International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 756376, 7 pages

http://dx.doi.org/10.1155/2014/756376

Research Article

Somewhat Slightly Generalized Double Fuzzy Semicontinuous Functions

Fatimah M. Mohammed,1,2 M. S. M. Noorani,¹ and A. Ghareeb3,4

¹School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

²College of Education, Tikrit University, Iraq

³Department of Mathematics, College of Science in Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia

⁴Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt

Received 18 April 2014; Revised 26 May 2014; Accepted 27 May 2014; Published 12 June 2014

Academic Editor: Nawab Hussain

Copyright © 2014 Fatimah M. Mohammed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to introduce the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions in double fuzzy topological spaces. Some interesting properties and characterizations of these functions are introduced and discussed. Furthermore, the relationships among the new concepts are discussed with some necessary examples.

1. Introduction

In 1968, Chang [1] was the first to introduce the concept of fuzzy topological spaces. These spaces and their generalization are later developed by Goguen [2], who replaced the closed interval by more general lattice . On the other hand, by the independent and parallel generalization of Kubiak and Šostak’s [3, 4], made topology itself fuzzy besides their dependence on fuzzy set in 1985.

Various generalizations of the concept of fuzzy set have been done by many authors. In [5–10], Atanassove introduced the notion of intuitionistic fuzzy sets. Later Çoker [11] defined intuitionistic fuzzy topology in Chang’s sense. Then, Mondal and Samanta [12] introduced the intuitionistic gradation of openness of fuzzy sets. Gutiérrez García and Rodabaugh [13], in 2005, replaced the term “intuitionistic” and concluded that the most appropriate work is under the name “double.”

In 1980, Jain [14] introduced the notion of slightly continuous functions. On the other hand, Nour [15] defined slightly semicontinuous functions as a weak form of slight continuity and investigated their properties. In [16], Noiri introduced the concept of slightly -continuous functions. Sudha et al. [17] introduced slightly fuzzy -continuous functions. Also in 2004, Ekici and Caldas [18] introduced the notion of slight -continuity (slight -continuity).

In this paper, the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions are introduced. Several interesting properties and characterizations are introduced and discussed. Furthermore, the relationships among the concepts are obtained and established with some interesting counter examples.

2. Preliminaries

Throughout this paper, let be a nonempty set, the unit interval , , and . The family of all fuzzy sets in is denoted by . is the family of all fuzzy points in . By and we denote the smallest and the greatest fuzzy sets on . For a fuzzy set , denotes its complement. Given a function , and defined the direct image and the inverse image of , defined by and for each , , and , respectively. All other notations are standard notations of fuzzy set theory.

Definition 1 (see [12, 13]). A double fuzzy topology on is a pair of maps , , which satisfies the following properties:

(O1) for each ;(O2) and for each ;(O3) and for each , .

The triplet is called double fuzzy topological spaces (dfts, for short). A fuzzy set is called an -fuzzy open (-fo, for short) if and , is called an -fuzzy closed (-fc, for short) if and only if is an -fo set, and is called -fuzzy clopen (-fco, for short) if and only if is -fo set and -fc set. Let and be two dfts's. A function is said to be a double fuzzy continuous if and only if and for each .

Before starting to present our results, there are two questions that we must ask ourselves. First, what is the difference between classical topology and double fuzzy topology? Secondly, where we can apply our results?

To answer the first question, we should know that double fuzzy sets and hence double fuzzy topological spaces deal with obscurities. In addition to that, we observed that the concept of double fuzzy topological spaces is a generalization of fuzzy topological spaces and classical topology. For example, when the first condition in Definition 1 does not hold, we get the definition of fuzzy topological spaces in Kubiak-Šostak’s sense [3, 4]. Also, in the same definition, when we replace with , we will get results in double gradation fuzzifying topological spaces [19]. Appropriate changes can be made to get results in the classical topological spaces.

With regard to applications, since double fuzzy topology forms an extension of fuzzy topology and general topology, we think that our results can be applied in the fuzzy mathematics, which has many applications in different branches of engineering and ICT. For example, recently double fuzzy topological spaces have been applied to study sensor bias [20] and there exist well-established applications of fuzzy topological spaces in the areas of digital topology [21], image processing [22], and geographic information systems (GIS) problems [23].

Theorem 2 (see [24, 25]). Let be a dfts. Then for each , , and , one defines an operator as follows:

For , and , the operator satisfies the following statements:

(C1) = ;(C2);(C3);(C4) if and ;(C5).

Theorem 3 (see [24, 25]). Let be a dfts. Then for each , , and , one defines an operator as follows:

For , and , the operator satisfies the following statements:

(I1);(I2);(I3);(I4);(I5) if and ;(I6);(I7)If , then .

Definition 4 (see [26]). Let be a dfts. For each , , and .

(1)A fuzzy set is called -fuzzy semiclosed (briefly, -fsc) if . is called -fuzzy semiopen (briefly, -fso) if and only if is an -fuzzy semiclosed set.(2)An -fuzzy semiclosure of is defined by = and is -fsc}.

Definition 5 (see [26]). Let be a dfts. For each , and .

(1)A fuzzy set is called -generalized fuzzy semiclosed (briefly, -gfsc) if , and , . is called -generalized fuzzy semiopen (briefly, -gfso) if and only if is -gfsc set.(2)An -fuzzy generalized semiclosure of is defined by = and is -gfsc}.(3)An -fuzzy generalized semi-interior of is defined by = and is -gfso}.

Definition 6 (see [27]). Let and be dfts's. A function is called

(1)slightly double fuzzy continuous (briefly, sdfc) if for every , , , and such that is -fco set and , there exists such that , , , and

(2)slightly generalized double fuzzy semicontinuous (briefly, sgdfsc) if for each , , , and such that is -fco set and , there exists an -gfso set such that and

3. Somewhat Slightly Generalized Double Fuzzy Semicontinuous Functions

Definition 7. Let and be dfts’s. A function is called somewhat slightly generalized double fuzzy semicontinuous (briefly, swsgdfsc) if for each fuzzy set , , , and such that and , there exists an -gfso set such that and

Definition 8. A fuzzy set in a dfts is called -generalized fuzzy semidense (resp., -fuzzy-) set if there exists no -gfsc (resp., -fco) set , , and such that

Example 9. (1) Let . Define and as follows:

And define and as follows:

So, if , , then there exists no -gfsc set in such that . Therefore, is -generalized fuzzy semidense set in .

(2) In (1), let and be defined as follows:

So, if , , then there exists no -fco set in such that . Therefore, is -fuzzy- set in .

Definition 10. Let be a dfts. For a fuzzy set , , and , and are defined as follows:

(1) and is -fco};(2) and is -fco}.

Proposition 11. Let and be dfts’s, and let be any function. Then the following are equivalent.

(1)  is swsgdfsc function.(2)If    is an  -fco set such that    and  , for each  , ,  and  ,  then there exists an  -gfsc set    such that   .(3)If     is  -gfs-dense set in  ,  then     is   -fuzzy-   set in     such that every  -fco set  ,  for each  ,  , and  .

Proof. Suppose is swsgdfsc function, and let be any -fco set in such that and , for each , , and . Then, is -fco in such that and . Then by the hypothesis, there exists an -gfso set , , and such that and . That is, is an -gfsc set and

Put . Then is an -gfsc set in such that .

Let be an -gfs-dense set in , and suppose that is not a fuzzy- set in , such that each -fco set , for each , , and . Then, there exists an -fco set such that

since

Now, is an -fco set such that and , for each , , and . Then by the hypothesis, there exists an -gfsc set such that .

But

That is, . Therefore, there exists an -gfsc set , , and such that , which is a contradiction. Therefore, is an -fuzzy set in such that

for each and -fco set .

Let be an -fco set such that and , for each , , and . Then, . Now, suppose that and . Then,

That is, is an -gfs-dense in . Then by (3), is an -fuzzy set such that there exists an -fco set , for each , , and .

But

since is an -fco and

That is,

which is a contradiction, since . Therefore, and . So is swsgdfsc.

4. Somewhat Slightly Generalized Double Fuzzy Semiopen Functions

Definition 12. Let and be dfts’s. A function is called  

(1)generalized double fuzzy semiopen (briefly, gdfso) if for each -gfso set , , and , is an -gfso in ;(2)slightly generalized double fuzzy semiopen (briefly, sgdfso) if for each -gfso set and each , , and such that , is an -fco set in and

(3)somewhat generalized double fuzzy semiopen (briefly, swgdfso) if for each -gfso set , , and , there exists an -gfso set such that

(4)somewhat slightly generalized double fuzzy semiopen (briefly, swsgdfso) if for each -gfso set such that and for each , , and , there exists an -fco set , such that

 That is, , and there exists an -fco set such that and , for each , , and .

Proposition 13. Let , , and be dfts’s. If and are swsgdfso functions, then is a swsgdfso function.

Proof. Let be an -gfso set and such that , for each fuzzy set , , and . Since is swsgdfso, then there exists an -fco set , and such that .

Now, is an -gfso in such that

for each .

Since is swsgdfso, then there exists an -fco set and such that

But

Thus, there exists an -fco set and

such that

Therefore, is swsgdfso.

Proposition 14. Let and be dfts’s, and let be a bijective function. Then the following are equivalent.

(1) is swsgdfso function.(2)If   is an  -gfsc set in    such that    and    for each  , then there exists an  -fco set  ,  , and    such that  .

Proof. Let be an -gfsc set in such that and , for each , , and . Then, is an -gfso set in such that and , for each . So

Since is a swsgdfso, then there exists an -fco set and such that

Now, is an -fco set in such that and such that

Take

so (2) is proved.

Let be any -gfso set in such that , for each . Then, is an -gfsc set in such that and for each and . Now,

For, if , then

Hence by the hypothesis, there exists an -fco set , , such that

That is,

such that

Let . Then, is an -fco set in such that and . Therefore, is swsgdfso function.

5. Interrelations

The following implication illustrates the relationships between different functions in Figure 1.

(a)

(b)

(c)

Figure 1

None of these implications is reversible where represents implies , as shown by the following examples.

Example 15. Let .

(1)Let be the identity function. Define , , , and as follows:

And define and as follows:

Then, is sgdfsc function but not sdfc.(2)In (1), is swsgdfsc function but not sdfc.(3)Let be a function defined by

Define , , and as follows:

and define and as follows:

Then, is swsgdfsc function but not sgdfsc.

(4)Let be the identity function. Define , , and as follows:

And define and as follows:

Then, is swgdfso function but not gdfso.(5)Let be the identity function. Define , , and as follows:

And define and as follows:

Then, is swsgdfso function but not sgdfso.

Conflict of Interests

The authors declare that there is no conflict of interests regarding this paper.

Acknowledgments

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant GUP-2013-040. The authors also wish to gratefully acknowledge all those who have generously given their time to referee their paper.

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