Research Article | Open Access

# -Fuzzy Ideals in Ordered Semigroups

**Academic Editor:**David Dobbs

#### Abstract

We introduce the concept of -fuzzy left (right) ideals in ordered semigroups and characterize ordered semigroups in terms of -fuzzy left (right) ideals. We characterize left regular (right regular) and left simple (right simple) ordered semigroups in terms of -fuzzy left (-fuzzy right) ideals. The semilattice of left (right) simple semigroups in terms of -fuzzy left (right) ideals is discussed.

#### 1. Introduction

A fuzzy subset of a given set is described as an arbitrary function , where is the usual closed interval of real numbers. This fundamental concept was first introduced by Zadeh in his pioneering paper [1] of 1965, which provides a natural framework for the generalizations of some basic notions of algebra, for example, set theory, group theory, ring theory, groupoids, real analysis, measure theory, topology, and differential equations, and so forth. Rosenfeld (see [2]) was the first who introduced the concept of a fuzzy set in a group. The concept of a fuzzy ideal in semigroups was first developed by Kuroki (see [3–8]). He studied fuzzy ideals, fuzzy bi-ideals, fuzzy quasi-ideals, and fuzzy semiprime ideals of semigroups. Fuzzy ideals and Green's relations in semigroups were studied by McLean and Kummer in [9]. Dib and Galham in [10] introduced the definitions of a fuzzy groupoid and a fuzzy semigroup and studied fuzzy ideals and fuzzy bi-ideals of a fuzzy semigroup. Ahsan et al. in [11] characterized semisimple semigroups in terms of fuzzy ideals. A systematic exposition of fuzzy semigroups by Mordeson et al. appeared in [12], where one can find theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy finite state machines, and fuzzy languages. The monograph by Mordeson and Malik (see [13]) deals with the applications of fuzzy approach to the concept of automata and formal languages. Fuzzy sets in ordered semigroups/ordered groupoids were first introduced by Kehayopulu and Tsingelis in [14]. They also introduced the concepts of fuzzy bi-ideals and fuzzy quasi-ideals in ordered semigroups in [14, 15].

In [16], Shabir and Khan introduced the concept of a fuzzy generalized bi-ideal of ordered semigroups and characterized different classes of ordered semigroups by using fuzzy generalized bi-ideals. They also gave the concept of fuzzy left (resp., bi-) filters in ordered semigroups and gave the relations of fuzzy bi-filters and fuzzy bi-ideal subsets of ordered semigroups in [17].

In this paper, we introduce the concept of -fuzzy left (resp., right) ideals and characterize regular, left, and right simple ordered semigroups and completely regular ordered semigroups in terms of -fuzzy left (resp., right) ideals. In this respect, we prove that a regular ordered semigroup is left simple if and only if every -fuzzy left ideal of is a constant function. We also prove that is left regular if and only if for every -fuzzy left ideal of we have for every . Next we characterize semilattice of left simple ordered semigroups in terms of -fuzzy left ideals. We prove that an ordered semigroup is a semilattice of left simple semigroups if and only if for every -fuzzy left ideal of , and for all .

#### 2. Preliminaries

By an *ordered semigroup* (or *po-semigroup*) we mean a structure in which

Let be an ordered semigroup. A nonempty subset of is called a *left* (resp., *right*) *ideal* of (see [14]) if

is called a *two-sided ideal* or simply an *ideal* if is both left and right ideal of .

For , denote for some . If , then we write instead of . Let , then , , and .

Let be an ordered semigroup and a fuzzy subset of . Then is called a *fuzzy left* (resp., *right*) *ideal* of if

A fuzzy left and right ideal of is called a *fuzzy two-sided ideal* or simply a *fuzzy ideal* of .

#### 3. -Fuzzy Left (Resp., Right) Ideals

Let be an ordered semigroup. By a *negative fuzzy subset* (briefly *-fuzzy subset*) of we mean a *mapping *.

*Definition 3.1. *Let be an ordered semigroups and an -fuzzy subset of . Then is called an -*fuzzy left* (resp., right) ideal of if(1)() (),(2)() ( (resp., )).

An -fuzzy left and right ideal of is called an -*fuzzy two-sided ideal* of .

For any -fuzzy subset of and the set
is called the -level subset of .

Theorem 3.2. *Let be an ordered semigroup. An -fuzzy subset of is an -fuzzy left (resp., right) ideal of if and only if it satisfies
*

*Proof. *Suppose that is an -fuzzy left ideal of . Let be such that . If , then . Since we have and and we have . Let be such that . Then , since . Then one has implies that . Thus .

Conversely, assume that for all such that , the set is a left ideal of . Let be such that . If , then since for all , we have . If , then and since , and is a left ideal of , we have and so . Let . If , then since for all we have . If , then and since is a left ideal of , we have . Then .

By left-right dual of the above theorem, we have the following theorem.

Theorem 3.3. *Let be an ordered semigroup. An -fuzzy subset of is an -fuzzy left (resp., right) ideal of if and only if it satisfies
*

*Example 3.4. *Let be the ordered semigroup defined by the multiplication and the order as follows:

Then left ideals of are and (see [18]). Define by

Then

Then is a left ideal of and by Theorem 3.2, is an -fuzzy left ideal of .

Let be an ordered semigroup and . The *characteristic **-function * of is defined by

Theorem 3.5. *Let be an ordered semigroup and . Then the followings are equivalent. *(i)* is a left (resp., right) ideal of .*(ii)*The characteristic -function
of is an -fuzzy left (resp., right) ideal of .*

*Proof. *(i)(ii) Suppose that is a left ideal of . Let , . If , then . Since and is a left ideal of , we have . Then and hence . If , then . Since is an -fuzzy subset of , we have for all . Hence .

Let and . Then . Since is a left ideal of and , we have . Then , and we have . If . Then . Since for all . Hence . Thus is an -fuzzy left ideal of .

(ii)(i) Assume that
is an -fuzzy left ideal of . Let , . If , then . Since , we have . Then and we have .

Let and . Then . Since is an -fuzzy left ideal of , we have . Hence and . Thus is a left ideal of .

A subset of an ordered semigroup is called *semiprime* (see [15]) if for every such that , we have . Equivalent definition: for each subset of such that , we have .

#### 4. Characterization of Left Simple and Left Regular Ordered Semigroups

Lemma 4.1 (cf. [15]). *Let be an ordered semigroup. Then the followings are equivalent.*(i)* is a left simple subsemigroup of , for every .*(ii)*Every left ideal of is a right ideal of and semiprime.**An ordered semigroup is regular (see [19]) if for every , there exists such that .**Equivalent definitions are*(1)*() (),*(2)*() ().**An ordered semigroup is left (resp., right ) simple (see [15]) if for every left (resp., right) ideal of , one has and is called simple if it is left simple and right simple.*

Lemma 4.2 (cf. [15]). *An ordered semigroup is left (resp., right) simple if and only if for every , (resp., ).*

Theorem 4.3. *For a regular ordered semigroup , the following conditions are equivalent.*(i)* is left simple.*(ii)*Every -fuzzy left ideal of is a constant -function.*

*Proof. *(i)(ii) Let be a left simple ordered semigroup, an -fuzzy left ideal of and . We consider the set

Then . In fact, since is regular and , there exists such that . It follows from (OS3) that
and so and hence .

(1) is a constant -function on . Let . Since is left simple and we have . Since , , so there exists such that . Hence . Since is an -fuzzy left ideal of , we have

Since , we have . Since is an -fuzzy left ideal of , we have . Thus . Besides, since is left simple and , we have . Since , so there exists such that . Hence . Since is an -fuzzy left ideal of , we have
On the other hand, , we have and so . Hence .

(2) is a constant -function on . Let . Since is regular, there exists such that . We consider the element . Then it follows by (OS3) that

Hence and by (1), we have . Besides, since is an -fuzzy left ideal of , we have . Thus . On the other hand, since is left simple and , . Since , we have for some . Since is an -fuzzy left ideal of , we have . Thus .

(ii)(i) Let . Then the set is a left ideal of . In fact, . If and , then . Since is a left ideal of , by Theorem 3.5, the characteristic -function of ,
is a fuzzy left ideal of . By hypothesis, is a constant -function; that is, there exists such that

Let and be such that . Then . On the other hand, since , we have , a contradiction to the fact that is a constant -mapping. Hence .

From left-right dual of Theorem 4.3, we have the following.

Theorem 4.4. *For a regular ordered semigroup. The following statements are equivalent.*(1)* is right simple.*(2)*Every -fuzzy right ideal of is a constant -function.**An ordered semigroup is left (resp., right ) regular [20] if for every there exists such that (resp., ).**Equivalent definitions are*(1)*() ( (resp., )),*(2)*() ( (resp., )).**An ordered semigroup is intraregular (see [16]) if for every , there exist such that .**Equivalent definitions are*(1)*() (),*(2)*() ().**An ordered semigroup is called completely regular (see [21]) if it is regular, left regular, and right regular.*

Lemma 4.5 (cf. [21]). *An ordered semigroup is completely regular if and only if for every . Equivalently, for every .*

Theorem 4.6. *An ordered semigroup is left regular if and only if for each -fuzzy left ideal of , one has
*

*Proof. *Suppose that is an -fuzzy left ideal of and let . Since is left regular, there exists such that . Since is an -fuzzy left ideal of , we have

Conversely, let . We consider the left ideal of , generated by . Then by Theorem 3.5, the characteristic -function
is an -fuzzy left ideal of . By hypothesis we have

Since , we have and . Then and for some . If , then and . If for some , then , and .

From left-right dual of Theorem 4.6, we have the following theorem.

Theorem 4.7. *An ordered semigroup is right regular if and only if for each -fuzzy right ideal of , one has
*

From [22] and by Theorems 4.6, 4.7, and Lemma 4.5, we have the following characterization theorem for completely regular ordered semigroups.

Theorem 4.8. *Let be an ordered semigroup. Then the following statements are equivalent.*(i)* is completely regular.*(ii)*For each -fuzzy bi-ideal of one has
*(iii)*For each -fuzzy left ideal and each -fuzzy right ideal of we have
**An ordered semigroup is called left (resp., right ) duo if every left (resp., right) ideal of is a two-sided ideal of . An ordered semigroup is called duo if it is both left and right duo.*

*Definition 4.9. *An ordered semigroup is called -*fuzzy left* (resp., *right*) *duo* if every -fuzzy left (resp., right) ideal of is an -fuzzy two-sided ideal of . An ordered semigroup is called -*fuzzy duo* if it is both -fuzzy left and -fuzzy right duo.

Theorem 4.10. *Let be a regular ordered semigroup. Then the followings are equivalent.*(i)* is left duo.*(ii)* is -fuzzy left duo.*

*Proof. *(i)(ii) Let be left duo and an -fuzzy left ideal of . Let . Then the set is a left ideal of . In fact, and if and , then . Since is left duo, then is a two-sided ideal of . Since is regular, there exists such that ,

Thus and for some . Since is an -fuzzy left ideal of , we have

Let be such that . Then , because is an -fuzzy left ideal of . Thus is an -fuzzy right deal of and is -fuzzy left duo.

(ii)(i) Let be -fuzzy left duo and a left ideal of . Then the characteristic -function of is an -fuzzy left ideal of . By hypothesis, is an -fuzzy right ideal of , and by Theorem 3.5, is a right ideal of . Thus is left duo.

By the left-right dual of Theorem 4.10, we have the following.

Theorem 4.11. *Let be a regular ordered semigroup. Then the followings are equivalent.*(i)* is right duo.*(ii)* is -fuzzy right duo.*

Theorem 4.12. *Let be a regular ordered semigroup. Then the followings are equivalent.*(i)*Every bi-ideal of is a right ideal of .*(ii)*Every -fuzzy bi-ideal of is an -fuzzy right ideal of .*

*Proof. *(i)(ii) Let and an -fuzzy bi-ideal of . Then is a bi-ideal of . In fact, , and if and , then . Since is a bi-ideal of , by hypothesis is right ideal of . Since and is regular, there exists such that , then

Then for some . Since is an -fuzzy bi-ideal of , we have

Let be such that . Then because is an -fuzzy bi-ideal of . Thus is an -fuzzy right ideal of .

(ii)(i) Let be a bi-ideal of . Then by Theorem 3.5, is an -fuzzy bi-ideal of . By hypothesis is an -fuzzy right ideal of . By Theorem 3.5, is a right ideal of .

By left-right dual of Theorem 4.12, we have the following.

Theorem 4.13. *Let be a regular ordered semigroup. Then the followings are equivalent.*(i)*Every bi-ideal of is a left ideal of .*(ii)*Every -fuzzy bi-ideal of is an -fuzzy left ideal of .*

#### 5. Characterization of Intraregular Ordered Semigroups in Terms of -Fuzzy Ideals

*Definition 5.1 (cf. [22]). *Let be an ordered semigroup and an -fuzzy subset of . Then is called an semiprime -fuzzy subset of if

Theorem 5.2. *Let be an ordered semigroup and . Then the followings are equivalent.*(i)* is semiprime.*(ii)*The characteristic -function of is an -fuzzy semiprime subset.*

*Proof. *(i)(ii) Suppose that is semiprime subset. Let . Since for all , thus . If , then . Since is semiprime, we have . Then and; hence .

(ii)(i) Assume that is -fuzzy semiprime subset. Let be such that . Then . Since , we have , hence .

Theorem 5.3. *Let be an ordered semigroup and let be an -fuzzy subsemigroup of . Then is an -fuzzy semiprime if and only if for every , one has
*

*Proof. *Suppose that is an -fuzzy subsemigroup of such that is semiprime. Let . Then

The converse is obvious.

Theorem 5.4. *An ordered semigroup is intraregular if and only if every -fuzzy ideal of is semiprime.*

*Proof. *Suppose that is intraregular and an -fuzzy ideal of . Let . Then . In fact, since is intraregular, there exist such that . Then

Assume that is an -fuzzy ideal of such that for all . Consider the ideal of generated by . Then by Theorem 3.5, the characteristic -function is an -fuzzy ideal of and by hypothesis, we have . Since , then and . Then for some . If , then and . If for some , then and . If , then and .

#### 6. Some Semilattices of Left Simple Ordered Semigroups in Terms of -Fuzzy Left Ideals

Let be an ordered semigroup. A subsemigroup of is called *filter* (see [15]) of if

For , we denote by the filter of generated by () (i.e., the least filter with respect to inclusion relation containing ). denotes the equivalence relation on defined by (see [15]).

*Definition 6.1 (cf. [15]). *Let be an ordered semigroup. An equivalence relation on is called *congruence* if implies and for every . A congruence on is called *semilattice congruence* if and for each . If is a semilattice congruence on , then the -class of containing is a subsemigroup of for every .

An ordered semigroup is called a semilattice of *left simple semigroups* if there exists a semilattice congruence on such that the -class of containing is a left simple subsemigroup of for every .

Equivalent definition: there exists a semilattice and a family of left simple subsemigroups of such that(1) for all , ,(2),(3) for all .

In ordered semigroups the semilattice congruences are defined exactly same as in the case of semigroups—without order—so the two definitions are equivalent (see [15]).

Lemma 6.2 (cf. [22]). *An ordered semigroup is a semilattice of left simple semigroups if and only if for all left ideals , of one has
*

Theorem 6.3. *An ordered semigroup is a semilattice of left simple semigroups if and only if for every -fuzzy left ideal of , one has
*

*Proof. * (A) Let be a semilattice of left simple semigroups. By hypothesis, there exists a semilattice and a family of left simple subsemigroups of such that(i) for all , ,(ii),(iii) for all .

Let be an -fuzzy left ideal of and . Then . In fact, by Theorem 4.6, it is enough to prove that for every . Let , then there exists such that . Since is left simple, we have and

Since , we have and for some . Thus we have
since , we have .

(B) Let . Then by (A), we have

By symmetry we can prove that . Hence .

Assume that for every fuzzy left ideal of , we have
by condition (1) and Theorem 4.6, we have that is left regular. Let be a left ideal of and let . Then , since is left regular there exists such that
then and . On the other hand, since is a left ideal of , we have , then . Let and be left ideals of and let then for some and . We consider the left ideal generated by , that is, the set . Then by Theorem 3.5, the characteristic -function of defined by
is a fuzzy left ideal of . By hypothesis, we have . Since , we have and and hence . Then or for some . If , then and . If , then and . Thus . By symmetry we can prove that . Therefore , and by Lemma 6.2, it follows that is a semilattice of left simple semigroups.

From left-right dual of Theorem 6.3, we have the following.

Theorem 6.4. *An ordered semigroup is a semilattice of right simple semigroups if and only if for every -fuzzy right ideal of , one has
*

Lemma 6.5. *Let be an ordered semigroup and an -fuzzy left (resp., right) ideal of , and such that . Then .*

*Proof. *Since and is an -fuzzy left ideal of , we have
and so .

#### 7. Conclusion

Here we provided the concept of an -fuzzy ideal in ordered semigroups and characterized some classes in terms of -fuzzy left (resp., right) ideals of ordered semigroups. In this regard, we provided the characterizations of left (resp., right) regular, left (resp., right) simple, and completely regular ordered semigroups in terms of -fuzzy left (resp., right) ideals.

In our future work we will consider -fuzzy prime ideals and -fuzzy filters in ordered semigroups and will establish the relations between them. We will also try to discuss the quotient structure of ordered semigroup in terms of -fuzzy ideals.

#### Acknowledgments

The authors would like to thank the learned referees for their valuable comments and suggestions which improved the presentation of the paper and for their interest in their work. We would also like to thank Professor Dost Muhammad for his excellent guidance and valuable suggestions during this work.

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Copyright © 2009 Asghar 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.