Abstract
The concept of non--quasi-coincidence of an interval valued ordered fuzzy point with an interval valued fuzzy set is considered. In fact, this concept is a generalized concept of the non--quasi-coincidence of a fuzzy point with a fuzzy set. By using this new concept, we introduce the notion of interval valued -fuzzy quasi-ideals of ordered semigroups and study their related properties. In addition, we also introduce the concepts of prime and completely semiprime interval valued -fuzzy quasi-ideals of ordered semigroups and characterize bi-regular ordered semigroups in terms of completely semiprime interval valued -fuzzy quasi-ideals. Furthermore, some new characterizations of regular and intra-regular ordered semigroups by the properties of interval valued -fuzzy quasi-ideals are given.
1. Introduction
The theory of fuzzy sets was first developed by Zadeh [1] and has been applied to many branches in mathematics. Rosenfeld [2] inspired the fuzzification of algebraic structures and introduced the notion of fuzzy subgroups. Later on, Zadeh [3] also introduced the concept of interval valued fuzzy set by considering the values of the membership functions as the intervals of numbers instead of the numbers alone. An interval valued fuzzy set is a generalized form of the ordinary fuzzy sets and provides a more adequate description of uncertainty than the ordinary fuzzy sets. The interval valued fuzzy subgroups were first defined and studied by Biswas [4] which are the subgroups of the same nature of the fuzzy subgroups defined by Rosenfeld. Interval valued fuzzy subsets have been applied extensively in different kinds of fields of mathematics (see [5–10]). A new kind of fuzzy subgroup, that is, the -fuzzy subgroup, was then introduced by Bhakat and Das [11, 12] by using the combined notions of “belongingness” and “quasicoincidence” of fuzzy points and fuzzy sets proposed by Pu and Liu in [13]. In fact, the -fuzzy subgroup is an important and useful generalization of Rosenfeld's fuzzy subgroup. Since then, many researchers used the idea of generalized fuzzy sets and gave several results in different branches of algebra. For more details, the reader is referred to [14–20].
In mathematics, an ordered semigroup is a semigroup together with a partial order that is compatible with the semigroup operation. Ordered semigroups have several applications in the theory of sequential machines, formal languages, computer arithmetics, and error-correcting codes. In [21], Kehayopulu and Tsingelis applied the concept of fuzzy sets to the theory of ordered semigroups. Then they defined “fuzzy” analogous of several notations, which appeared to be useful in the theory of ordered semigroups. The theory of fuzzy sets on ordered semigroups has been recently developed (see [22–30]). Recently, Khan et al. [31] introduced the concept of interval valued -fuzzy biideals of ordered semigroups by using the idea of a quasicoincidence of an interval valued fuzzy point with an interval valued fuzzy set. In particular, they studied the properties of interval valued -fuzzy biideals in detail. Furthermore, Davvaz et al. [32] introduced the concept of interval valued generalized fuzzy filters of an ordered semigroup and obtained many interesting related results.
As we know, quasi-ideals play an important role in the study of ring, semigroup, and ordered semigroup structures. The concept of a quasi-ideal in rings and semigroups was studied by Stienfeld in [33]. Furthermore, Kehayopulu and Tsingelis extended the concept of quasi-ideals in ordered semigroups as a nonempty subset of an ordered semigroup such that and if and , then (see [34]). The fuzzy quasi-ideals in ordered semigroups were studied in [23, 27], where the basic properties of ordered semigroups in terms of fuzzy quasi-ideals are given. Motivated by the study of fuzzy quasi-ideals in rings, semigroups and ordered semigroups, and also motivated by Davvaz and Khan’s works in ordered semigroups in terms of interval valued fuzzy subsets, we attempt in the present paper to study a new type of interval valued fuzzy quasi-ideals in an ordered semigroup in detail. The rest of this paper is organized as follows. In Section 2, we recall some basic definitions and results of ordered semigroups which will be used throughout this paper. In Section 3, we introduce the concept of interval valued -fuzzy quasi-ideals of an ordered semigroup by the interval valued ordered fuzzy points of and investigate some related properties. In particular, we discuss the relationships interval valued fuzzy quasi-ideals and interval valued -fuzzy quasi-ideals of an ordered semigroup. The idea of prime interval valued -fuzzy quasi-ideals in ordered semigroups is given and the related theorems are provided in Section 4. In Section 5, we first give some characterizations of regular and intra-regular ordered semigroups by the properties of interval valued -fuzzy quasi-ideals. Furthermore, we introduce the notion of completely semiprime interval valued -fuzzy quasi-ideals of ordered semigroups and characterize biregular ordered semigroups in terms of completely semiprime interval valued -fuzzy quasi-ideals. Some conclusions are given in the last Section.
2. Preliminaries and Some Notations
Recall that an ordered semigroup is a semigroup with an order relation “” such that implies and for any .
Let be an ordered semigroup. For , we define For , we write instead of . For two subsets , of , we have ; if , then ; ; ; (see [35]).
By a subsemigroup of we mean a nonempty subset of such that . A quasi-ideal of an ordered semigroup is called prime if for any two elements , of , such that , then or . We denote by the quasi-ideal of generated by . Then .
An ordered semigroup is called regular if, for each , there exists such that . Equivalent definitions: , . , (see [34]). An ordered semigroup is called intra-regular if, for each , there exist such that . Equivalent definitions: , . , (see [34]).
We next state some fuzzy logic concepts. Recall that a fuzzy subset of an ordered semigroup is a function from into the real closed interval , that is, . A fuzzy subset of an ordered semigroup is called a fuzzy quasi-ideal of if for all , and (see [23]).
By an interval number we mean an interval , where . The set of all interval numbers is denoted by . The interval number can be simply identified by the number . In particular, we define the set , where . For the interval numbers , , we define the operations “”, “”, “”, “”, “”, “+” and “−” in as follows.(1). In particular, .(2). In particular, .(3), , where .(4) and .(5) and .(6) and .(7).(8), where .
Then, it is clear that forms a complete lattice with as its least element and as its greatest element.
Let be an ordered semigroup. A mapping is called an interval valued fuzzy subset of , where for all , and are two ordinary fuzzy sets of such that for all . The set of all interval valued fuzzy sets of is denoted by . Let . Then we define(1) and for all ;(2) and for all ;(3) for all ;(4) for all .
One can easily show that forms a complete lattice with the maximum element and the minimum element , where the interval valued fuzzy subsets and of are defined by
Let be an ordered semigroup. For , we define . For any , the product of and is defined by for all .
Lemma 1. Let be an ordered semigroup and . Then the following statements are true:(1).(2)If and , then , .
Proof. The proof is similar to that of Theorem in [36].
By Lemma 1, we can easily see that this operation “” on the set is associative and forms an ordered semigroup.
Definition 2. An interval valued fuzzy subset of an ordered semigroup is called comparable if and are comparable for all ; that is, or .
Throughout this paper, let be a comparable interval valued fuzzy subset of unless otherwise specified.
Definition 3. Let be an interval valued fuzzy subset of an ordered semigroup and . Then the crisp set is called a level subset of .
Definition 4. Let be an ordered semigroup, and . An interval valued ordered fuzzy point of is defined by the rule that for any . It is accepted that is a mapping from into , then an interval valued ordered fuzzy point of is an interval valued fuzzy subset of . For any interval valued fuzzy subset of , we also denote by in the sequel.
Definition 5. An interval valued ordered fuzzy point of an ordered semigroup is said to be not belonging to (resp., not -quasi-coincident with) an interval valued fuzzy subset of , written as (resp., ), if (resp., ), where . If or , then we write . The symbol means that does not hold.
Let be a nonempty subset of an ordered semigroup . We denote by the interval valued characteristic function of , which is the mapping of into defined by Clearly, is an interval valued fuzzy subset of .
The reader is referred to [28, 37, 38] for notation and terminology not defined in this paper.
3. Interval Valued -Fuzzy Quasi-Ideals of Ordered Semigroups
Throughout this paper, let be an ordered semigroup and let denote an arbitrary element of unless otherwise specified. In this section, we will introduce and study interval valued -fuzzy quasi-ideals of ordered semigroups, which is a new generalized form of interval valued fuzzy quasi-ideals.
In what follows, we emphasize that any element in must satisfy the following condition: or ,where . For any , we say that satisfies the condition if satisfies the condition for any .
In the following, we first extend the concept of fuzzy quasi-ideals to the concept of interval valued fuzzy quasi-ideals in an ordered semigroup as follows.
Definition 6. Let be an ordered semigroup and an interval valued fuzzy subset of . Then is called an interval valued fuzzy quasi-ideal of if the following conditions are satisfied:(1) for all .(2).
Theorem 7. Let be an ordered semigroup and an interval valued fuzzy subset of . Then is an interval valued fuzzy quasi-ideal of if and only if all nonempty level subsets of are quasi-ideals of for all .
Proof. Suppose that is an interval valued fuzzy quasi-ideal of and such that . Let be such that . Then and , and we have and for some and . Then and . By hypothesis, we have Since , we have and . Then , and so . Thus . Furthermore, let , . Then . Indeed, since , , and is an interval valued fuzzy quasi-ideal of , we have , so . Therefore, is a quasi-ideal of .
Conversely, assume that for every such that the set is a quasi-ideal of . Let . Then . In fact, if , then there exists such that , and so and . Then and . Since is comparable, and there exist with and such that and , then and so and . Hence and . By hypothesis, , and so . Then . This is a contradiction. Thus, . Moreover, let . If , then . Indeed, let . Then . Since is a quasi-ideal of , we have . Then . Therefore, is an interval valued fuzzy quasi-ideal of .
Example 8. We consider a set with the following multiplication “” and the order “”: Then is an ordered semigroup and , , , , , , , , and are all quasi-ideals of (see [39]). Let be an interval valued fuzzy subset of such that , , , , and . Then By Theorem 7, we can verify that is an interval valued fuzzy quasi-ideal of .
Definition 9. An interval valued fuzzy subset of an ordered semigroup is called an interval valued -fuzzy quasi-ideal of , if, for all and , the following conditions hold:(1) and .(2), , and or .
Theorem 10. Let be an ordered semigroup and an interval valued fuzzy subset of . Then is an interval valued -fuzzy quasi-ideal of if and only if satisfies the following conditions:(1) for all .(2) for all .
Proof. Suppose that is an interval valued -fuzzy quasi-ideal of . Let such that . If , then , , but does not hold. Thus, by Definition 9, we have . It follows that , which is a contradiction with . Hence ≥ for all with . Furthermore, for all . Indeed, if for some , then there exists such that . Since , , there exist such that and , and , . Then , but and do not hold. Thus, by Definition 9, we have or . Then or , which is impossible. Therefore, for all .
Conversely, assume that the conditions (1) and (2) hold. Let and be such that . If , then . Since satisfies the condition , we can consider the following two cases.
Case 1. If , then, by the condition (1), , which means that . Thus .
Case 2. If , then, by hypothesis, we have . If , then . Now let . Then, , which implies that , and thus .
Thus, in both cases, we have . Furthermore, let and be such that , , and . Then , and
Since satisfies the condition , we can consider the following two cases.
Case 1. If , then . Also, since is comparable, we have or . It thus follows that or , which implies that or .
Case 2. If , then, by hypothesis, we have . If or , then or . Now let and . Then, since is comparable, we have or . It follows that or , and thus or .
Thus, in both cases, we have or . Therefore, is an interval valued -fuzzy quasi-ideal of .
From Theorem 10, we can see that the interval valued -fuzzy quasi-ideal is characterized in terms of the multiplications and . A natural question is if an interval valued -fuzzy quasi-ideal can be defined using only the interval valued fuzzy subset itself. The theorems below give the answer.
Theorem 11. Let be an ordered semigroup. Then an interval valued fuzzy subset of is an interval valued -fuzzy quasi-ideal of if and only if the following conditions are satisfied:(1) for all .(2) and for all .
Proof. : Let be such that and . Since is an interval valued -fuzzy quasi-ideal of and , by Theorem 10, we have
Since , we have ; then
By in a similar way we can get . It thus follows that .
: Assume that the conditions (1) and (2) hold. Let . Then . Indeed, if , then , and
Let . Then
Since is comparable and satisfies the condition , we can consider the following two cases.
Case 1. If , then .
Case 2. Let . Then, by (14), there exists such that . Since , we have
So we can show that for any . In fact, since , we have and . Since , we have and . Since there exist such that and , by hypothesis, we have
Since is comparable, we have or . If , then , which is impossible by (15). Hence we have
for any . Then we have
and the proof is completed by Theorem 10.
Theorem 12. Let be an ordered semigroup. Then an interval valued fuzzy subset of is an interval valued -fuzzy quasi-ideal of if and only if the following conditions are satisfied:(1).(2) and ≥ for all .
Proof. The proof is straightforward verification by Theorem 11 and we omit it.
By Theorem 10, we can easily observe that every interval valued fuzzy quasi-ideal of an ordered semigroup is an interval valued -fuzzy quasi-ideal of . However, the converse is not true, in general, as shown in the following example.
Example 13. Consider the ordered semigroup given in Example 8 and define an interval valued fuzzy subset of by , , , and . Then By Theorem 11, we can easily show that is an interval valued -fuzzy quasi-ideal of for any . But is not an interval valued fuzzy quasi-ideal of . In fact, since is not a quasi-ideal of , we deduce that is not an interval valued fuzzy quasi-ideal of by Theorem 7.
In the following theorem, we give a condition for an interval valued -fuzzy quasi-ideal of an ordered semigroup to be an interval valued fuzzy quasi-ideal of .
Theorem 14. Let be an ordered semigroup and an interval valued -fuzzy quasi-ideal of . If for all , then is an interval valued fuzzy quasi-ideal of .
Proof. It is obvious by Theorem 10.
Theorem 15. Let be an ordered semigroup. If and , then every interval valued -fuzzy quasi-ideal of is an interval valued -fuzzy quasi-ideal of .
Proof. The proof is straightforward and we omit it.
The following example shows that the converse of Theorem 15 does not hold, in general.
Example 16. Consider the ordered semigroup given in Example 8 and define an interval valued fuzzy subset of by , , , and . Then is an interval valued -fuzzy quasi-ideal of for any . But is not an interval valued -fuzzy quasi-ideal of , where . In fact, since and , where . Thus, by Theorem 11, is not an interval valued -fuzzy quasi-ideal of .
Now we will characterize the interval valued -fuzzy quasi-ideals by using their level subsets.
Theorem 17. Let be an ordered semigroup and an interval valued fuzzy subset of . Then is an interval valued -fuzzy quasi-ideal of if and only if the level subset of is a quasi-ideal of for all with .
Proof. Suppose that is an interval valued -fuzzy quasi-ideal of . Let , be such that for some . Then . It follows from Theorem 11(1) that
Note that , and we conclude that , which implies that . Furthermore, let for some . Then and , and there exist and such that and . Then and . It follows from Theorem 11(2) that
It implies that , according to . Therefore, is a quasi-ideal of for all with .
Conversely, assume that is a quasi-ideal of for all with . If there exist with such that , then , , and . Since is a quasi-ideal of for any and , we have and , which contradicts with . Hence . Furthermore, let such that and . If possible, for some such that and , then there exists such that . Then . By hypothesis, we have , and so , which is impossible. Thus . Therefore, is an interval valued -fuzzy quasi-ideal of by Theorem 11.
Theorem 18. Let be a nonempty subset of an ordered semigroup and an interval valued fuzzy subset of defined by for any , where . Then is a quasi-ideal of if and only if is an interval valued -fuzzy quasi-ideal of .
Proof. Suppose that is a quasi-ideal of and let be the level subset of . Then it is easy to see that for all . It thus follows from Theorem 17 that is an interval valued -fuzzy quasi-ideal of .
Conversely, assume that is an interval valued -fuzzy quasi-ideal of . Let , and . Then . By Theorem 11, ≥ . Thus, by the definition of , we have , that is, . Furthermore, let . Then and , and there exist and such that and . Then . It thus follows from Theorem 11 that , which implies that , that is, . Therefore, is a quasi-ideal of .
Lemma 19. Let be an ordered semigroup and . Then is a quasi-ideal of if and only if the interval valued characteristic function of is an interval valued -fuzzy quasi-ideal of .
Proof. The proof is straightforward by Theorem 18.
Theorem 20. Let be a family of interval valued -fuzzy quasi-ideals of an ordered semigroup . Then is an interval valued -fuzzy quasi-ideal of , where .
Proof. Let such that and . Then, since each is an interval valued -fuzzy quasi-ideal of , we have Furthermore, if , then . Indeed, since each is an interval valued -fuzzy quasi-ideal of , we have for all . Thus Therefore, is an interval valued -fuzzy quasi-ideal of by Theorem 11.
Suppose that is a family of interval valued -fuzzy quasi-ideals of an ordered semigroup . Is it true that is an interval valued -fuzzy quasi-ideal of , where . The following example gives a negative answer to the above question.
Example 21. We consider the ordered semigroup defined by the following multiplication “” and the order “”: Let and be two interval valued fuzzy subsets of such that Then, by Theorem 11, and are both interval valued -fuzzy quasi-ideals of for any . But is not an interval valued -fuzzy quasi-ideal of for any . In fact, since and , By Theorem 11, is not an interval valued -fuzzy quasi-ideal of .
The following theorem can be obtained under the assumption of an additional condition.
Theorem 22. Let be a family of interval valued -fuzzy quasi-ideals of an ordered semigroup . Then is an interval valued -fuzzy quasi-ideal of , where .
Proof. Let such that and . Then we have
In the following we show that holds. It is obvious that . Assume that . Then there exists such that < . Since or for all , there exists such that . On the other hand, for all , which is a contradiction. Thus .
Furthermore, if , then . Indeed, since is an interval valued -fuzzy quasi-ideal of for all , we have for any . Thus
Therefore, is an interval valued -fuzzy quasi-ideal of by Theorem 11.
4. Prime Interval Valued -Fuzzy Quasi-Ideals of Ordered Semigroups
In this section, we study mainly the prime interval valued -fuzzy quasi-ideals of ordered semigroups and discuss their related properties.
Definition 23. An interval valued -fuzzy quasi-ideal of an ordered semigroup is called prime if for all and ,
Example 24. Consider the ordered semigroup given in Example 8 and define an interval valued fuzzy subset of by , , and . By routine calculations, is a prime interval valued -fuzzy quasi-ideal of for any .
Theorem 25. Let be an ordered semigroup and an interval valued -fuzzy quasi-ideal of . Then is prime if and only if satisfies for all .
Proof. Let be a prime interval valued -fuzzy quasi-ideal of and . Then . Indeed, if < for some , then there exists such that . Then , but , + , so , which is a contradiction. Hence for all .
Conversely, assume that for all . Let be such that . Then and . Now we consider the following two cases.
Case 1. If , then, since is comparable, by hypothesis, we have , that is, . Thus, .
Case 2. Let . Then we have ≤ ≤ = . Hence ≤ + , that is, . Therefore, .
This proves that is prime.
Remark 26. The hypothesis that the interval valued fuzzy subset is comparable cannot be removed in the above theorem. Otherwise, the sufficiency of Theorem 25 is not true. We can illustrate it by the following example.
Example 27. We consider the ordered semigroup defined by the following multiplication “” and the order “”: Let be an interval valued fuzzy subset of such that , , and . By routine calculations, is an interval valued -fuzzy quasi-ideal of , and satisfies for all . But is not prime, since and , while
Now we will characterize the prime interval valued -fuzzy quasi-ideals of ordered semigroups by using their level subsets.
Theorem 28. Let be an ordered semigroup and an interval valued fuzzy subset of . Then is a prime interval valued -fuzzy quasi-ideal of if and only if the level subset of is a prime quasi-ideal of for all with .
Proof. Let be a prime interval valued -fuzzy quasi-ideal of and . Then, by Theorem 17, is a quasi-ideal of for all . To prove that is prime, let . By Theorem 25, we have
Note that and satisfies the condition . Then . Since is comparable, we have or . Thus, or . This shows that is a prime quasi-ideal of for all with .
Conversely, assume that for every such that the set is a prime quasi-ideal of . Then, by Theorem 17, is an interval valued -fuzzy quasi-ideal of . Let and . Then . Since is prime, we have ; that is, . Thus, . Therefore, is a prime interval valued -fuzzy quasi-ideal of .
5. Characterizations of Some Types of Ordered Semigroups
Throughout the remaining paper, let denote the set of all positive integers. In this section we first study the properties of -upper parts of interval valued -fuzzy quasi-ideals of ordered semigroups. Furthermore, we give characterizations of regular, intra-regular, or bi-regular ordered semigroups in terms of interval valued -fuzzy quasi-ideals.
Definition 29. Let be an interval valued fuzzy subset of an ordered semigroup . Then we define the -upper part of as follows: for all .
Clearly, is still an interval valued fuzzy subset of . Furthermore, for any , the interval valued fuzzy subsets and of are defined as follows: for all .
Lemma 30. Let be an ordered semigroup and . Then the following statements are true:(1).(2)If and , then .(3).(4), , and . Furthermore, if , then , , and .
Proof. The proof is straightforward and we omit it.
Lemma 31. Let , be any nonempty subsets of an ordered semigroup . Then , where , , and are the interval valued characteristic function of , , and , respectively.
Proof. Let . If , then and for some and . Thus , and On the other hand, since for all , we have = . Therefore, . If , then . We now prove that . Indeed, if , then = = , and . If , then for all . If and , then , so , which is impossible. Thus or . If , then . Since , we have . If , then, as in the previous case, we have . Thus we have This completes the proof.
In Example 13, we have shown that an interval valued -fuzzy quasi-ideal of an ordered semigroup is not necessarily an interval valued fuzzy quasi-ideal of . In the following theorem, we show that if is an interval valued -fuzzy quasi-ideal of , then the -upper part of is an interval valued fuzzy quasi-ideal of .
Theorem 32. Let be an ordered semigroup and an interval valued -fuzzy quasi-ideal of . Then the -upper part of is an interval valued fuzzy quasi-ideal of .
Proof. Let be an interval valued -fuzzy quasi-ideal of and . Then, by Theorem 10, , and we have It implies that . Moreover, if , then . Indeed, since is an interval valued -fuzzy quasi-ideal of , we have , and so = = ≥ . Therefore, is an interval valued fuzzy quasi-ideal of .
Theorem 33. Let be an ordered semigroup and an interval valued fuzzy subset of . Then is an interval valued -fuzzy quasi-ideal of if and only if satisfies the following assertions:(1) for all .(2).
Proof. It is obvious by Lemma 30 and Theorem 10.
Lemma 34 (cf. [37]). An ordered semigroup is regular and intra-regular if and only if for every quasi-ideal of .
Now we give characterizations of an ordered semigroup which is both regular and intra-regular by interval valued -fuzzy quasi-ideals.
Theorem 35. Let be an ordered semigroup. Then the following conditions are equivalent:(1) is regular and intra-regular.(2) for every interval valued -fuzzy quasi-ideal of .(3) for any interval valued -fuzzy quasi-ideals and of .
Proof. (1)(3). Let and be interval valued -fuzzy quasi-ideals of and . Then, since is both regular and intra-regular, there exists such that , and there exist such that . Thus
that is, . Since is an interval valued -fuzzy quasi-ideal of and , , by Theorem 11, we have
By , , in a similar way we can get . Thus we have
which means that . In the same way, we can show that . Hence, .
(3)(1). Take in (3) and we get . On the other hand, by Lemma 30(2) and Theorem 33, we have . Thus we deduce that .
(2)(1). Let be a quasi-ideal of . By Lemma 19, the interval valued characteristic function of is an interval valued -fuzzy quasi-ideal of . Then, by hypothesis and Lemma 31, we have
from which we deduce that . By Lemma 34, is regular and intra-regular.
An ordered semigroup is called left (resp., right) regular if for each there exists such that (resp., , that is, (resp., ) (see [24]). An ordered semigroup is called bi-regular if it is both left regular and right regular. Clearly, an ordered semigroup is bi-regular if and only if for every .
Definition 36. Let be an ordered semigroup and an interval valued -fuzzy quasi-ideal of . Then is called completely semiprime if for any .
Now we will give some characterizations of bi-regular ordered semigroups in terms of interval valued -fuzzy quasi-ideals.
Theorem 37. Let be an ordered semigroup. Then the following statements are equivalent:(1) is bi-regular.(2)For each interval valued -fuzzy quasi-ideal of , for any .(3)For each interval valued -fuzzy quasi-ideal of , for any , .(4)Every interval valued -fuzzy quasi-ideal of is completely semiprime.
Proof. (1)(2). Let be a bi-regular ordered semigroup and an interval valued -fuzzy quasi-ideal of and let . Since is bi-regular, we have and . Then there exist such that and . Then, by Theorem 11, we have
On the other hand, since and , similar to the above proof, we can show that . Therefore, for any .
(2)(3). Let and . Then, by Lemma 30(4), we have
Similarly, it can be shown that . Thus, by hypothesis and Theorem 33, we have
On the other hand, let . Then we have
for any . Similarly, . Since is an interval valued -fuzzy quasi-ideal of , by hypothesis, we have
(3)(2) and are obvious.
(4)(1). Let . We consider the quasi-ideal of generated by . By Lemma 19, the interval valued characteristic function of is an interval valued -fuzzy quasi-ideal of . By hypothesis, . By Definition 29, for all . Thus , and we have
Then for some . If , then , that is, . If , then . Therefore, is bi-regular.
6. Conclusion
It is well known that ordered semigroups are basic structures in many applied branches like automata and formal languages, coding theory, finite state machines, and others. Due to these possibilities of applications, semigroups and related structures are presently, extensively investigated in fuzzy settings. In particular, generalized fuzzy ordered semigroups play an important role in the mentioned applications. Interval valued fuzzy quasi-ideals of an ordered semigroup with special properties always play an important role in the study of ordered semigroups structure. The interval valued ordered fuzzy points of an ordered semigroup are key to describe the algebraic subsystems of ordered semigroups. In this paper we have introduced the concept of interval valued -fuzzy quasi-ideals and prime (completely semiprime) interval valued -fuzzy quasi-ideals of an ordered semigroup and investigated their related properties. Furthermore, we have given some characterizations of regular, intra-regular, or bi-regular ordered semigroups in terms of interval valued -fuzzy quasi-ideals. The results obtained in this paper are fundamental for the advanced study on ordered semigroups and (generalized) fuzzy ordered semigroups, which we will present in the next paper. As an application of the results of this paper, the corresponding results of semigroups (without order) are also obtained.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Natural Science Foundation of China (no. 11371177, 11271040, and 11361027) and the Anhui Provincial Excellent Youth Talent Foundation (no. 2012SQRL115ZD).