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Journal of Mathematics
Volume 2013 (2013), Article ID 475190, 5 pages
http://dx.doi.org/10.1155/2013/475190
Research Article

On Minimal Fuzzy Ideals of Semigroups

1Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan
2Department of Applied Mathematics, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
3School of Chemical & Materials Engineering (SCME), National University of Sciences & Technology, H-12, Islamabad 44000, Pakistan

Received 18 September 2012; Accepted 27 September 2012

Academic Editor: Baoding Liu

Copyright © 2013 Madad Khan 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 present paper contains the sufficient condition of a fuzzy semigroup to be a fuzzy group using fuzzy points. The existence of a fuzzy kernel in semigroup is explored. It has been shown that every fuzzy ideal of a semigroup contains every minimal fuzzy left and every minimal fuzzy right ideal of semigroup. The fuzzy kernel is the class sum of minimal fuzzy left (right) ideals of a semigroup. Every fuzzy left ideal of a fuzzy kernel is also a fuzzy left ideal of a semigroup. It has been shown that the product of minimal fuzzy left ideal and minimal fuzzy right ideal of a semigroup forms a group. The representation of minimal fuzzy left (right) ideals and also the representation of intersection of minimal fuzzy left ideal and minimal fuzzy right ideal are shown. The fuzzy kernel of a semigroup is basically the class sum of all the minimal fuzzy left (right) ideals of a semigroup. Finally the sufficient condition of fuzzy kernel to be completely simple semigroup has been proved.

1. Introduction

Zadeh in 1965 introduced the fundamental concept of a fuzzy set in his paper [1] which provides a useful mathematical tool for describing the behavior of systems that are too complex or ill-defined to admit precise mathematical analysis by classical methods. The literature in fuzzy set theory and its practicability has been functioning quickly uptil now. The applications of these concepts can now be seen in a variety of disciplines like artificial intellegence, computer science, control engineering, expert systems, operation research, management science, and robotics.

 Mordeson [2] has demonstrated the basic exploration of fuzzy semigroups. He also gave a theoratical exposition of fuzzy semigroups and their application in fuzzy coding, fuzzy finite state machines, and fuzzy languages. The role of fuzzy theory in automata and daily language has extensively been discussed in [2].

Fuzzy sets are considered with respect to a nonempty set . The main idea is that each element of is assigned a membership grade in , with corresponding to nonmembership, to partial membership, and to full membership. Mathematically, a fuzzy subset of a set is a function from into the closed interval . The fuzzy theory has provided generalization in many fields of mathematics such as algebra, topology, differential equation, logic, and set theory. The fuzzy theory has been studied in the structure of groups and groupoids by Rosenfeld [3], where he has defined the fuzzy subgroupoid and the fuzzy left (right, two-sided) ideals of a groupoid . He has used the characteristic mapping of a subset of a groupoid to show that is subgroupoid or left (right, two-sided) ideal of if and only if is the fuzzy subgroupoid or fuzzy left (right, two-sided) ideal of . Fuzzy semigroups were first considered by Kuroki [4] in which he studied the bi-ideals in semigroups. He also considered the fuzzy interior ideal of a semigroup, characterization of groups, union of groups, and semilattices of groups by means of fuzzy bi-ideals. N. Kehayopulu, X. Y. Xie, and M. Tsingelis have produced several papers on fuzzy ideals in semigroup.

A nonempty subset of a semigroup is a left (right) ideal of if and is ideal if is both left and right ideal of . An ideal of is called minimal ideal of if does not properly contains any other ideal of . If the intersection of all the ideals of a semigroup is nonempty then we shall call the kernel of . Huntington in [5] has shown the simplified definition of a semigroup to be a group. We have used this concept in fuzzy semigroups and have shown that the product of minimal fuzzy left ideal and minimal fuzzy right ideal forms a group. Clifford in [6] had given the representation of minimal left (right) ideal of a semigroup. We have used this ideas to extend the fuzzy ideal theory of minimal fuzzy ideal in semigroups and have given a representation of fuzzy kernel in terms of minimal fuzzy left (right) ideals of a semigroup.

For a semigroup , will denote the collection of all fuzzy subsets of . Let and be two fuzzy subsets of a semigroup . The operation “” in is defined by; For simplicity, we will denote as .

2. Sufficient Condition of a Semigroup to Be a Group

The very first definition of fuzzy point is given by Pu and Liu [7]. Let be a nonempty set and . A fuzzy point of means that is a fuzzy subset of defined by

Each fuzzy point of is the fuzzy subset of a set [7]. Every fuzzy set can be expressed as the union of all the fuzzy points in . It is to be noted that for any and in , , and in the fuzzy points and have the inclusion relation that is if and only if and .

In the following text for , and in , we denote , and as the fuzzy points of different fuzzy subsets of .

Theorem 1. If in a semigroup , the following conditions hold:(i) for any fuzzy points and of there exists in , such that ,(ii) for any fuzzy points and of there exists in , such that , then is a fuzzy group.

Proof. First of all we will show that the fuzzy points and existing in the hypothesis are uniquely determined. To show that in condition (i) of the theorem is unique, let us assume on contrary that there exists two fuzzy points and such that, and . Thus, The condition (ii) in the theorem implies that for the fuzzy points and of there exists a fuzzy point of such that . Hence (4) implies that, Now, we will show that if for any fuzzy point of , then for each fuzzy point of , . By the condition (ii) for fuzzy points and of there exists a fuzzy point of such that . Now by hypothesis, Therefore (6) implies that , and so . It has been shown that the left cancellative law holds in .
Similarly, we can show that for any fuzzy points and of there exists a unique fuzzy point in , such that . Similarly, we can show that is a right cancellative.
Hence the cancellation law holds in a semigroup and thus for each fuzzy point of we have, and . Now there exists fuzzy points and of such that and . Let be any arbitrary fuzzy point of then is in , that is there exists in such that . Thus, , which implies that is the right identity in . Similarly we can show that is the left identity in . Now since is the left identity so and is the right identity so . Hence , which implies that has an identity element in . Now for in , there exists and in such that and . Now
Hence there exists an inverse element for each fuzzy point of . The uniqueness of identity and inverse elements can be proved easily.

The following corollary can be easily asserted from the proof of Theorem 1.

Corollary 2. A fuzzy semigroup is a fuzzy group if and only if and for all fuzzy points of .

3. Minimal Fuzzy Ideals of a Semigroup

The nonempty intersection of fuzzy ideals of semigroup is called the fuzzy kernel of a semigroup . The fuzzy ideal of is called minimal fuzzy ideal if contains no proper fuzzy ideal of . A semigroup is called a fuzzy simple if it does not contain any proper fuzzy ideal of .

The following lemmas are in our previous knowledge [2].

Lemma 3. For any nonempty subsets and of semigroup , we have if and only if .

Lemma 4. Let be a nonempty subset of a semigroup , then is an ideal of if and only if is a fuzzy ideal of .

Lemma 5. Let and be fuzzy ideals of a semigroup , then and are also fuzzy ideals of .

Lemma 6. Let and be fuzzy ideals of a semigroup , then and are also fuzzy ideals of .

Theorem 7. A nonempty subset of a semigroup is minimal ideal if and only if is minimal fuzzy ideal of .

Proof. Let be a minimal ideal of , then by Lemma 4, is a fuzzy ideal of . Suppose that is not minimal fuzzy ideal of , then there exists some fuzzy ideal of such that, . Hence by Lemma 3, , where is an ideal of . This is a contradiction to the fact that is minimal ideal of . Thus is the minimal fuzzy ideal of . Conversely, let be the minimal fuzzy ideal of , then by Lemma 4, is the ideal of . Suppose that is not minimal ideal of , then there exists some ideal of such that . Now by Lemma 3, , where is the fuzzy ideal of . This is a contradiction to the fact that is the minimal fuzzy ideal of . Thus is the minimal ideal of .

Theorem 8. If a semigroup contains a minimal fuzzy ideal of then is the fuzzy kernel of .

Proof. Let be any fuzzy ideal of then for a minimal fuzzy ideal of ,. Thus is non-empty. Since by Lemma 6, is a fuzzy ideal of and , which implies that . But then , so contains in every fuzzy ideal of and hence is a fuzzy kernel of .

Theorem 9. If a semigroup has fuzzy kernel then is a simple subsemigroup of .

Proof. Since is a fuzzy ideal of , so is a fuzzy subsemigroup of , since . To show is simple, let be any fuzzy ideal of , then is fuzzy ideal of , since and . Also , but by Theorem 8, is minimal fuzzy ideal of because every fuzzy kernel of , if exists, is a minimal fuzzy ideal of . Hence . Also , which implies that . Thus , implies that is simple subsemigroup of .

Lemma 10. If is a minimal left ideal of a semigroup and be any fuzzy point of for in then is also a minimal fuzzy left ideal of .

Proof. Since for a minimal fuzzy left ideal of ,, which implies that is a fuzzy left ideal of . Suppose is the fuzzy left ideal of and let . Let for in , be a fuzzy point of then for in , is in and so . Now , which implies that . Hence is fuzzy left ideal of contained in minimal fuzzy left ideal of and so . Thus for all in ,, which implies that . Hence , and so is a minimal fuzzy left ideal of .

Lemma 11. A fuzzy ideal of a semigroup contains every minimal fuzzy left ideal of .

Proof. Let be any minimal fuzzy left ideal of , then is fuzzy left ideal of since, . Also as is a fuzzy left ideal of . But is minimal so . Hence every minimal fuzzy left ideal of is contained in every fuzzy ideal of .

Theorem 12. If a semigroup contains at least one minimal fuzzy left ideal then it has a fuzzy kernel of , where fuzzy kernel is the class sum of all the minimal fuzzy left ideals of .

Proof. Let be the union of all the minimal fuzzy left ideals of . Since contains at least one minimal fuzzy left ideal of , so is non-empty. Let be all the minimal fuzzy left ideals of then, , which implies that is the fuzzy left ideal of . Now it has to be shown that is minimal. Let, for in , and be arbitrary fuzzy points of and , respectively. By definition of , the fuzzy point is in some for some , such that . Now since is minimal fuzzy left ideal of so by Lemma 10, is also minimal fuzzy left ideal of and hence contained in . Thus, and so . Hence is fuzzy right ideal of and so fuzzy ideal of . Now by Lemma 11, every fuzzy ideal contains every minimal fuzzy left ideal of and so being union of all minimal fuzzy left ideals of is contained in every fuzzy ideal. Hence is the fuzzy kernel of .

Theorem 13. Let a semigroup contain at least one minimal left ideal then every fuzzy left ideal of fuzzy kernel is also a fuzzy left ideal of .

Proof. Let be the fuzzy kernel of , then by Theorem 12, is the class sum of all the minimal fuzzy left ideals of . Let be a fuzzy left ideal of , then each fuzzy point for in , of belongs to some minimal fuzzy left ideal of . Now is a fuzzy left ideal of , since . Also , where is minimal fuzzy left ideal so . Thus implies that , where is minimal fuzzy left ideal of and so . Hence, is the fuzzy left ideal of .

Remark 14. Every minimal fuzzy left ideal of is a minimal fuzzy left ideal of fuzzy kernel of and vice versa.

Theorem 15. Let a semigroup contain at least one minimal fuzzy left ideal of then every fuzzy left ideal of contains at least one minimal fuzzy left ideal of .

Proof. Let be the fuzzy left ideal of , then by Theorem 12, has a fuzzy kernel , where is the class sum of all the minimal left ideals of . Now is a fuzzy left ideal of , since , and also . Thus is one of the minimal fuzzy left ideals of or contains some minimal fuzzy left ideals of . But is also contained in . Hence contains at least one minimal fuzzy left ideal of .

Remark 16. Lemmas 10 and 11 and Theorems 12, 13, and 15 are also applicable for fuzzy right ideal of a semigroup.

Theorem 17. If a semigroup contains at least one minimal fuzzy left ideal and one minimal fuzzy right ideal of , then the class sum of all the minimal fuzzy left ideals of coincide with the class sum of all the minimal fuzzy right ideals and constitutes the fuzzy kernel of .

Proof. By Theorem 12 the class sum of all the minimal fuzzy left ideals of is a fuzzy kernel of . Similarly the class sum of all the minimal fuzzy right ideals of is also the fuzzy kernel of . But fuzzy kernel of a semigroup is unique since it is the intersection of all the fuzzy ideals of .

From now on we assume that and will denote the minimal fuzzy right ideal and the minimal fuzzy left ideal of a semigroup , respectively. A fuzzy idempotent point of semigroup is said to be under another fuzzy idempotent point if . The fuzzy idempotent is called fuzzy primitive if there is no fuzzy idempotent under . A simple fuzzy semigroup is said to be completely simple if every fuzzy idempotent point of is fuzzy primitive and for each fuzzy point of there exists fuzzy idempotents and such that .

Lemma 18. Let be in and be in , then there exists a fuzzy point in such that .

Proof. Note that is a fuzzy right ideal of since, and also . But is the minimal fuzzy right ideal of so . Hence .

Lemma 19. Let be in and be in , then there exists a fuzzy point in such that .

Proof. Since is a fuzzy left ideal of since, and also . But is the minimal fuzzy left ideal of so . Hence .

Theorem 20. is a group.

Proof. Since , which implies that is a semigroup. Let and be any two fuzzy points of then and are in both and , since and . Thus by Lemmas 18 and 19 there exists and in such that, and . Hence by Theorem 1, is a group.

Lemma 21. Let be the identity element of the group , then and .

Proof. Since identity element is in since . Now , which implies that is a fuzzy right ideal of and is contained in minimal fuzzy right ideal of and hence . Similarly we can show that . Now , also .

Theorem 22. For minimal fuzzy right ideal and minimal fuzzy left ideal of a semigroup ,.

Proof. We only need to show that . Let be arbitrary fuzzy point of and let be identity element of then by Lemma 18 there exists fuzzy point of such that . Let be the inverse fuzzy element of in then, . Now , so by Lemma 21, . Hence and so is in .

Lemma 23. The identity fuzzy point of the fuzzy group is a fuzzy primitive.

Proof. Let be a fuzzy idempotent under , then . Further, Lemma 21 implies that is in . By Theorem 22   is in the fuzzy group , which can contain only one fuzzy idempotent point, namely the identity element , so .

Theorem 24. Let a semigroup have at least one minimal fuzzy left ideal and at least one minimal fuzzy right ideal of then the fuzzy kernel of is completely simple semigroup.

Proof. Every fuzzy kernel of a semigroup is simple. Each fuzzy point of the fuzzy kernel of belongs to exactly one minimal fuzzy left ideal and to exactly one minimal fuzzy right ideal of , otherwise and are not the minimal fuzzy left and minimal fuzzy right ideals of . So the fuzzy point belongs to unique fuzzy group . Thus is the union of the disjoint fuzzy groups . Each fuzzy idempotent point of must be in one of these fuzzy groups, which can only have the fuzzy idempotent point , namely the identity point. But by Lemma 23 the identity point is fuzzy primitive. The second condition for to be completely simple is straightforward as each fuzzy point of is in one of the fuzzy groups having fuzzy idempotent such that . Hence is completely simple semigroup.

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