Abstract
A new trend of using fuzzy algebraic structures in various applied sciences is becoming a central focus due to the accuracy and nondecoding nature. The aim of the present paper is to develop a new type of fuzzy subsystem of an ordered semigroup This new type of fuzzy subsystem will overcome the difficulties faced in fuzzy ideal theory of an ordered semigroup up to some extent. More precisely, we introduce -fuzzy left (resp., right, quasi-) ideals of These concepts are elaborated through appropriate examples. Further, we are bridging ordinary ideals and -fuzzy ideals of an ordered semigroup through level subset and characteristic function Finally, we characterize regular ordered semigroups in terms of -fuzzy left (resp., right, quasi-) ideals.
1. Introduction
The breakthrough of Zadeh’s pioneering idea [1] of fuzzy sets changed our traditional tools for formal modeling, reasoning, and computing. These traditional tools are crisp in nature. Crisp means dichotomous, i.e., yes or no type rather than more or less type. In set theory, an element can either belong to a set or not. Zadeh’s seminal paper on fuzzy set opens a new direction for researchers to use fuzzy sets in applied sciences like automata theory, computer science, artificial intelligence, coding theory, robotics, and much more. Keeping this inspiration in view, Rosenfeld [2] introduced fuzzy subgroups. The inception of fuzzy subgroups provides a platform for other researchers to use this pioneering idea in other algebraic structures along with several diverse applications. Among other algebraic structures, semigroups (especially ordered semigroups) are having a lot of applications in error correcting codes, control engineering, performance of super computer, and information sciences. Mordeson et al.’s idea in [3] gives birth to an up to date account of fuzzy subsemigroups and fuzzy ideals of semigroups while Kehayopulu and Tsingelis [4–6] used fuzzy sets in ordered semigroups to develop fuzzy ideal theory. Shabir and Khan [7] gave a characterization of ordered semigroups by the properties of fuzzy ideals and fuzzy generalized bi-ideals.
In 1996, the idea of a quasi-coincidence of a fuzzy point with a fuzzy set [8, 9] is presented which played a vital role to generate different types of fuzzy subgroups. Bhakat and Das [8] gave the concept of -fuzzy subgroups and introduced -fuzzy subgroups by using the “belongs to” relation and “quasi-coincident with” relation between a fuzzy point and a fuzzy subgroup. In fact this is an important and useful generalization of Rosenfeld’s idea of fuzzy subgroup [2]. These new generalizations provide a motivation to other researchers for further development in other algebraic structures. Since then a variety of research has been carried out using this icebreaking idea. More precisely, the aforementioned notion is used by Ma et al. [10, 11] in -algebras and Davvaz and Mozafar [12] in Lie algebra, and Davvaz and coauthors used the idea of generalized fuzzy sets in rings [13–15], Jun et al. [16], Khan and Shabir [17], and Khan et al. [18] in ordered semigroups, and Shabir et al. [19, 20] in semigroups. For further details, the readers are referred to [21–26].
In 2009, Jun [27] initiated a more general form of quasi-coincident with relation and provided , where The notion is further strengthened by other researchers to use in various algebraic structures [28–31].
Recently, Jun et al. [32] presented another comprehensive generalization of -fuzzy subgroups and discussed fuzzy subgroups in light of generalized quasi-coincident with relation. Further, Khan et al. [33] elaborated ordered semigroups in terms of fuzzy generalized bi-ideals using this idea [32]. Also, Khan et al. [34] determined fuzzy filters of ordered semigroups for the said notion.
In this paper, we apply Jun et al.’s idea [32] to ordered semigroups and discuss ordered semigroups through fuzzy left (resp., right) ideals of type , where and , and obtained several important results. More precisely, -fuzzy left (resp., right) ideals are introduced. Ordinary fuzzy ideals and -fuzzy ideals of an ordered semigroup are linked. Finally, regular ordered semigroups by the properties of these newly developed fuzzy left (resp., right) ideals are characterized.
2. Preliminaries
Algebraic structures like semigroups, ordered semigroups, and monids are extensively used in applied fields such as automata theory, coding theory, and theoretical computer science. The present section discusses various fundamental definitions and results essential for the proceeding article.
A system , in which is a semigroup, is a poset with both left and right compatibility, i.e., , for all , is known as ordered semigroup. For , we denote for some If , then we write instead of . For , we denote . If , then , and [18]. Later, will denote an ordered semigroup, unless otherwise stated. For a nonempty subset of , and , then is called a right (resp., left) ideal [18] of if (resp., ). If is both a left and a right ideal of then is called a two-sided ideal or simply an ideal of An ordered semigroup is regular [22] if for every there exists such that Let be an ordered semigroup and . Then the characteristic function of is defined by
By a fuzzy subset of an ordered semigroup , we mean a function .
A fuzzy subset of is called a fuzzy left (resp., right) ideal of if, for all ,(i),(ii) (resp. .
A fuzzy subset of is called a fuzzy ideal of if it is both a fuzzy left and a fuzzy right ideal of For a nonempty subset of , define (see [18]).
Transfer principle in fuzzy set theory [35] plays a remarkable role by providing the connection between classical and fuzzy subsets through level subset. A subset of the form where , is known as level subset of For the said purpose, the following lemma is used.
Lemma 1 (see [35]). A fuzzy subset defined on has the property if and only if all nonempty level subsets have the property
As a consequence of the above property, the following useful theorem is given.
Theorem 2 (see [14]). A fuzzy subset of is a fuzzy left (resp., right) ideal of if and only if each nonempty level subset , for all , is a left (resp., right) ideal of
We denote by the set of all fuzzy subsets of and define an order relation “” on the fuzzy subsets of as follows:If , , the intersection , union , and product can be defined as follows.
For all ,
Theorem 3 (see [4]). A nonempty subset of an ordered semigroup is a left (resp., right) ideal of if and only if the characteristic function of is a fuzzy left (resp., right) ideal of .
3. Fuzzy Ideals Based on -Quasi-Coincident with Relation
A verity of latest research, conducting multiple conferences and workshops on fuzzy mathematics, shows the importance of this particular field of study. It is worth mentioning that the researchers are more interested to use algebraic structures in upcoming completed problems not even in mathematics but in an engineering, computer science, robotics coding theory, and others. The idea of quasi-coincident with relation was based on the aforementioned concept, where Jun’s [27] generalization of quasi-coincident with relation played a vital role in generating several types of fuzzy subsystems which have already been used in various applied fields which leads to a verity of productive research [20, 28–31, 33, 34]. Jun et al. [32] further generalized his idea [27] and initiated an essential generalization of -fuzzy subgroups. Keeping in view Jun et al.’s idea [32], we introduce a new generalization called -quasi-coincident with relation. In this section, new classification of ordered semigroups based on -fuzzy ideals, where is an arbitrary element of and , is determined. A fuzzy subset of of the form is fuzzy point [36] 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. Generalizing the concept of , in ordered semigroups, we define , as , where and . Note that implies , but the converse of the statement is not always true which will later be elaborated through examples. Particularly, if , then every -quasi-coincident with relation will lead to quasi-coincident with relation, symbolically Also, means that or
Definition 4. A fuzzy subset of is called an -fuzzy left (resp., right) ideal of if it satisfies the conditions:(1)(2)Note that every -fuzzy left (resp., right) ideal with is an -fuzzy left (resp., right) ideal of [30].
Example 5. Consider an ordered semigroup with multiplication Table 1 and order relation: This order relation is represented by Figure 1.
Let be a fuzzy subset of , for , and define a fuzzy subset as follows:
Then is an -fuzzy ideal of with
Obviously, every -fuzzy left (resp., right) ideal is an -fuzzy left (resp., right) ideal of , but the converse in not true in general. Because in the aforementioned example, if and then is an -fuzzy ideal of with but not -fuzzy ideal. but If , then is not an -fuzzy ideal of
Theorem 6. Let be a left (resp., right) ideal of and a fuzzy subset in defined by Then(1) is a -fuzzy ideal of .(2) is an -fuzzy ideal of .
Proof. (1) We discuss only the case of left ideals since the case for right ideals can be proved similarly. Let , , and be such that . Then and so Since is a left ideal of and , we have . Thus If , then and so If , then and so Therefore
Let and be such that Then so Since is a left ideal of , we have . Thus Now, if , then and so If , then and so Therefore,
(2) Let , , and be such that . Then and Since is a left ideal of and , we have . Thus If , then and so If , then and so Therefore
If and be such that Then and so . Thus If , then and so If , then and we have Therefore Consequently, is an -fuzzy left ideal of .
If we take in Theorem 6, then we get the following corollary.
Corollary 7 (see [30]). Let be a left (resp., right) ideal of and , a fuzzy subset in defined by Then(1) is a -fuzzy ideal of .(2) is an -fuzzy ideal of .
Theorem 8. A fuzzy subset of is an -fuzzy left (resp., right) ideal of if and only if(1),(2).
Proof. Suppose that is an -fuzzy ideal of On the contrary, assume that there exist , such that Choose such that Then , but and , so , a contradiction. Hence for all with .
If there exist such that , choose such that . Then but and , so . Thus, , a contradiction. Therefore, for all .
Conversely, let for some Then . Now, . If , then and , and follows. If , then and so Thus,
Let , then and so . If , then and and so If , then implies that Thus and consequently, is an -fuzzy left ideal of . Similarly we can prove that is an -fuzzy right ideal of
If we take in Theorem 8, then we have the following corollary [30].
Corollary 9. A fuzzy subset of is an -fuzzy left (resp. right) ideal of if and only if(1),(2).Using Theorem 8, we have the following characterization of fuzzy left (resp., right) ideals of ordered semigroups.
Proposition 10. Let be an ordered semigroup and Then is a left (resp., right) ideal of if and only if the characteristic function of is an -fuzzy left (resp., right) ideal of .
Corollary 11. A fuzzy subset of an ordered semigroup is an -fuzzy left (resp., right) ideal of if and only if it satisfies conditions (1) and (2) of Theorem 8.
Remark 12. Every fuzzy left (resp., right) ideal of an ordered semigroup is an -fuzzy left (resp., right) ideal of . But the converse is not true.
Example 13. Consider the ordered semigroup as given in Example 5, and define a fuzzy subset as follows: Then is an -fuzzy ideal with but for all is not an ideal of . Hence by Theorem 2, is not a fuzzy ideal of for all .
Theorem 14. Every -fuzzy left (resp., right) ideal is an -fuzzy left (resp., right) ideal of .
Proof. The proof is straightforward and is omitted here.
The converse of the above theorem is not true in general as shown in Example 15.
Example 15. Consider the ordered semigroup as given in Example 5, and define a fuzzy subset as follows: Then is an -fuzzy ideal but not an -fuzzy ideal of . This is because but .
Theorem 16. Every -fuzzy left (resp., right) ideal is an -fuzzy left (resp., right) ideal of .
Proof. Suppose that is an -fuzzy left ideal of . Let , and be such that , , then So by hypothesis, we have . Now let , and be such that . Then Hence . Similarly we can prove that .
If in Theorem 16, then the following corollary [30] can be obtained.
Corollary 17. Every -fuzzy left (resp., right) ideal is an -fuzzy left (resp., right) ideal of .
The importance of the following theorem is to provide the bridge between ordinary ideals and fuzzy ideals of type This linkage is very much essential for the characterization of ordered semigroups. Some time it is quite difficult to understand weather a particular ideal is fuzzy (resp., ordinary) ideal or not. In such a situation, the following result plays its role by providing the required information.
Theorem 18. A fuzzy subset of is an -fuzzy left (resp., right) ideal of if and only if is left (resp., right) ideal of for all .
Proof. Assume that is an -fuzzy left ideal of . Let , be such that for some Then and by hypothesis Hence Let be such that for some Then and we have Thus
Conversely, suppose that is a left ideal of for all . If there exist , such that , choose such that , then, implies that but , a contradiction. Hence, . If such that , then choose , and . Thus, but , a contradiction. Hence, for all , and . Therefore, is an -fuzzy left ideal of . By a similar way, we can prove that is an -fuzzy right ideal of
Note that Therefore, if in Theorem 18, then this new result can produce the following corollary of the literature [30].
Corollary 19. A fuzzy subset of is an -fuzzy left (resp., right) ideal of if and only if is left (resp., right) ideal of for all
Proposition 20. Let be a nonzero -fuzzy left (resp., right) ideal of Then the set is a left (resp., right) ideal of .
Proof. Let be an -fuzzy left ideal of . If , , and , then, Since is an -fuzzy left ideal of , we have Thus and so Let , then, and by hypothesis, Thus Consequently, is a left ideal of . Similarly, we can prove that is a right ideal of
If , then Proposition 20 leads to the following result [30].
Result 1. Let be a nonzero -fuzzy left (resp., right) ideal of Then the set is a left (resp., right) ideal of .
In the following lemma, we establish a relationship between -fuzzy left (resp., right) ideals and -fuzzy left (resp., right) ideals of
Lemma 21. Suppose that is an -fuzzy left (resp., right) ideal of such that for all Then is an -fuzzy left (resp., right) ideal of
Proof. Let and be such that , Then Since is an -fuzzy left ideal and , we have Hence, If and be such that Then and we have Thus, By a similar way, we can prove that
If , then Lemma 21 leads to the following result [30].
Result 2. If is an -fuzzy left (resp., right) ideal of such that for all , then is an -fuzzy left (resp., right) ideal of .
For any fuzzy subset of and , we denote as follows: Clearly
We call an -fuzzy level left (resp., right) ideal and a -level left (resp., right) ideal of .
We give another characterization of -fuzzy left (resp., right) ideals by using
Theorem 22. A fuzzy subset of is an -fuzzy left ideal of if and only if is a left (resp., right) ideal of for all
Proof. Assume is an -fuzzy left (resp., right) ideal of Let , , and be such that Then , that is, or Since is an -fuzzy left ideal of and , we have . We discuss the following cases.
Case 1 . If , then and so so If , then . Hence
Case 2 . If , then that is, Thus If , then Therefore Hence Let and for Then , that is, or . Since is an -fuzzy left ideal of , we have .
Case 1 . If , then and so so If , then Hence
Case 2 . If , then that is, Thus If , then Therefore, Hence Thus is left ideal. Similarly, we can prove that Consequently, is left (resp., right) ideal of
Conversely, assume that is a fuzzy subset of and let be such that is left ideal of If there exist , , and such that then Hence , since is left ideal and Thus and we have or , a contradiction. Hence for all with Let be such that for some Then Hence , since is left ideal. Thus and we have or , a contradiction. Hence for all Thus is an -fuzzy left ideal of Similarly we can prove that is an -fuzzy right ideal of
If , then, a fuzzy subset of is an -fuzzy left ideal of if and only if is a left (resp., right) ideal of for all [30].
From Theorems 2 and 22, and are left ideals of for all , but is not left ideal of for in general, as shown in the next example.
Example 23. Consider an ordered semigroup as given in Example 5, and define a fuzzy subset as follows: Then is left ideal for , , and but neither or is left ideal because of but .
4. Characterization of Regular Ordered Semigroups by -Quasi-Coincident with Relation
Regular ordered semigroups are known to be the most important classes of semigroup theory. In this section, fuzzy quasi-ideals of type are introduced. Further, the connection between -fuzzy left (resp., right) ideals and -fuzzy quasi-ideals is provided. The classification of regular ordered semigroups by the properties of -fuzzy left (resp., right) ideals is also the milestone of the present section.
Definition 24. A fuzzy subset of is called an -fuzzy quasi-ideal of if it satisfies the conditions:(1),(2).
Theorem 25. Let be a nonzero -fuzzy quasi-ideal of , then the set is a quasi-ideal of
Proof. Let be such that If , then Since is an -fuzzy quasi-ideal of , therefore Thus and so Let , then and Hence there exist and such that , and Then Since , we have Hence implies that
The proof of the following proposition is straightforward and is omitted.
Proposition 26. Every -fuzzy left (resp., right) or two-sided ideal of is an -fuzzy quasi-ideal of .
In the following example, it is shown that the converse of Proposition 26 is not true in general.
Example 27. Consider an ordered semigroup with multiplication Table 2 and order relation: The order relation is given by Figure 2.
Then is a quasi-ideal of , but not a left (resp., right) ideal of
Define a fuzzy subset as follows: Then is fuzzy quasi-ideal of but not an -fuzzy left ideal of for and since .
Definition 28. Let be an ordered semigroup and fuzzy subsets of Then the -product of and is defined by
Proposition 29. Let be an ordered semigroup and fuzzy subsets of such that and Then
Lemma 30. If and are -fuzzy left (resp., right) ideals of an ordered semigroup , then is an -fuzzy left (resp., right) ideal of , where for all
Proof. Let such that Since and are -fuzzy left ideals of , hence and . Therefore Hence .
Let . Since and are -fuzzy left ideals of Then, and . Thus Hence is an -fuzzy left ideal of Similarly we can prove that is an -fuzzy right ideal of .
Let and be fuzzy subsets of ; we define as follows: for all
Putting in Lemma 30 leads to a result [30]: “If and are -fuzzy left (resp., right) ideals of an ordered semigroup , then is an -fuzzy left (resp., right) ideal of ”.
Lemma 31. If is an -fuzzy right of and is an -fuzzy left ideal of , respectively, then
Proof. Let If , then . Assume that , then Since , and is an -fuzzy right and is an -fuzzy left ideal of , thus Hence,
Regular ordered semigroups are characterized by right and left ideals in [5]. In the following we give a characterization of regular ordered semigroups in terms of -fuzzy right and -fuzzy left ideals.
Lemma 32 (see [5]). Let be an ordered semigroup. Then the following statements are equivalent:(1) is regular.(2) for every right ideal and every left ideal of
Lemma 33. Let be an ordered semigroup and , Then, we have(i) for all (ii).(iii).
Theorem 34. An ordered semigroup is regular if and only if, for every -fuzzy right ideal and every -fuzzy left ideal of , one has
Proof. Let be regular ordered semigroup and Then there exists such that Hence and we have Since is an -fuzzy right ideal of , therefore . Thus Hence . On the other hand, by Lemma 31, we have Therefore .
Conversely, suppose for every -fuzzy right ideal and every -fuzzy left ideal of we have By Lemma 32, we need to show that for every right ideal and left ideal of Let , then and . Since is right ideal and is left ideal of , thus by Lemma 33, is a fuzzy right and is a fuzzy left ideal Therefore by Proposition 10, is an -fuzzy right ideal and is an -fuzzy left ideal of By hypothesis, . Since and so and , then It follows that . By Lemma 33, . Thus It follows that Hence but , always holds. Therefore Consequently is regular.
5. Conclusion
Since the traditional mathematical tools are dichotomous in nature, i.e., yes or no type rather than more or less type, therefore, these tools can not be used to handle complex problems involving uncertainties of various applied fields like decision-making problems, control engineering, coding theory, theoretical physics, computer sciences, and fuzzy automata theory. For the said problems, fuzzy sets and their various generalizations are used. A quasi-coincident with relation is one of the most important relations used in algebraic structures along with a verity of applications in the aforementioned areas. The aim of the present paper was to introduce a comprehensive type of relation known as -quasi-coincident with relation which can lead to the existing systems and also some new types of subsystems involving quasi-coincident with relation. More precisely, -fuzzy left (resp., right, quasi-) ideals of are developedwhich were elaborated through examples. The bridging of ordinary ideals and -fuzzy ideals of an ordered semigroup is constructed while regular ordered semigroups were characterized through -fuzzy left (resp., right, quasi-) ideals. As , therefore for particular value , our algebraic system reduced to [30].
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.