On Minimal Fuzzy Ideals of Semigroups

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.

Madad Khan 1 and Feng Feng 2 and M. Nouman Aslam Khan 3

Academic Editor:Baoding Liu

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

Received 18 September 2012; Accepted 27 September 2012

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.

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 [figure omitted; refer to PDF] . The main idea is that each element [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is assigned a membership grade [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] corresponding to nonmembership, [figure omitted; refer to PDF] to partial membership, and [figure omitted; refer to PDF] to full membership. Mathematically, a fuzzy subset [figure omitted; refer to PDF] of a set [figure omitted; refer to PDF] is a function from [figure omitted; refer to PDF] into the closed interval [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] . He has used the characteristic mapping [figure omitted; refer to PDF] of a subset [figure omitted; refer to PDF] of a groupoid [figure omitted; refer to PDF] to show that [figure omitted; refer to PDF] is subgroupoid or left (right, two-sided) ideal of [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] is the fuzzy subgroupoid or fuzzy left (right, two-sided) ideal of [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] of a semigroup [figure omitted; refer to PDF] is a left (right) ideal of [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and is ideal if [figure omitted; refer to PDF] is both left and right ideal of [figure omitted; refer to PDF] . An ideal [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is called minimal ideal of [figure omitted; refer to PDF] if [figure omitted; refer to PDF] does not properly contains any other ideal of [figure omitted; refer to PDF] . If the intersection [figure omitted; refer to PDF] of all the ideals of a semigroup [figure omitted; refer to PDF] is nonempty then we shall call [figure omitted; refer to PDF] the kernel of [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] , [figure omitted; refer to PDF] will denote the collection of all fuzzy subsets of [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two fuzzy subsets of a semigroup [figure omitted; refer to PDF] . The operation " [figure omitted; refer to PDF] " in [figure omitted; refer to PDF] is defined by; [figure omitted; refer to PDF] For simplicity, we will denote [figure omitted; refer to PDF] as [figure omitted; refer to PDF] .

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 [figure omitted; refer to PDF] be a nonempty set and [figure omitted; refer to PDF] . A fuzzy point [figure omitted; refer to PDF] of [figure omitted; refer to PDF] means that [figure omitted; refer to PDF] is a fuzzy subset of [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF]

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

In the following text for [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , we denote [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] as the fuzzy points of different fuzzy subsets of [figure omitted; refer to PDF] .

Theorem 1.

If in a semigroup [figure omitted; refer to PDF] , the following conditions hold:

(i) for any fuzzy points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of [figure omitted; refer to PDF] there exists [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] ,

(ii) for any fuzzy points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of [figure omitted; refer to PDF] there exists [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is a fuzzy group.

Proof.

First of all we will show that the fuzzy points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] existing in the hypothesis are uniquely determined. To show that [figure omitted; refer to PDF] in condition (i) of the theorem is unique, let us assume on contrary that there exists two fuzzy points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] The condition (ii) in the theorem implies that for the fuzzy points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of [figure omitted; refer to PDF] there exists a fuzzy point [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Hence (4) implies that, [figure omitted; refer to PDF] Now, we will show that if for any fuzzy point [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] then for each fuzzy point [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . By the condition (ii) for fuzzy points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of [figure omitted; refer to PDF] there exists a fuzzy point [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Now by hypothesis, [figure omitted; refer to PDF] Therefore (6) implies that [figure omitted; refer to PDF] , and so [figure omitted; refer to PDF] . It has been shown that the left cancellative law holds in [figure omitted; refer to PDF] .

Similarly, we can show that for any fuzzy points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of [figure omitted; refer to PDF] there exists a unique fuzzy point [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] . Similarly, we can show that [figure omitted; refer to PDF] is a right cancellative.

Hence the cancellation law holds in a semigroup [figure omitted; refer to PDF] and thus for each fuzzy point [figure omitted; refer to PDF] of [figure omitted; refer to PDF] we have, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Now there exists fuzzy points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be any arbitrary fuzzy point of [figure omitted; refer to PDF] then [figure omitted; refer to PDF] is in [figure omitted; refer to PDF] , that is there exists [figure omitted; refer to PDF] in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] is the right identity in [figure omitted; refer to PDF] . Similarly we can show that [figure omitted; refer to PDF] is the left identity in [figure omitted; refer to PDF] . Now since [figure omitted; refer to PDF] is the left identity so [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the right identity so [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] has an identity element [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . Now for [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Now [figure omitted; refer to PDF]

Hence there exists an inverse element for each fuzzy point of [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] is a fuzzy group if and only if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all fuzzy points [figure omitted; refer to PDF] of [figure omitted; refer to PDF] .

3. Minimal Fuzzy Ideals of a Semigroup

The nonempty intersection of fuzzy ideals of semigroup [figure omitted; refer to PDF] is called the fuzzy kernel of a semigroup [figure omitted; refer to PDF] . The fuzzy ideal [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is called minimal fuzzy ideal if [figure omitted; refer to PDF] contains no proper fuzzy ideal of [figure omitted; refer to PDF] . A semigroup [figure omitted; refer to PDF] is called a fuzzy simple if it does not contain any proper fuzzy ideal of [figure omitted; refer to PDF] .

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

Lemma 3.

For any nonempty subsets [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of semigroup [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] .

Lemma 4.

Let [figure omitted; refer to PDF] be a nonempty subset of a semigroup [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is an ideal of [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] is a fuzzy ideal of [figure omitted; refer to PDF] .

Lemma 5.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be fuzzy ideals of a semigroup [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are also fuzzy ideals of [figure omitted; refer to PDF] .

Lemma 6.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be fuzzy ideals of a semigroup [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are also fuzzy ideals of [figure omitted; refer to PDF] .

Theorem 7.

A nonempty subset [figure omitted; refer to PDF] of a semigroup [figure omitted; refer to PDF] is minimal ideal if and only if [figure omitted; refer to PDF] is minimal fuzzy ideal of [figure omitted; refer to PDF] .

Proof.

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

Theorem 8.

If a semigroup [figure omitted; refer to PDF] contains a minimal fuzzy ideal [figure omitted; refer to PDF] of [figure omitted; refer to PDF] then [figure omitted; refer to PDF] is the fuzzy kernel of [figure omitted; refer to PDF] .

Proof.

Let [figure omitted; refer to PDF] be any fuzzy ideal of [figure omitted; refer to PDF] then for a minimal fuzzy ideal [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] is non-empty. Since by Lemma 6, [figure omitted; refer to PDF] is a fuzzy ideal of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] . But then [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] contains in every fuzzy ideal of [figure omitted; refer to PDF] and hence is a fuzzy kernel of [figure omitted; refer to PDF] .

Theorem 9.

If a semigroup has fuzzy kernel [figure omitted; refer to PDF] then [figure omitted; refer to PDF] is a simple subsemigroup of [figure omitted; refer to PDF] .

Proof.

Since [figure omitted; refer to PDF] is a fuzzy ideal of [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] is a fuzzy subsemigroup of [figure omitted; refer to PDF] , since [figure omitted; refer to PDF] . To show [figure omitted; refer to PDF] is simple, let [figure omitted; refer to PDF] be any fuzzy ideal of [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is fuzzy ideal of [figure omitted; refer to PDF] , since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Also [figure omitted; refer to PDF] , but by Theorem 8, [figure omitted; refer to PDF] is minimal fuzzy ideal of [figure omitted; refer to PDF] because every fuzzy kernel of [figure omitted; refer to PDF] , if exists, is a minimal fuzzy ideal of [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] . Also [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] , implies that [figure omitted; refer to PDF] is simple subsemigroup of [figure omitted; refer to PDF] .

Lemma 10.

If [figure omitted; refer to PDF] is a minimal left ideal of a semigroup [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be any fuzzy point of [figure omitted; refer to PDF] for [figure omitted; refer to PDF] in [figure omitted; refer to PDF] then [figure omitted; refer to PDF] is also a minimal fuzzy left ideal of [figure omitted; refer to PDF] .

Proof.

Since for a minimal fuzzy left ideal [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] is a fuzzy left ideal of [figure omitted; refer to PDF] . Suppose [figure omitted; refer to PDF] is the fuzzy left ideal of [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] . Let for [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be a fuzzy point of [figure omitted; refer to PDF] then for [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is in [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] . Now [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is fuzzy left ideal of [figure omitted; refer to PDF] contained in minimal fuzzy left ideal [figure omitted; refer to PDF] of [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] . Thus for all [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] , and so [figure omitted; refer to PDF] is a minimal fuzzy left ideal of [figure omitted; refer to PDF] .

Lemma 11.

A fuzzy ideal [figure omitted; refer to PDF] of a semigroup [figure omitted; refer to PDF] contains every minimal fuzzy left ideal of [figure omitted; refer to PDF] .

Proof.

Let [figure omitted; refer to PDF] be any minimal fuzzy left ideal of [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is fuzzy left ideal of [figure omitted; refer to PDF] since, [figure omitted; refer to PDF] . Also [figure omitted; refer to PDF] as [figure omitted; refer to PDF] is a fuzzy left ideal of [figure omitted; refer to PDF] . But [figure omitted; refer to PDF] is minimal so [figure omitted; refer to PDF] . Hence every minimal fuzzy left ideal [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is contained in every fuzzy ideal of [figure omitted; refer to PDF] .

Theorem 12.

If a semigroup [figure omitted; refer to PDF] contains at least one minimal fuzzy left ideal then it has a fuzzy kernel of [figure omitted; refer to PDF] , where fuzzy kernel is the class sum of all the minimal fuzzy left ideals of [figure omitted; refer to PDF] .

Proof.

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

Theorem 13.

Let a semigroup [figure omitted; refer to PDF] contain at least one minimal left ideal then every fuzzy left ideal of fuzzy kernel is also a fuzzy left ideal of [figure omitted; refer to PDF] .

Proof.

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

Remark 14.

Every minimal fuzzy left ideal of [figure omitted; refer to PDF] is a minimal fuzzy left ideal of fuzzy kernel of [figure omitted; refer to PDF] and vice versa.

Theorem 15.

Let a semigroup [figure omitted; refer to PDF] contain at least one minimal fuzzy left ideal of [figure omitted; refer to PDF] then every fuzzy left ideal of [figure omitted; refer to PDF] contains at least one minimal fuzzy left ideal of [figure omitted; refer to PDF] .

Proof.

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

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 [figure omitted; refer to PDF] contains at least one minimal fuzzy left ideal and one minimal fuzzy right ideal of [figure omitted; refer to PDF] , then the class sum of all the minimal fuzzy left ideals of [figure omitted; refer to PDF] coincide with the class sum of all the minimal fuzzy right ideals and constitutes the fuzzy kernel of [figure omitted; refer to PDF] .

Proof.

By Theorem 12 the class sum of all the minimal fuzzy left ideals of [figure omitted; refer to PDF] is a fuzzy kernel of [figure omitted; refer to PDF] . Similarly the class sum of all the minimal fuzzy right ideals of [figure omitted; refer to PDF] is also the fuzzy kernel of [figure omitted; refer to PDF] . But fuzzy kernel of a semigroup [figure omitted; refer to PDF] is unique since it is the intersection of all the fuzzy ideals of [figure omitted; refer to PDF] .

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

Lemma 18.

Let [figure omitted; refer to PDF] be in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be in [figure omitted; refer to PDF] , then there exists a fuzzy point [figure omitted; refer to PDF] in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Proof.

Note that [figure omitted; refer to PDF] is a fuzzy right ideal of [figure omitted; refer to PDF] since, [figure omitted; refer to PDF] and also [figure omitted; refer to PDF] . But [figure omitted; refer to PDF] is the minimal fuzzy right ideal of [figure omitted; refer to PDF] so [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] .

Lemma 19.

Let [figure omitted; refer to PDF] be in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be in [figure omitted; refer to PDF] , then there exists a fuzzy point [figure omitted; refer to PDF] in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Proof.

Since [figure omitted; refer to PDF] is a fuzzy left ideal of [figure omitted; refer to PDF] since, [figure omitted; refer to PDF] and also [figure omitted; refer to PDF] . But [figure omitted; refer to PDF] is the minimal fuzzy left ideal of [figure omitted; refer to PDF] so [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] .

Theorem 20.

[figure omitted; refer to PDF] is a group.

Proof.

Since [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] is a semigroup. Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be any two fuzzy points of [figure omitted; refer to PDF] then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are in both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus by Lemmas 18 and 19 there exists [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in [figure omitted; refer to PDF] such that, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Hence by Theorem 1, [figure omitted; refer to PDF] is a group.

Lemma 21.

Let [figure omitted; refer to PDF] be the identity element of the group [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Proof.

Since identity element is in [figure omitted; refer to PDF] since [figure omitted; refer to PDF] . Now [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] is a fuzzy right ideal of [figure omitted; refer to PDF] and is contained in minimal fuzzy right ideal [figure omitted; refer to PDF] of [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] . Similarly we can show that [figure omitted; refer to PDF] . Now [figure omitted; refer to PDF] , also [figure omitted; refer to PDF] .

Theorem 22.

For minimal fuzzy right ideal [figure omitted; refer to PDF] and minimal fuzzy left ideal [figure omitted; refer to PDF] of a semigroup [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Proof.

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

Lemma 23.

The identity fuzzy point [figure omitted; refer to PDF] of the fuzzy group [figure omitted; refer to PDF] is a fuzzy primitive.

Proof.

Let [figure omitted; refer to PDF] be a fuzzy idempotent under [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] . Further, Lemma 21 implies that [figure omitted; refer to PDF] is in [figure omitted; refer to PDF] . By Theorem 22 [figure omitted; refer to PDF] is in the fuzzy group [figure omitted; refer to PDF] , which can contain only one fuzzy idempotent point, namely the identity element [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] .

Theorem 24.

Let a semigroup [figure omitted; refer to PDF] have at least one minimal fuzzy left ideal and at least one minimal fuzzy right ideal of [figure omitted; refer to PDF] then the fuzzy kernel [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is completely simple semigroup.

Proof.

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

References

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