Journal of Mathematics

Volume 2013 (2013), Article ID 592708, 7 pages

http://dx.doi.org/10.1155/2013/592708

Research Article

Semigroups Characterized by Their Generalized Fuzzy Ideals

Madad Khan and Saima Anis

Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

Received 14 January 2013; Accepted 27 February 2013

Academic Editor: Feng Feng

Copyright © 2013 Madad Khan and Saima Anis. 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.

We have characterized right weakly regular semigroups by the properties of their -fuzzy ideals.

1. Introduction

Usually the models of real world problems in almost all disciplines like in engineering, medical science, mathematics, physics, computer science, management sciences, operations research, and artificial intelligence are mostly full of complexities and consist of several types of uncertainties while dealing with them in several occasion. To overcome these difficulties of uncertainties, many theories had been developed such as rough sets theory, probability theory, fuzzy sets theory, theory of vague sets, theory of soft ideals, and the theory of intuitionistic fuzzy sets. Zadeh discovered the relationships of probability and fuzzy set theory in [1] which has appropriate approach to deal with uncertainties. Many authors have applied the fuzzy set theory to generalize the basic theories of Algebra. The concept of fuzzy sets in structure of groups was given by Rosenfeld [2]. The theory of fuzzy semigroups and fuzzy ideals in semigroups was introduced by Kuroki in [3, 4]. The theoretical exposition of fuzzy semigroups and their application in fuzzy coding, fuzzy finite state machines, and fuzzy languages was considered by Mordeson. The concept of belongingness of a fuzzy point to a fuzzy subset by using natural equivalence on a fuzzy subset was considered by Murali [5]. By using these ideas, Bhakat and Das [6, 7] gave the concept of -fuzzy subgroups by using the “belongs to” relation and “quasi-coincident with” relation between a fuzzy point and a fuzzy subgroup and introduced the concept of an -fuzzy subgroups, where and . In particular, -fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. These fuzzy subgroups are further studied in [8, 9]. The concept of -fuzzy subgroups is a viable generalization of Rosenfeld’s fuzzy subgroups. Davvaz defined -fuzzy subnearrings and ideals of a near ring in [10]. Jun and Song initiated the study of -fuzzy interior ideals of a semigroup in [11] which is the generalization of fuzzy interior ideals [12]. In [13], Kazanci and Yamak studied -fuzzy bi-ideals of a semigroup.

In this paper we have characterized right regular semigroups by the properties of their right ideal, bi-ideal, generalized bi-ideal, and interior ideal. Moreover we characterized right regular semigroups in terms of their -fuzzy right ideal, -fuzzy bi-ideal, -fuzzy generalized bi-ideal, -fuzzy bi-ideal, and -fuzzy interior ideals.

Throughout this paper denotes a semigroup. A nonempty subset of is called a subsemigroup of if . A nonempty subset of is called a left ideal of if . is called a two-sided ideal or simply an ideal of if it is both left and right ideal of . A nonempty subset of is called a generalized bi-ideal of if . A nonempty subset of is called a bi-ideal of if it is both a subsemigroup and a generalized bi-ideal of . A subsemigroup of is called an interior ideal of if .

An semigroup is called a right weakly regular if for every there exist such that .

Definition 1. For a fuzzy set of a semigroup and , the crisp set such that is called level subset of .

Definition 2. A fuzzy subset of a semigroup of the form

is said to be a fuzzy point with support and value and is denoted by .

A fuzzy point is said to belong to (resp., quasi-coincident with) a fuzzy set , written as resp., , if resp., . If or , then we write . The symbol means does not hold. For any two fuzzy subsets and of , means that, for all , .

Generalizing the concept of , Jun [12, 14] defined , where , as . if or .

2. -Fuzzy Ideals in Semigroups

Definition 3. A fuzzy subset of is called an -fuzzy subsemigroup of if for all and the following condition holds: and imply .

Lemma 4 (see [15]). Let be a fuzzy subset of . Then is an -fuzzy subsemigroup of if and only if .

Definition 5. A fuzzy subset of is called an -fuzzy left ideal of if for all and the following condition holds: implies .

Lemma 6 (see [15]). Let be a fuzzy subset of . Then is an -fuzzy left ideal of if and only if .

Definition 7. A fuzzy subsemigroup of a semigroup is called an -fuzzy bi-ideal of if for all and the following condition holds: and imply .

Lemma 8 (see [15]). A fuzzy subset of is an -fuzzy bi-ideal of if and only if it satisfies the following conditions:

(i) for all and ;(ii) for all and .

Definition 9. A fuzzy subset of a semigroup is called an -fuzzy generalized bi-ideal of if for all and the following condition holds: and imply .

Lemma 10 (see [15]). A fuzzy subset of is an -fuzzy generalized bi-ideal of if and only if for all and .

Definition 11. A fuzzy subsemigroup of a semigroup is called an -fuzzy interior ideal of if for all and the following condition holds: imply .

Lemma 12 (see [15]). A fuzzy subset of is an -fuzzy interior ideal of if and only if it satisfies the following condition:

(i) for all and ;(ii) for all and .

Example 13. Let be a semigroup with binary operation “,” as defined in the following Cayley table:

Clearly is regular semigroup and , , and are left ideals of . Let us define a fuzzy subset of as

Then clearly is an -fuzzy ideal of .

Lemma 14 (see [15]). A nonempty subset of a semigroup is right (left) ideal if and only if is an -fuzzy right (left) ideal of .

Lemma 15. A nonempty subset of a semigroup is an interior ideal if and only if is an -fuzzy interior ideal of .

Lemma 16. A nonempty subset of a semigroup is bi-ideal if and only if is an -fuzzy bi-ideal of .

Lemma 17. Let and be any fuzzy subsets of semigroup . Then following properties hold:

(i),(ii).

Proof. It is straightforward.

Lemma 18. Let and be any nonempty subsets of a semigroup . Then the following properties hold:

(i), (ii).

Proof. It is straightforward.

3. Characterizations of Regular Semigroups

Theorem 19. For a semigroup , the following conditions are equivalent:

(i) is regular;(ii) for left ideals , , and bi-ideal of a semigroup .(iii), for some in ;

Proof. : Let be regular semigroup, then for an element there exists such that . Let , where is a bi-ideal and and are left ideals of . So , , and .

As . Thus .

is obvious.

: As and are left ideal and bi-ideal of generated by , respectively, thus by assumption we have

Thus or or , for some in . Hence is regular semigroup.

Theorem 20. For a semigroup , the following conditions are equivalent:

(i) is regular;(ii)  for every right ideal and bi-ideal of a semigroup ;(iii), for some in .

Proof. : Let be regular semigroup, then for an element there exists such that . Let , where is right ideal and , and are left ideals of . So , and . As . Thus .

is obvious.

: As is right ideal and is left ideal of generated by , respectively, thus by assumption we have

Thus or , for some in . Hence is regular semigroup.

Theorem 21. For a semigroup , the following conditions are equivalent:

(i) is regular;(ii) for every -fuzzy right ideal , -fuzzy left ideals , and of a semigroup .

Proof. : Let be -fuzzy right ideal, and any -fuzzy left ideals of . Since is regular, therefore for each there exists such that

Thus

: Let be right ideal, and let and be any two left ideals of generated by , respectively.

Then is any -fuzzy right ideal, and and are any -fuzzy left ideals of semigroup , respectively. Let and . Then , , and . Now

Thus . Therefore .

So by Theorem 20, is regular.

4. Characterizations of Right Weakly Regular Semigroups in Terms of -Fuzzy Ideals

Theorem 22. For a semigroup , the following conditions are equivalent:

(i) is right weakly regular;(ii) for every right ideal, left ideal, and interior ideal of , respectively;(iii).

Proof. : Let be right weakly regular semigroup, and let , , and be right ideal, left ideal, and interior ideal of , respectively. Let then , , and . Since is right weakly regular semigroup so for there exist such that

Therefore . So .

is obvious.

: As , , and are right ideal, left ideal, and interior ideal of generated by an element of , respectively, thus by assumption, we have

Thus or or , for some in . Hence is right weakly regular semigroup.

Theorem 23. For a semigroup , the following conditions are equivalent:

(i) is right weakly regular;(ii) for every fuzzy right ideal, fuzzy left ideal, and fuzzy interior ideal of , respectively.

Proof. : Let ,  , and be any -fuzzy right ideal, -fuzzy generalized bi-ideal, and -fuzzy interior ideal of . Since is right weakly regular therefore for each there exist such that

Then

Therefore .

Now

: Let , , and be right ideal, left ideal, and interior ideal of generated by , respectively.

Then , , and are -fuzzy right ideal, -fuzzy left ideal, and -fuzzy interior ideal of semigroup . Let and . Then , , and . Now

Thus . Therefore . Hence by Theorem 22, is right weakly regular semigroup.

Theorem 24. For a semigroup , the following conditions are equivalent:

(i) is right weakly regular;(ii) for every bi-ideal, left ideal, and interior ideal of , respectively;(iii).

Proof. : Let be right weakly regular semigroup, and , , and be bi-ideal, left ideal, and interior ideal of , respectively. Let then , , and . Since is right weakly regular semigroup so for there exist such that

Therefore .So .

is obvious.

: As , , and are bi-ideal, left ideal, and interior ideal of generated by an element of , respectively, thus by assumption we have

Thus or or or , for some in . Hence is right weakly regular semigroup.

Theorem 25. For a semigroup , the following conditions are equivalent:

(i) is right weakly regular;(ii) for every fuzzy bi-ideal, fuzzy left ideal and fuzzy interior ideal of , respectively;(iii) for every fuzzy generalized bi-ideal, fuzzy left ideal, and fuzzy interior ideal of , respectively.

Proof. : Let ,  , and be any -fuzzy generalized bi-ideal, -fuzzy left ideal, and -fuzzy interior ideal of . Since is right weakly regular for each there exist such that

Then

Therefore .

is obvious.

: Let , , and be bi-ideal, left ideal, and interior ideal of generated by , respectively.

Then , , and are -fuzzy bi-ideal, -fuzzy left ideal, and -fuzzy interior ideal of semigroup . Let and . Then , , and . Now

Thus . Therefore . Hence by Theorem 24, is right weakly regular semigroup.

Theorem 26. For a semigroup , the following conditions are equivalent:

(i) is right weakly regular;(ii) for every quasi-ideal , left ideal , and interior ideal of , respectively;(iii).

Proof. : Let be right weakly regular semigroup, and let , , and be quasi-ideal, left ideal, and interior ideal of , respectively. Let then , , and . Since is right weakly regular semigroup so for there exist such that

Therefore . So .

is obvious.

: As , , and are quasi-ideal, left ideal, and interior ideal of generated by an element of , respectively, thus by assumption we have

Thus or or or , for some in . Hence is right weakly regular semigroup.

Theorem 27. For a semigroup , the following conditions are equivalent:

(i) is right weakly regular;(ii) for every fuzzy quasi-ideal, fuzzy left ideal, and fuzzy interior ideal of , respectively.

Proof. : Let ,  , and be any -fuzzy quasi-ideal, -fuzzy left ideal, and -fuzzy interior ideal of . Since is right weakly regular therefore for each there exist such that

Then

Therefore .

is obvious.

: Let , , and be quasi-ideal, left ideal and interior ideal of generated by , respectively.

Then , , and are -fuzzy quasi-ideal, -fuzzy left ideal, and -fuzzy interior ideal of semigroup . Let and let . Then , , and . Now

Thus . Therefore . Hence by Theorem 26, is right weakly regular semigroup.

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