þ is a F-topology on A and ðA; d1A; d2AÞ is called theF-subspace of the F-bts ðX ; d1; d2Þ with underlying set A.

Definition 1.11 [10]. Let ðX ; d; cÞ be a F-bts, and A � X . Then the F-subspaceðA; dA; cAÞ is called a F-pairwise retract of ðX ; d; cÞ, briefly, FPR if there exists a

F-pc mapping

r : ðX ; d; cÞ ! ðA; dA; cAÞ s:t: rðaÞ ¼ a 8a 2 A:

In this case r is called a F-pairwise retraction.

Definition 1.12 [10]. Let ðX ; d; cÞ be a F-bts, and A � X . Then the F-subspace

ðA; dA; cAÞ is called a F-pairwise neighborhood retract (or F-pairwise nbd re-

tract) of ðX ; d; cÞ, briefly, FPNR if ðA; dA; cAÞ is a F-pairwise retract of a F-

pairwise open subspace ðU ; dU ; cUÞ of X which containing A.

Simply by X ;A we will mean a F-bts ðX ; d; cÞ and a F-subspace ðA; dA; cAÞ of

the F-bts ðX ; d; cÞ.

2. F-pairwise semiretracts and F-pairwise preretracts

Definition 2.1. Let ðX ; d; cÞ be a F-bts and A � X . Then the F-subspace

ðA; dA; cAÞ is called a F-pairwise semiretract (resp. F-pairwise preretract) of

ðX ; d; cÞ, briefly, FPSR, FPPR if there exists a F-psc (resp. F-ppc) mapping

r : ðX ; d; cÞ ! ðA; dA; cAÞ s:t: rðaÞ ¼ a 8a 2 A:

In this case r is called a F-pairwise semiretraction (resp. F-pairwise prere-

traction).

Remark 2.1. Every F-pairwise retract A of a F-bts X is a F-pairwise semiretract

of X but the converse is not true in general, as shown by the following example.

Example 2.1. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k1, k2 be F-sets of X de-

fined as:

k1 ¼ a0:5 _ b0:5 _ c0:1;

k2 ¼ a0:6 _ b0:6 _ c0:2:

F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54 43

Consider the F-bts ðX ; d1; d2Þ where

d1 ¼ f0; 1; k1g; d2 ¼ f0; 1; k2g:

It is easy to show that A is a F-pairwise semiretract of X but not a F-pairwise

retract of X .

Remark 2.2. Every F-pairwise retract A of a F-bts X is a F-pairwise preretract

of X but the converse is not true in general, as shown by the following ex-

ample.

Example 2.2. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k1, k2 be F-sets of X de-

fined as:

k1 ¼ a0:2 _ b0:5 _ c1;

k2 ¼ a0:5:

Consider the F-bts ðX ; d1; d2Þ where

d1 ¼ f0; 1; k1g; d2 ¼ f0; 1; k2g:

It is easy to show that A is a F-pairwise preretract of X but not a F-pairwise

retract of X .

Theorem 2.1. Let ðX ; d1; d2Þ and ðY ; c1; c2Þ be two F-tbs such that X and Y arecontained in one productivity class. If A � X is a F-pairwise semiretract (resp. F-pairwise preretract) of X and B � Y is a F- pairwise semiretract (resp. F-pairwisepreretract) of Y then, A B is a F-pairwise semiretract (resp. F-pairwise pre-retract) of X Y .

Proof. Since A � X is a F-pairwise semiretract of X then there exists a F-pscmapping

r1 : ðX ; d1; d2Þ ! ðA; d1A; d2AÞ s:t: r1ðaÞ ¼ a 8a 2 A:

Also B � Y is a F-pairwise semiretract of Y then there exists a F-psc mapping

r2 : ðY ; c1; c2Þ ! ðB; c1B; c2BÞ s:t: r2ðbÞ ¼ b 8b 2 B:

Consider the mapping

r1 r2 : ðX Y ; h1; h2Þ ! ðA B;K1;K2Þ;

where hi (resp. KiÞ is the F-product topology generated by di and ci (resp. diA

and ciAÞ ði ¼ 1; 2Þ. Let k ¼W

k;lðmk llÞ where, mk �s are diA-F-o sets of A and

ll�s are ciB-F-o sets of B. Then, k is a Ki-F-o set of A B, by Lemma 1.3 we

have,

44 F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54

ðr1 r2Þ ðkÞ ¼_

k;l

ðr1½ r2Þ ðmk llÞ� ¼_

k;l

r1 ðmkÞ�

r2 ðllÞ�:

Since r1 and r2 are F-psc mappings then, r1 ðmkÞ�s are ðdi; djÞ-F-so sets of Xand r2 ðllÞ�s are ðci; cjÞ- F-so sets of Y . By Theorem 2.3 and Theorem 2.6 of

[13] we have ðr1 r2Þ ðkÞ is a ðhi; hjÞ-F-so set of X Y thus r1 r2 is F-psc,

and

ðr1 r2Þða; bÞ ¼ ðr1ðaÞ; r2ðbÞÞ ¼ ða; bÞ8ða; bÞ 2 A B:

Hence A B is a F-pairwise semiretract of X Y . The part of F-pairwise

preretract is the same way. h

Theorem 2.2. Let ðX ; d1; d2Þ be a F-bts, A � X and r : ðX ; d1; d2Þ ! ðA; d1A; d2AÞis a mapping s.t. rðaÞ ¼ a 8a 2 A. If the graph g : ðX ; d1; d2Þ ! ðX A; h1; h2Þ ofr which is defined by gðxÞ ¼ ðx; rðxÞÞ is F-psc (resp. F-ppc) then, r is a F-pairwisesemiretraction (resp. F-pairwise preretraction) where, hi is the F-product topol-ogy generated by di and diA ði ¼ 1; 2Þ.

Proof. Suppose that g : ðX ; d1; d2Þ ! ðX A; h1; h2Þ is F-psc and we will prove

that, r : ðX ; d1; d2Þ ! ðA; d1A; d2AÞ is a F-pairwise semiretraction. So let l be a

diA-F-o set in A, by Lemma 1.1 we have,

r ðlÞ ¼ 1 ^ r ðlÞ ¼ g ð1 lÞ:

Since g is F-psc and 1 l is a hi-F-o set in X A. Then, r ðlÞ ¼ g ð1 lÞis a ðdi; djÞ-F-so set in X , hence r is F-psc and therefore r is a F-pairwise

semiretraction. The part of F-pairwise preretraction is the same way. h

Theorem 2.3. Let ðX ; d1; d2Þ be a F-bts and Z � Y � X . If Z is a F-pairwiseretract of Y and, Y is a F-pairwise semiretract (resp. F-pairwise preretract) of Xthen, Z is a F-pairwise semiretract (resp. F-pairwise preretract) of X .

Proof. Since Z is a F-pairwise retract of Y then, there exists a F-pc mapping

r1 : Y ! Z s:t: r1ðzÞ ¼ z 8z 2 Z:

Also, Since Y is a F-pairwise semiretract of X then, there exists a F-psc

mapping

r2 : X ! Y s:t: r2ðyÞ ¼ y 8y 2 Y :

Consider the composed mapping

r ¼ r1r2 : X ! Z:

Let k be a F-o set of Z. Then

r ðkÞ ¼ ðr1r2Þ ðkÞ ¼ r2 ðr1 ðkÞÞ

F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54 45

since r1 is F-pc then, r1 ðkÞ is a F-open set in Y and, since r2 is F-psc then,

r ðkÞ ¼ r2 ðr1 ðkÞÞ is a ðdi; djÞ-F-so set in X . Then, r a F-psc mapping and,

rðzÞ ¼ r1r2ðzÞ ¼ r1ðr2ðzÞÞ ¼ r1ðzÞ ¼ z 8z 2 Z:

Hence Z is a F-pairwise semiretract of X . The part of F-pairwise preretract is

the same way. h

Definition 2.2. Let ðX ; d; cÞ be a F-bts and A � X . Then, ðA; dA; cAÞ is said to be

a F-pairwise nbd semiretract (resp. F-pairwise nbd preretract) of ðX ; d; cÞ,briefly, FPNSR, FPNPR if ðA; dA; cAÞ is a F-pairwise semiretract (resp. F-

pairwise preretract) of a F-pairwise open subspace ðU ; dU ; cUÞ of X which

contains A.

Remark 2.3. Every F-pairwise semiretract is also a F-pairwise nbd semiretract,

but the converse is not true in general, as shown by the following example.

Example 2.3. Let X ¼ fa; bg, A ¼ fag � X and k1, k2, k3 be F-sets of X defined

as:

k1 ¼ 1A;

k2 ¼ a0:4 _ b0:9;

k3 ¼ a0:6 _ b1:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1; k2; k1 _ k2; k1 ^ k2f g; d2 ¼ 0; 1; k1; k3; k1 ^ k3f g:

It is easy to show that A is a F-pairwise nbd semiretract of X , but not a F-

pairwise semiretract of X .

Remark 2.4. Every F-pairwise preretract is also a F-pairwise nbd preretract,

but the converse is not true in general, as shown by the following example.

Example 2.4. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k1, k2, k3 be F-sets of Xdefined as:

k1 ¼ 1A;

k2 ¼ a1 _ b0:3;

k3 ¼ b0:5 _ c0:5:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1; k2f g; d2 ¼ 0; 1; k1; k3; k1 _ k3; k1 ^ k3f g:

It is easy to verify that A is a F-pairwise nbd preretract of X , but not a F-

pairwise preretract of X .

46 F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54

Theorem 2.4. Let ðX ; d1; d2Þ and ðY ; c1; c2Þ be two F-tbs such that X and Y arecontained in one productivity class. If A � X is a F-pairwise nbd semiretract(resp. F-pairwise nbd preretract) of X and B � Y is a F-pairwise nbd semire-tract (resp. F-pairwise nbd preretract) of Y then, A B is a F-pairwise nbdsemiretract (resp. F-pairwise nbd preretract) of X Y .

Proof. Since A � X is a F-pairwise nbd semiretract of X then, A is a F-pairwise

semiretract of a F-pairwise open subspace U of X so, there exists a F-pscmapping

r1 : ðU ; d1U ; d2U Þ ! ðA; d1A; d2AÞ s:t: r1ðaÞ ¼ a 8a 2 A:

Also B � Y is a F-pairwise nbd semiretract of Y then, B is a F-pairwise

semiretract of a F-pairwise open subspace V of Y then, there exists a F-psc

mapping,

r2 : ðV ; c1V ; c2V Þ ! ðB; c1B; c2BÞ s:t: r2ðbÞ ¼ b 8b 2 B:

It is easy to prove that,

r1 r2 : ðU V ; h1; h2Þ ! ðA B;K1;K2Þ

is F-psc where hi (resp. Ki) is the F-product topology generated by diU and ciV(resp. diA and ciB) ði ¼ 1; 2Þ. Also, since U is a F-pairwise open subspace of Xand V is a F-pairwise open subspace of Y then, U V is a F-pairwise open

subspace of X Y and,

ðr1 r2Þða; bÞ ¼ ðr1ðaÞ; r2ðbÞÞ ¼ ða; bÞ 8ða; bÞ 2 A B:

Hence A B is a F-pairwise nbd semiretract of X Y . The part of F-pairwise

nbd preretract is the same way. h

3. F-pairwise strongly semiretracts and F-pairwise less strongly semiretracts

Definition 3.1. Let ðX ; d; cÞ be a F-bts and A � X . Then the F-subspace

ðA; dA; cAÞ is called a F-pairwise strongly semiretract (resp. F-pairwise less

strongly semiretract) of ðX ; d; cÞ, briefly, FPSSR, FPLSSR if there exists a F-pssc (resp. F-plssc) mapping

ðX ; d; cÞ ! ðA; dA; cAÞ s:t: rðaÞ ¼ a 8a 2 A:

In this case r is called a F-pairwise strongly semiretraction (resp. F-pairwise less

strongly semiretraction).

Remark 3.1. Every F-pairwise retract A of a F-bts X is also a F-pairwise

strongly semiretract of X , but the converse is not true in general, as shown by

the following example.

F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54 47

Example 3.1. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k1, k2 be F-sets of X de-

fined as:

k1 ¼ a0:5 _ b0:6;

k2 ¼ 1A:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1f g; d2 ¼ 0; 1; k2f g:

It is easy to show that A is a F-pairwise strongly semiretract of X , but not a F-pairwise retract of X .

Remark 3.2. Every F-pairwise retract A of a F-bts X is also a F-pairwise less

strongly semiretract of X , but the converse is not true in general, as shown bythe following example.

Example 3.2. In Example 3.1, A is a F-pairwise less strongly semiretract of X ,

but not a F-pairwise retract of X .

Remark 3.3. According to Remark 1.1 the property FPLSSR has no corre-

spondence in F-topological spaces.

Remark 3.4. The concepts of F-pairwise strongly semiretracts and F-pairwise

less strongly semiretracts are independent, as shown by the following examples.

Example 3.3. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k be a F-set of X defined

as:

k ¼ a0:5 _ b0:5 _ c0:3:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; kf g; d2 ¼ 0; 1; 0:5f g:

It is easy to verify that A is a F-pairwise less strongly semiretract of X , but not a

F-pairwise strongly semiretract of X .

Example 3.4. Let X ¼ fa; bg, A ¼ fag � X and k1, k2, k3 be F-sets of X defined

as:

k1 ¼ a0:7 _ b0:2;

k2 ¼ a1 _ b0:7;

k3 ¼ b0:2:

48 F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1; k2f g; d2 ¼ 0; 1; k3f g:

It is easy to show that A is a F-pairwise strongly semiretract of X , but not a

F-pairwise less strongly semiretract of X .

Remark 3.5. Every F-pairwise strongly semiretract A of a F-bts X is also a

F-pairwise semiretract of X , but the converse is not true in general, as shown

by the following example.

Example 3.5. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k1, k2 be two F-sets of Xdefined as:

k1 ¼ a0:2 _ b0:8 _ c0:1;

k2 ¼ a0:8 _ b0:1 _ c0:1:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1f g; d2 ¼ 0; 1; k2f g:

It is easy to show that A is a F-pairwise semiretract of X , but not a F-pairwise

strongly semiretract of X .

Remark 3.6. Every F-pairwise strongly semiretract A of a F-bts X is also a F-

pairwise preretract of X , but the converse is not true in general, as shown by

the following example.

Example 3.6. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k1, k2 be two F-sets of Xdefined as:

k1 ¼ a0:2 _ b0:1 _ c0:5;

k2 ¼ 1A:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1f g; d2 ¼ 0; 1; k2f g:

It is easy to show that A is a F-pairwise preretract of X , but not a F-pairwise

strongly semiretract of X .

Remark 3.7. Every F-pairwise less strongly semiretract A of a F-bts X is also a

F-pairwise semiretract of X , but the converse is not true in general, as shownby the following example.

Example 3.7. In Example 3.5, A is a F-pairwise semiretract of X , but not a

F-pairwise less strongly semiretract of X .

F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54 49

Remark 3.8. The concepts of F-pairwise less strongly semiretracts and F-

pairwise preretracts are independent as shown by the following example.

Example 3.8. In Example 3.3 A is a F-pairwise less strongly semiretract of Xbut not a F-pairwise preretract of X and, in Example 3.6 A is a F-pairwisepreretract of X but not a F-pairwise less strongly semiretract of X .

Theorem 3.1. Let ðX ; d1; d2Þ be a F-bts, A � X . If the map

r : ðX ; d1; d2Þ ! ðA; d1A; d2AÞ s:t: rðaÞ ¼ a 8a 2 A

is F-psc and F-ppc then, A is a F-pairwise strongly semiretract of X .

Proof. Let k be a diA-F-o set of A then, r ðkÞ is a ðdi; djÞ-F-so and ðdi; djÞ-F-po

set of X hence,

r ðkÞ6 dj clðdi int r ðkÞÞ and r ðkÞ6 di intðdj cl r ðkÞÞ

then, dj cl r ðkÞ6 dj clðdi int r ðkÞÞ this implies that, di intðdj cl r ðkÞÞ6 di intðdj clðdi int r ðkÞÞÞ so,

r ðkÞ6 di intðdj cl r ðkÞÞ6 di intðdj clðdi int r ðkÞÞÞ:

Hence, r ðkÞ is a ðdi; djÞ-F-sso set of X thus r is a F-pssc therefore, A is a

F-pairwise strongly semiretract of X . h

Theorem 3.2. Let ðX ; d1; d2Þ and ðY ; c1; c2Þ be two F-bts such that X and Y arecontained in one productivity class. If A � X is a F-pairwise strongly semiretract(resp. F-pairwise less strongly semiretract) of X and B � Y is a F-pairwisestrongly semiretract (resp. F-pairwise less strongly semiretract) of Y then, A Bis a F-pairwise strongly semiretract (resp. F-pairwise less strongly semiretract) ofX Y .

Proof. It is similar to that of Theorem 2.1. h

Theorem 3.3. Let ðX ; d1; d2Þ be a F-bts, A � X and r : ðX ; d1; d2Þ ! ðA; d1A; d2AÞis a mapping s.t. rðaÞ ¼ a 8a 2 A. If the graph g : ðX ; d1; d2Þ ! ðX A; h1; h2Þ ofr which is defined by gðxÞ ¼ ðx; rðxÞÞ is F-pssc (resp. F-plssc) then, r is aF-pairwise strongly semiretraction (resp. F-pairwise lesss strongly semiretrac-tion) where, hi is F-product topology generated by di and diA ði ¼ 1; 2Þ.

Proof. It is similar to that of Theorem 2.2. h

Theorem 3.4. Let ðX ; d1; d2Þ be a F-bts and Z � Y � X . If Z is a F-pairwiseretract of Y and, Y is a F-pairwise strongly semiretract (resp. F-pairwise less

50 F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54

strongly semiretract) of X then, Z is a F-pairwise strongly semiretract (resp. F-pairwise less strongly semiretract) of X .

Proof. It is similar to that of Theorem 2.3. h

Definition 3.2. Let ðX ; d; cÞ be a F-bts and A � X . Then, ðA; dA; cAÞ is said to be

a F-pairwise nbd strongly semiretract (resp. F-pairwise nbd less strongly

semireretract) of ðX ; d; cÞ, briefly, FPNSSR, FPNLSSR if ðA; dA; cAÞ is aF-pairwise strongly semiretract (resp. F-pairwise less strongly semietract) of a

F-pairwise open subspace ðU ; dU ; cU Þ of X which contains A.

Remark 3.9. Every F-pairwise strongly semiretract is also a F-pairwise nbd

strongly semiretract retract of X but the converse is not true in general, as

shown by the following example.

Example 3.9. Let X ¼ fa; bg, A ¼ fag � X and k1, k2, k3 be F-sets of X defined

as:

k1 ¼ 1A;

k2 ¼ a0:3 _ b0:5;

k3 ¼ a0:7 _ b1:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1; k2; k1 _ k2; k1 ^ k2f g; d2 ¼ 0; 1; k1; k3; k1 ^ k3f g:

It is easy to show that A is a F-pairwise nbd strongly semiretract of X , but not a

F-pairwise strongly semiretract of X .

Remark 3.10. Every F-pairwise less strongly semiretract is also a F-pairwise

nbd less strongly semiretract retract but the converse is not true in general, as

shown by the following example.

Example 3.10. In Example 3.9 A is a F-pairwise nbd less strongly semiretract of

X but not a F-pairwise less strongly semiretract of X .

Theorem 3.5. Let ðX ; d1; d2Þ and ðY ; c1; c2Þ be two F-tbs such that X and Y arecontained in one productivity class. If A � X is a F-pairwise nbd strongly semi-retract (resp. F-pairwise nbd less strongly semiretract) of X and B � Y is aF-pairwise nbd strongly semiretract (resp. F-pairwise nbd less strongly semire-tract) of Y then, A B is a F-pairwise nbd strongly semiretract (resp. F-pairwisenbd less strongly semiretract) of X Y .

Proof. It is similar to that of Theorem 2.4. h

F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54 51

4. F-pairwise semi-preretracts

Definition 4.1. Let ðX ; d; cÞ be a F-bts and A � X . Then the F-subspace

ðA; dA; cAÞ is called a F-pairwise semi-preretract of ðX ; d; cÞ, briefly, FPSPR if

there exists a F-pspc mapping

r : ðX ; d; cÞ ! ðA; dA; cAÞ s:t: rðaÞ ¼ a 8a 2 A:

In this case r is called a F-pairwise semi-preretraction.

Remark 4.1. Every F-pairwise semiretract A of a F-bts X is also a F-pairwise

semi-preretract of X , but the converse is not true in general, as shown by the

following example.

Example 4.1. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k1, k2, k3 be F-sets of Xdefined as:

k1 ¼ 1A;

k2 ¼ a1 _ b0:5;

k3 ¼ b0:5 _ c1:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1; k2f g; d2 ¼ 0; 1; k3f g:

It is easy to show that A is a F-pairwise semi-preretract of X , but not a F-

pairwise semiretract of X .

Remark 4.2. Every F-pairwise preretract A of a F-bts X is also a F-pairwise

semi-preretract of X , but the converse is not true in general, as shown by the

following example.

Example 4.2. Let X ¼ fa; b; cg, A ¼ fa; bg � X and k1, k2 be F-sets of X de-

fined as:

k1 ¼ a0:9 _ b0:5;

k2 ¼ a0:1 _ b0:5 _ c0:1:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1f g; d2 ¼ 0; 1; k2f g:

It is easy to verify that A is a F-pairwise semi-preretract of X , but not a F-

pairwise preretract of X .

Theorem 4.1. Let ðX ; d1; d2Þ and ðY ; c1; c2Þ be two F-bts such that X and Y arecontained in one productivity class. If A � X is a F-pairwise semi-preretract of X

52 F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54

and B � Y is a F-pairwise semi-preretract of Y then, A B is a F-pairwise semi-preretract of X Y .

Proof. It is similar to that of Theorem 2.1. h

Theorem 4.2. Let ðX ; d1; d2Þ be a F-bts, A � X and r : ðX ; d1; d2Þ ! ðA; d1A; d2AÞis a mapping s.t. rðaÞ ¼ a 8a 2 A. If the graph g : ðX ; d1; d2Þ ! ðX A; h1; h2Þ ofr which is defined by gðxÞ ¼ ðx; rðxÞÞ is F-pairwise semi-precontinuous then, r is aF-pairwise semi-preretraction where, hi is F-product topology generated by di anddiA ði ¼ 1; 2Þ.

Proof. It is similar to that of Theorem 2.2. h

Theorem 4.3. Let ðX ; d1; d2Þ be a F-bts and Z � Y � X . If Z is a F-pairwiseretract of Y and, Y is a F-pairwise semi-preretract of X then, Z is a F-pairwisesemi-preretract of X .

Proof. It is similar to that of Theorem 2.3. h

Definition 4.2. Let ðX ; d; cÞ be a F-bts and A � X . Then, the F-subspace

ðA; dA; cAÞ is called a F-pairwise nbd semi-preretract of ðX ; d; cÞ, briefly,

FPNSPR if ðA; dA; cAÞ is a F-pairwise semi-preretract of a F-pairwise open

subspace ðU ; dU ; cU Þ of X which contains A.

Remark 4.3. Every F-pairwise semi-preretract is also a F-pairwise nbd semi-

preretract, but the converse is not true in general, as shown by the following

example.

Example 4.3. Let X ¼ fa; bg, A ¼ fag � X and k1, k2, k3, k4, k5 be F-sets of Xdefined as:

k1 ¼ 1A;

k2 ¼ a0:9;

k3 ¼ a0:9 _ b1;

k4 ¼ a0:1 _ b0:9;

k5 ¼ 0:1:

Consider the F-bts ðX ; d1; d2Þ where,

d1 ¼ 0; 1; k1; k2; k3; k1 _ k3f g;d2 ¼ 0; 1; k1; k4; k5; k1 _ k4; k1 ^ k4; k1 _ k5f g:

F.S. Mahmoud, A.A. Abd-Ellatif / Appl. Math. Comput. 155 (2004) 39–54 53

It is easy to show that A is a F-pairwise nbd semi-preretract of X , but not a F-

pairwise semi-preretract of X .

Theorem 4.4. Let ðX ; d1; d2Þ and ðY ; c1; c2Þ be two F-bts’ such that X and Y arecontained in one productivity class. If A � X is a F-pairwise nbd semi-preretractof X and B � Y is a F-pairwise nbd semi-preretract of Y then, A B is a F-pairwise nbd semi-preretract of X Y .

Proof. It is similar to that of Theorem 2.4. h

Remark 4.4. The following diagram illustrates the relationship between the

various retracts discussed in this paper

FPSPR FPSSR ) FPNSSR

* . * &FPNPR ( FPPR ( FPR ) FPSR ) FPNSR

+ % +FPNLSSR ( FPLSSR FPSPR ) FPNSPR

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