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.

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 𝑆. The main idea is that each element 𝑥 of 𝑆 is assigned a membership grade 𝑓(𝑥) in [0,1], with 𝑓(𝑥)=0 corresponding to nonmembership, 0<𝑓(𝑥)<1 to partial membership, and 𝑓(𝑥)=1 to full membership. Mathematically, a fuzzy subset 𝑓 of a set 𝑆 is a function from 𝑆 into the closed interval [0,1]. 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 𝑆. He has used the characteristic mapping𝐶_𝐴=1if𝑥∈𝐴0if𝑥∉𝐴 (1) of a subset 𝐴 of a groupoid 𝑆 to show that 𝐴 is subgroupoid or left (right, two-sided) ideal of 𝑆 if and only if 𝐶_𝐴 is the fuzzy subgroupoid or fuzzy left (right, two-sided) ideal of 𝑆. 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 𝐴 of a semigroup 𝑆 is a left (right) ideal of 𝑆 if 𝑆𝐴⊆𝐴(𝐴𝑆⊆𝐴) and is ideal if 𝐴 is both left and right ideal of 𝑆. An ideal 𝐴 of 𝑆 is called minimal ideal of 𝑆 if 𝐴 does not properly contains any other ideal of 𝑆. If the intersection 𝐾 of all the ideals of a semigroup 𝑆 is nonempty then we shall call 𝐾 the kernel of 𝑆. 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 𝑆, 𝐹(𝑆) will denote the collection of all fuzzy subsets of 𝑆. Let 𝑓 and 𝑔 be two fuzzy subsets of a semigroup 𝑆. The operation “∘” in 𝐹(𝑆) is defined by; (𝑥)=⋁fog^{𝑥=𝑦𝑧}𝑓𝑦∧𝑔(𝑧),if∃𝑦,𝑧∈𝑆,suchthat𝑥=𝑦𝑧,0,otherwise. (2) For simplicity, we will denote 𝐹(𝑆) as 𝑆.

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 𝐴 be a nonempty set and 𝜆∈[0,1]. A fuzzy point 𝑎_𝜆 of 𝐴 means that 𝑎_𝜆 is a fuzzy subset of 𝐴 defined by 𝑎_𝜆(𝑥)=𝜆,if𝑥=𝑎,0,if𝑥≠𝑎. (3)

Each fuzzy point 𝑎_𝜆 of 𝐴 is the fuzzy subset of a set 𝐴 [7]. Every fuzzy set 𝐴 can be expressed as the union of all the fuzzy points in 𝐴. It is to be noted that for any 𝑥 and 𝑦 in 𝐴, 𝜆, and 𝜇 in [0,1] the fuzzy points 𝑥_𝜆 and 𝑦_𝜇 have the inclusion relation that is 𝑥_𝜆⊆𝑦_𝜇 if and only if 𝑥=𝑦 and 𝜆≤𝜇.

In the following text for 𝜆,𝜇,𝑡,𝛾,𝛿,𝛼,𝛽, and 𝜌 in [0,1], we denote 𝑎_𝜆,𝑏_𝜇,𝑥_𝑡,𝑥_𝜇,𝑒_𝛼,𝑓_𝛽, and 𝑐_𝜌 as the fuzzy points of different fuzzy subsets of 𝑆.

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

Proof. First of all we will show that the fuzzy points 𝑥_𝛾 and 𝑦_𝛾 existing in the hypothesis are uniquely determined. To show that 𝑥_𝛾 in condition (i) of the theorem is unique, let us assume on contrary that there exists two fuzzy points 𝑥_𝛾 and 𝑧_𝛿 such that, 𝑎_𝜆∘𝑥_𝛾=𝑏_𝜇 and 𝑎_𝜆∘𝑧_𝛿=𝑏_𝜇. Thus, 𝑎_𝜆∘𝑥_𝛾=𝑎_𝜆∘𝑧_𝛿. (4) The condition (ii) in the theorem implies that for the fuzzy points 𝑥_𝛾 and 𝑧_𝛿 of 𝑆 there exists a fuzzy point 𝑐_𝛾 of 𝑆 such that 𝑧_𝛿∘𝑐_𝛾=𝑥_𝛾. Hence (4) implies that, 𝑎_𝜆∘𝑧_𝛿∘𝑐_𝛾=𝑎_𝜆∘𝑧_𝛿𝑎_𝜆,∘𝑧_𝛿∘𝑐_𝛾=𝑎_𝜆∘𝑧_𝛿. (5)(6) Now, we will show that if for any fuzzy point 𝑎_𝜆 of 𝑆,𝑎_𝜆∘𝑥_𝛾=𝑎_𝜆 then for each fuzzy point 𝑏_𝜇 of 𝑆, 𝑏_𝜇∘𝑥_𝛾=𝑏_𝜇. By the condition (ii) for fuzzy points 𝑎_𝜆 and 𝑏_𝜇 of 𝑆 there exists a fuzzy point 𝑦_𝜆 of 𝑆 such that 𝑦_𝛾∘𝑎_𝜆=𝑏_𝜇. Now by hypothesis, 𝑦_𝛾∘𝑥_𝛾∘𝑎_𝜆=𝑏_𝜇,𝑦_𝛾∘𝑥_𝛾∘𝑎_𝜆=𝑏_𝜇,𝑏_𝜇∘𝑎_𝜆=𝑏_𝜇. (7) Therefore (6) implies that 𝑧_𝛿∘𝑐_𝛾=𝑧_𝛿, and so 𝑥_𝛾=𝑧_𝛿. It has been shown that the left cancellative law holds in 𝑆.
Similarly, we can show that for any fuzzy points 𝑎_𝜆 and 𝑏_𝜇 of 𝑆 there exists a unique fuzzy point 𝑦_𝛾 in 𝑆, such that 𝑦_𝛾∘𝑎_𝜆=𝑏_𝜇. Similarly, we can show that 𝑆 is a right cancellative.
Hence the cancellation law holds in a semigroup 𝑆 and thus for each fuzzy point 𝑎_𝜆 of 𝑆 we have, 𝑎_𝜆∘𝑆=𝑆 and 𝑆∘𝑎_𝜆=𝑆. Now there exists fuzzy points 𝑒_𝛼 and 𝑓_𝛽 of 𝑆 such that 𝑓_𝛽∘𝑎_𝜆=𝑎_𝜆 and 𝑎_𝜆∘𝑒_𝛼=𝑎_𝜆. Let 𝑥_𝛾 be any arbitrary fuzzy point of 𝑆 then 𝑥_𝛾 is in 𝑆∘𝑎_𝜆, that is there exists 𝑏_𝜇 in 𝑆 such that 𝑥_𝛾=𝑏_𝜇∘𝑎_𝜆. Thus, 𝑥_𝛾∘𝑒_𝛼=(𝑏_𝜇∘𝑎_𝜆)∘𝑒_𝛼=𝑏_𝜇∘(𝑎_𝜆∘𝑒_𝛼)=𝑏_𝜇∘𝑎_𝜆=𝑥_𝛾, which implies that 𝑒_𝛼 is the right identity in 𝑆. Similarly we can show that 𝑓_𝛽 is the left identity in 𝑆. Now since 𝑓_𝛽 is the left identity so 𝑓_𝛽∘𝑒_𝛼=𝑒_𝛼 and 𝑒_𝛼 is the right identity so 𝑓_𝛽∘𝑒_𝛼=𝑓_𝛽. Hence 𝑓_𝛽=𝑒_𝛼, which implies that 𝑆 has an identity element 𝑒_𝛼 in 𝑆. Now for 𝑒_𝛼 in 𝑆, there exists 𝑥_𝛾 and 𝑦_𝜌 in 𝑆 such that 𝑥_𝛾∘𝑎_𝜆=𝑒_𝛼 and 𝑎_𝜆∘𝑦_𝜌=𝑒_𝛼. Now 𝑥_𝛾∘𝑎_𝜆∘𝑦_𝜌=𝑒_𝛼∘𝑦_𝜌,𝑥_𝛾∘𝑎_𝜆∘𝑦_𝜌=𝑦_𝜌,𝑥_𝛾∘𝑒_𝛼=𝑦_𝜌,𝑥_𝛾=𝑦_𝜌. (8)
Hence there exists an inverse element for each fuzzy point of 𝑆. 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 𝑆 is a fuzzy group if and only if 𝑎_𝜆∘𝑆=𝑆 and 𝑆∘𝑎_𝜆=𝑆 for all fuzzy points 𝑎_𝜆 of 𝑆.

3. Minimal Fuzzy Ideals of a Semigroup

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

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

Lemma 3. For any nonempty subsets 𝐴 and 𝐵 of semigroup 𝑆, we have 𝐴⊆𝐵 if and only if 𝐶_𝐴⊆𝐶_𝐵.

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

Lemma 5. Let 𝑓 and 𝑔 be fuzzy ideals of a semigroup 𝑆, then 𝑓∘𝑔 and 𝑔∘𝑓 are also fuzzy ideals of 𝑆.

Lemma 6. Let 𝑓 and 𝑔 be fuzzy ideals of a semigroup 𝑆, then 𝑓∩𝑔 and 𝑔∩𝑓 are also fuzzy ideals of 𝑆.

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

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

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

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

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

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

Lemma 10. If 𝑓 is a minimal left ideal of a semigroup 𝑆 and 𝑐_𝜆 be any fuzzy point of 𝑆 for 𝜆 in [0,1] then 𝑓∘𝑐_𝜆 is also a minimal fuzzy left ideal of 𝑆.

Proof. Since for a minimal fuzzy left ideal 𝑓 of 𝑆,𝑆∘(𝑓∘𝑐_𝜆)=(𝑆∘𝑓)∘𝑐_𝜆⊆𝑓∘𝑐_𝜆, which implies that 𝑓∘𝑐_𝜆 is a fuzzy left ideal of 𝑆. Suppose 𝑔 is the fuzzy left ideal of 𝑓∘𝑐_𝜆 and let ℎ={𝑎_𝜇∈𝑓∶𝑎_𝜇∘𝑐_𝜆⊆𝑔for𝜇∈[0,1]}. Let for 𝑡 in [0,1],𝑏_𝑡 be a fuzzy point of 𝑆 then for 𝑎_𝜇∘𝑐_𝜆 in ℎ,𝑏_𝑡∘𝑎_𝜇 is in 𝑓 and so 𝑏_𝑡∘𝑎_𝜇∘𝑐_𝜆⊆𝑔. Now 𝑏_𝑡∘𝑎_𝜇⊆ℎ, which implies that 𝑆∘ℎ⊆ℎ. Hence ℎ is fuzzy left ideal of 𝑆 contained in minimal fuzzy left ideal 𝑓 of 𝑆 and so ℎ=𝑓. Thus for all 𝑎_𝜇 in 𝑓,𝑎_𝜇∘𝑐_𝜆⊆𝑔, which implies that 𝑓∘𝑐_𝜆⊆𝑔. Hence 𝑓∘𝑐_𝜆=𝑔, and so 𝑓∘𝑐_𝜆 is a minimal fuzzy left ideal of 𝑆.

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

Proof. Let 𝑔 be any minimal fuzzy left ideal of 𝑆, then 𝑓∘𝑔 is fuzzy left ideal of 𝑆 since, 𝑆∘(𝑓∘𝑔)=(𝑆∘𝑓)∘𝑔⊆𝑓∘𝑔. Also 𝑓∘𝑔⊆𝑔 as 𝑔 is a fuzzy left ideal of 𝑆. But 𝑔 is minimal so 𝑔=𝑓∘𝑔⊆𝑓. Hence every minimal fuzzy left ideal 𝑔 of 𝑆 is contained in every fuzzy ideal of 𝑆.

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

Proof. Let 𝑓 be the union of all the minimal fuzzy left ideals of 𝑆. Since 𝑆 contains at least one minimal fuzzy left ideal of 𝑆, so 𝑓 is non-empty. Let 𝑓₁,𝑓₂,𝑓₃,… be all the minimal fuzzy left ideals of 𝑆 then, 𝑆∘𝑓=𝑆∘(𝑓₁∪𝑓₂∪𝑓₃∪⋯)⊆𝑆∘𝑓₁∪𝑆∘𝑓₂∪𝑆∘𝑓₃∪⋯⊆𝑓₁∪𝑓₂∪𝑓₃∪⋯=𝑓, which implies that 𝑓 is the fuzzy left ideal of 𝑆. Now it has to be shown that 𝑓 is minimal. Let, for 𝜆,𝜇 in [0,1],𝑎_𝜆 and 𝑏_𝜇 be arbitrary fuzzy points of 𝑆 and 𝑓, respectively. By definition of 𝑓, the fuzzy point 𝑏_𝜇 is in some 𝑓_𝑖 for some 𝑖∈ℕ, such that 𝑏_𝜇∘𝑎_𝜆⊆𝑓_𝑖∘𝑎_𝜆. Now since 𝑓_𝑖 is minimal fuzzy left ideal of 𝑆 so by Lemma 10, 𝑓_𝑖∘𝑎_𝜆 is also minimal fuzzy left ideal of 𝑆 and hence contained in 𝑓. Thus, 𝑏_𝜇∘𝑎_𝜆⊆𝑓 and so 𝑓∘𝑆⊆𝑓. Hence 𝑓 is fuzzy right ideal of 𝑆 and so fuzzy ideal of 𝑆. Now by Lemma 11, every fuzzy ideal contains every minimal fuzzy left ideal of 𝑆 and so 𝑓 being union of all minimal fuzzy left ideals of 𝑆 is contained in every fuzzy ideal. Hence 𝑓 is the fuzzy kernel of 𝑆.

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

Proof. Let 𝑓 be the fuzzy kernel of 𝑆, then by Theorem 12, 𝑓 is the class sum of all the minimal fuzzy left ideals of 𝑆. Let 𝑔 be a fuzzy left ideal of 𝑓, then each fuzzy point 𝑎_𝜆 for 𝜆 in [0,1], of 𝑓 belongs to some minimal fuzzy left ideal 𝑓_𝑖,𝑖∈ℕ of 𝑆. Now 𝑓∘𝑎_𝜆 is a fuzzy left ideal of 𝑆, since 𝑆∘(𝑓∘𝑎_𝜆)=(𝑆∘𝑓)∘𝑎_𝜆⊆𝑓∘𝑎_𝜆. Also 𝑓∘𝑎_𝜆⊆𝑓∘𝑓_𝑖⊆𝑓_𝑖, where 𝑓_𝑖 is minimal fuzzy left ideal so 𝑓∘𝑎_𝜆=𝑓_𝑖. Thus 𝑎_𝜆⊆𝑓∘𝑎_𝜆 implies that 𝑓⊆𝑓∘𝑎_𝜆, where 𝑓∘𝑎_𝜆 is minimal fuzzy left ideal of 𝑆 and so 𝑓=𝑓∘𝑎_𝜆. Hence, 𝑓 is the fuzzy left ideal of 𝑆.

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

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

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

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 𝑆 contains at least one minimal fuzzy left ideal and one minimal fuzzy right ideal of 𝑆, then the class sum of all the minimal fuzzy left ideals of 𝑆 coincide with the class sum of all the minimal fuzzy right ideals and constitutes the fuzzy kernel of 𝑆.

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

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

Lemma 18. Let 𝑎_𝜆 be in 𝑓_𝑅 and 𝑏_𝜇 be in 𝑓_𝑅∘𝑓_𝐿, then there exists a fuzzy point 𝑥_𝑡 in 𝑓_𝑅∘𝑓_𝐿 such that 𝑎_𝜆∘𝑥_𝑡=𝑏_𝜇.

Proof. Note that 𝑎_𝜆∘𝑓_𝑅 is a fuzzy right ideal of 𝑆 since, (𝑎_𝜆∘𝑓_𝑅)∘𝑆=𝑎_𝜆∘(𝑓_𝑅∘𝑆)⊆𝑎_𝜆∘𝑓_𝑅 and also 𝑎_𝜆∘𝑓_𝑅⊆𝑓_𝑅∘𝑓_𝑅⊆𝑓_𝑅. But 𝑓_𝑅 is the minimal fuzzy right ideal of 𝑆 so 𝑎_𝜆∘𝑓_𝑅=𝑓_𝑅. Hence 𝑎_𝜆∘𝑓_𝑅∘𝑓_𝐿=𝑓_𝑅∘𝑓_𝐿.

Lemma 19. Let 𝑎_𝜆 be in 𝑓_𝐿 and 𝑏_𝜇 be in 𝑓_𝑅∘𝑓_𝐿, then there exists a fuzzy point 𝑥_𝑡 in 𝑓_𝑅∘𝑓_𝐿 such that 𝑥_𝑡∘𝑎_𝜆=𝑏_𝜇.

Proof. Since 𝑓_𝐿∘𝑎_𝜆 is a fuzzy left ideal of 𝑆 since, 𝑆∘(𝑓_𝐿∘𝑎_𝜆)=(𝑆∘𝑓_𝐿)∘𝑎_𝜆⊆𝑓_𝐿∘𝑎_𝜆 and also 𝑓_𝐿∘𝑎_𝜆⊆𝑓_𝐿∘𝑓_𝐿⊆𝑓_𝐿. But 𝑓_𝐿 is the minimal fuzzy left ideal of 𝑆 so 𝑓_𝐿∘𝑎_𝜆=𝑓_𝐿. Hence 𝑓_𝑅∘𝑓_𝐿∘𝑎_𝜆=𝑓_𝑅∘𝑓_𝐿.

Theorem 20. 𝑓_𝑅∘𝑓_𝐿 is a group.

Proof. Since (𝑓_𝑅∘𝑓_𝐿)∘(𝑓_𝑅∘𝑓_𝐿)⊆(𝑓_𝑅∘𝑓_𝐿)∘𝑓_𝐿⊆𝑓_𝑅∘𝑓_𝐿, which implies that 𝑓_𝑅∘𝑓_𝐿 is a semigroup. Let 𝑎_𝜆 and 𝑏_𝜇 be any two fuzzy points of 𝑓_𝑅∘𝑓_𝐿 then 𝑎_𝜆 and 𝑏_𝜇 are in both 𝑓_𝑅 and 𝑓_𝐿, since 𝑓_𝑅∘𝑓_𝐿⊆𝑓_𝑅 and 𝑓_𝑅∘𝑓_𝐿⊆𝑓_𝐿. Thus by Lemmas 18 and 19 there exists 𝑥_𝛾 and 𝑦_𝜌 in 𝑆 such that, 𝑎_𝜆∘𝑥_𝛾=𝑏_𝜇 and 𝑦_𝜌∘𝑎_𝜆=𝑏_𝜇. Hence by Theorem 1, 𝑓_𝑅∘𝑓_𝐿 is a group.

Lemma 21. Let 𝑒_𝛼 be the identity element of the group (𝑓_𝑅∘𝑓_𝐿,∘), then 𝑓_𝑅=𝑒_𝛼∘𝑆,𝑓_𝐿=𝑆∘𝑒_𝛼 and 𝑓_𝑅∩𝑓_𝐿=𝑒_𝛼∘𝑆∘𝑒_𝛼.

Proof. Since identity element is in 𝑓_𝑅 since 𝑓_𝑅∘𝑓_𝐿⊆𝑓_𝑅. Now (𝑒_𝜆∘𝑆)∘𝑆=𝑒_𝜆∘(𝑆∘𝑆)⊆𝑒_𝜆∘𝑆, which implies that 𝑒_𝜆∘𝑆 is a fuzzy right ideal of 𝑆 and is contained in minimal fuzzy right ideal 𝑓_𝑅 of 𝑆 and hence 𝑒_𝜆∘𝑆=𝑓_𝑅. Similarly we can show that 𝑆∘𝑒_𝜆=𝑓_𝐿. Now 𝑓_𝑅∘𝑓_𝐿=(𝑒_𝜆∘𝑆)∘(𝑆∘𝑒_𝜆)=𝑒_𝜆∘(𝑆∘𝑆)∘𝑒_𝜆⊆𝑒_𝜆∘𝑆∘𝑒_𝜆, also 𝑒_𝜆∘𝑆∘𝑒_𝜆⊆𝑓_𝑅∘𝑆∘𝑓_𝐿⊆𝑓_𝑅∘𝑓_𝐿.

Theorem 22. For minimal fuzzy right ideal 𝑓_𝑅 and minimal fuzzy left ideal 𝑓_𝐿 of a semigroup 𝑆,𝑓_𝑅∘𝑓_𝐿=𝑓_𝑅∩𝑓_𝐿.

Proof. We only need to show that 𝑓_𝑅∩𝑓_𝐿⊆𝑓_𝑅∘𝑓_𝐿. Let 𝑎_𝜆 be arbitrary fuzzy point of 𝑓_𝑅∩𝑓_𝐿 and let 𝑒_𝛼 be identity element of 𝑓_𝑅∘𝑓_𝐿 then by Lemma 18 there exists fuzzy point 𝑥_𝛾 of 𝑓_𝑅∘𝑓_𝐿 such that 𝑎_𝜆∘𝑥_𝛾=𝑒_𝛼. Let 𝑦_𝜌 be the inverse fuzzy element of 𝑥_𝛾 in 𝑓_𝑅∘𝑓_𝐿 then, 𝑎_𝜆∘𝑒_𝛼=𝑎_𝜆∘(𝑥_𝛾∘𝑦_𝜌)=(𝑎_𝜆∘𝑥_𝛾)∘𝑦_𝜌=𝑒_𝛼∘𝑦_𝜌=𝑦_𝜌. Now 𝑎_𝜆⊆𝑓_𝑅∘𝑓_𝐿⊆𝑓_𝐿, so by Lemma 21, 𝑎_𝜆∘𝑒_𝛼=𝑒_𝛼. Hence 𝑎_𝜆=𝑦_𝜌 and so 𝑎_𝜆 is in 𝑓_𝑅∘𝑓_𝐿.

Lemma 23. The identity fuzzy point 𝑒_𝜆 of the fuzzy group 𝑓_𝑅∘𝑓_𝐿 is a fuzzy primitive.

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

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

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