Abstract
The notions of an -hesitant fuzzy -ideal and a -hesitant fuzzy -ideal, which are a generalization of an interval-valued fuzzy -ideal, of a -semigroup are introduced and some properties are investigated. Characterizations of the notions are provided in terms of sets, fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets. Furthermore, characterizations of a -ideal of a -semigroup are given in terms of -hesitant and -hesitant fuzzy -ideals.
1. Introduction
The notion of a fuzzy set, proposed by Zadeh [1], has provided a useful mathematical tool and method for describing the behavior of complex and ill-defined systems. The notion has huge applications in decision making, artificial intelligence, automata theory, control engineering, finite state machine, expert, graph theory, robotics, and many branches of pure and applied mathematics (cf. [2]). Nevertheless, there are limitations for using the notion to deal with vague and imprecise information when different sources of vagueness appear simultaneously. In order to overcome such limitations, Torra and Narukawa [3, 4] proposed an extension of the notion so-called a hesitant fuzzy set which is a function from a reference set to a power set of the unit interval. Hesitant fuzzy set theory has been applied to several practical problems, primarily in the area of decision making (see [5–9]) and different algebraic structures; for example, Jun and Ahn [10] introduced hesitant fuzzy subalgebras and hesitant fuzzy ideals of BCK/BCI-algebras and investigated related properties. Mosrijai et al. [11–13] studied hesitant fuzzy sets on UP-algebras. Kim et al. [14] studied the concepts and properties of a hesitant fuzzy subgroupoid (left ideal, right ideal, and ideal) of a groupoid, a hesitant fuzzy subgroup (normal subgroup and quotient subgroup) of a group, and a hesitant fuzzy subring (left ideal, right ideal, and ideal) of a ring. Jittburus and Julatha [15] proposed the concepts of a -hesitant fuzzy ideal of a semigroup and its -hesitant fuzzy translations and -hesitant fuzzy extensions. They showed that the -hesitant fuzzy ideal is a general concept of a hesitant fuzzy ideal and an interval-valued fuzzy ideal and gave its characterizations in terms of sets, fuzzy sets, hesitant fuzzy sets, and interval-valued fuzzy sets. Julatha and Iampan [16] introduced -types of hesitant fuzzy sets based on ideal theory of ternary semigroups and examined their properties via a fuzzy set, an interval-valued fuzzy set, and a hesitant fuzzy set.
In 1981, Sen [17] introduced the concept and notion of the -semigroup as a generalization of the plain semigroup and ternary semigroup. Many classical notions and results of (ternary) semigroups have been extended and generalized to -semigroups, by many mathematicians, for instance, Siripitukdet and Iampan [18, 19], Siripitukdet and Julatha [20], Dutta and Adhikari [21, 22], Saha and Sen [23–25], Hila [26, 27], and Chinram [28, 29]. Simuen, Iampan, Chinram, Sardar, Majumder, Dutta, and Davvaz [30–35] studied theory of -semigroups via fuzzy subsets. Uckun et al. [36] studied theory of -semigroup via intuitionistic fuzzy subsets. Abbasi et al. [37] introduced hesitant fuzzy left (resp., right, bi-, interior, and two-sided) -ideals of -semigroups and characterized simple -semigroups by hesitant fuzzy sets. Julatha and Iampan [38] introduced a -hesitant fuzzy -ideal, which is a general concept of an interval-valued fuzzy -ideal and a hesitant fuzzy -ideal, of a -semigroup and studied its properties via level sets, fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets.
In this paper, the notions of an -hesitant fuzzy -ideal and a -hesitant fuzzy -ideal, which are a general notion of an interval-valued fuzzy -ideal, of a -semigroup are introduced and their properties are investigated. Equivalent conditions for a hesitant fuzzy set to be an -hesitant fuzzy -ideal and a -hesitant fuzzy -ideal are provided in terms of sets, fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets. We show that every interval-valued fuzzy set on a -semigroup is an interval-valued fuzzy -ideal if and only if it is a -hesitant fuzzy -ideal. Furthermore, characterizations of a -ideal of a -semigroup are given in terms of -hesitant and -hesitant fuzzy -ideals.
2. Preliminaries
We will introduce some definitions and results that are important for study in this paper.
First, we recall the definition of -semigroups which is defined by Sen and Saha [25]. By a -semigroup, we mean a nonempty set with a nonempty set and a mapping , written as satisfying the identity for all and . From now on throughout this paper, is represented as a -semigroup and a nonempty set unless otherwise specified. For nonempty subsets and of , let . By a -ideal of , we mean a nonempty subset of such that and . Then, a nonempty subset of is a Id of if and only if for all , , and .
A fuzzy subset (FS) [1] of is a function from into the unit segment of the real line . A FS of is called a fuzzy-ideal of if
An intuitionistic fuzzy set (IFS) [39] in is an object having the form , where the functions and define the degree of membership and the degree of nonmembership, respectively, and for all . An IFS in can be identified to an ordered pair in . For a FS of , we define a FS by for all . Then, is an IFS in for all FSs and of . An IFS in is called an intuitionistic fuzzy-ideal [36] of if the following two conditions hold:(i) for all and (ii) for all and
By an interval number , we mean an interval , where . The set of all interval numbers is denoted by . Especially, we denoted and . For two elements and in , define the operations and in case of two elements in as follows:(i) and (ii) and (iii) and (iv)
Denote the case that or by . A function is called an interval-valued fuzzy set (IvFS) [40] on , where for all and and are FSs of such that for all . Let IvFS(X) be the set of all IvFSs on . An IvFS on is called an interval-valued fuzzy-ideal of if
Then, is an of if and only if and for all and .
A hesitant fuzzy set (HFS) [3, 4] on in terms of a function is that when applied to returns a subset of , that is, , where denotes the set of all subsets of . Let be the set of all HFSs on , that is, and let . Then, . A HFS on is called a hesitant fuzzy-ideal [37] of if
Then, is a of if and only if for all and .
For and , we define the element of [0, 1], the subset of , the fuzzy subset of , and the hesitant fuzzy set on [15, 16] by(1)(2)(3) for all (4) for all
We denote by , and then, .
Julatha and Iampan [38] introduced a -hesitant fuzzy -ideal, which is a generalization of the concepts of an and a , of a -semigroup and studied its properties in terms of FSs, IFSs, HFSs, and IvFSs in the following.
Definition 1 (see [38]). Given , a HFS on is called a -hesitant fuzzy-ideal ofrelated to (briefly, --hesitant fuzzy -ideal of ) if the set is a Id of . We say that is a -hesitant fuzzy-ideal (-) of if is a --hesitant fuzzy -ideal of for all when .
Lemma 1 (see [38]). Every of is a - of .
Lemma 2 (see [38]). Every of is a - of .
Theorem 1 (see [38]). For , the following are equivalent:(1) is a - of (2) is a of (3) is a of (4) is an of (5) is a - of (6) is a of for all
Given , the HFS , defined by for all , is called the supremum complement [13, 38] of on . Then, for all and is an IFS in .
Theorem 2 (see [38]). A HFS on is a - of if and only if is an of .
For every HFS on and every element of , the setis called a -upper-level subset [13, 38] of .
Theorem 3 (see [38]). A HFS on is a - of if and only if is either empty or a of for all .
For a subset of and two elements of , define the characteristic interval-valued fuzzy set (CIvFS) , the characteristic hesitant fuzzy set (CHFS) , and the hesitant fuzzy set by for all ,
Then, the HFS is a general concept of the CHFS and CIvFS, that is, and . Julatha and Iampan [38] gave conditions for a nonempty subset of to be a by using the CIvFS , the CHFS , and as the following theorem.
Theorem 4 (see [38]). For a nonempty subset of , the following are equivalent:(1) is a of (2)The CIvFS is a - of (3)The CHFS is a - of (4) is a - of for all with
3. inf-Hesitant Fuzzy -Ideals
For a HFS on and an element , define and by
Note that for all and for all , we have . Now, we introduce the notion of an -hesitant fuzzy -ideal of a -semigroup in the following definition.
Definition 2. A HFS on is called an -hesitant fuzzy -ideal (-) of if the set is a Id of for all when .
Example 1. Let be the set of all negative integers, , and . Then, is a -semigroup with respect to usual multiplication.(1)Define a HFS on by for all . Then, is an - of but not a - of because(2)Define a HFS on byfor all . Then, is a of but not an - of because the nonempty subset of is not a of , that is,
By Example 1 and Lemma 2, we obtain that an - of is not a - and a of and a - of is not an - of .
Lemma 3. Every of is an - of .
Proof. Suppose that is an of and such that is a nonempty set. Let , , and . Since is an of , we get and . Thus,which implies that . Hence, is a of . Therefore, is an - of .
In the following example, it is shown that the converse of Lemma 3 is not generally true.
Example 2. Let be a -semigroup defined in Example 1. Define an IvFS on by for all ,Then, is an - of but not an of because
By Lemma 3 and Example 2, we obtain that an - of a -semigroup is a general concept of an of .
For every HFS on , define the FS of by for all . A HFS on is called an infimum complement of on if for all . Let be the set of all infimum complements of . Define the HFS by for all , and then we have , , and for all . Note that for all .
Lemma 4. If is a of , then is an - of for all .
Proof. Suppose that is a of and . Let , , , and . Then, and , and since , we getHence, we have . Therefore, we obtain that is a Id of . Consequently, is an - of .
Lemma 5. For , the following are equivalent:(1) is an - of (2) is a of (3) for all and (4) for all , , and (5) for all and
Proof. . Let and . Then, and . By assumption (1), we get and . Thus, .
and . They are clear.
. Let , , , and . By assumption (2), we haveThen, . Thus, is a Id of . Therefore, we have that is an - of .
. Let , , and . By assumption (3), we haveand then,. Let and . By using assumption (5), we get and then
Theorem 5. If an IvFS of is an of , then is a of .
Proof. It follows from Lemmas 3 and 5.
Theorem 6. If is a of , then is a of for all .
Proof. It follows from Lemmas 4 and 5.
For and , the setsare called an -upper-level subset and an -lower-level subset of , respectively.
Theorem 7. A HFS of is an - of if and only if for all , a nonempty subset of is a of .
Proof. Let and . Choose such that , and we get . Since is an - of , we get that is a of .
Conversely, let and . Choose , and by the assumption, we obtain that is a of . Therefore, is an - of .
Corollary 1. Let be an of . Then, for all , a nonempty subset of is a of .
Proof. It follows from Lemma 3 and Theorem 7.
Theorem 8. Let and . Then, is an - of if and only if for all , a nonempty subset of is a of .
Proof. Let and . There exists such that and then . Since is an - of , we obtain that is an of .
Conversely, let be such that . Choose , and by the assumption, we obtain that is a of . Hence, is an - of .
Corollary 2. If is a of, then for all , a nonempty subset of is a of .
Proof. It follows from Lemma 4 and Theorem 8.
In the following theorem, we give conditions for a HFS of a -semigroup to be an inf-HFΓId via IFSs.
Theorem 9. For , the following are equivalent:(1) is an - of (2) is an of for all (3) is an of
Proof. It follows from Lemma 5.
Corollary 3. If an IvFS of is an of , then is an of for all .
Proof. It follows from Lemma 3 and Theorem 9.
Corollary 4. If is a of , then is an of for all .
Proof. It follows from Lemma 4 and Theorem 9.
For and , we define the HFS on byand we denote by . Then, the following statements hold:(1) for all (2)(3) for all
In the following theorem, we give conditions for a HFS of a -semigroup to be an - in terms of IvFSs and HFSs.
Theorem 10. For , the following are equivalent:(1) is an - of (2) is a of for all (3) is a of (4) is an of
Proof. . Let , , , and . Then, and . By assumption (1) and Lemma 5, we getThus . Hence, . Therefore, we have that is a of .
. It is clear.
. Let and . Then, and . By assumption (3), we get . Thus,Since for all , we have and soTherefore, is an of .
. Let and . By assumption (4), we get . Then,Therefore, it follows from Lemma 5 that is an - of .
Corollary 5. Let be an of . Then, the following hold:(1) is a of for all (2) is both a and an of
Proof. It follows from Lemma 3 and Theorem 10.
Corollary 6. Let be a of . Then, the following hold:(1) is a of for all and (2) is both a and an of for all
Proof. It follows from Lemma 4 and Theorem 10.
In the following theorem, we give one characterization of a of a -semigroup in terms of a HFS.
Theorem 11. Let be a nonempty subset of and with . Then, is a of if and only if is an - of .
Proof. Assume that is a of . Suppose that is not an - of . By Lemma 5, there exist and such thatThus, , which implies that or . Since is a of , we get and sowhich is a contradiction. Therefore, is an - of .
Conversely, let , and . Then, and so . Since is an - of and Lemma 5, we get and . Thus, . Therefore, is a of .
Theorem 12. A nonempty subset of is a of if and only if the CIvFS is an - of .
Proof. It follows from Theorem 11.
Remark 1. If is a subset of , then the CHFS is an - of .
Definition 3. A HFS on is called a -hesitant fuzzy-ideal (-) of if is both an - and a - of .
In Theorem 13, equivalent conditions for a hesitant fuzzy set to be a - are given in terms of level sets, FSs, IFSs, IvFSs, and HFSs.
Theorem 13. For , the following are equivalent:(1) is a - of (2) and are of (3) and are of (4) and are of (5) is an of (6) is an of (7) is an of for all (8) and are of for all (9)For all , nonempty subsets and of are of (10) and for all and
Proof. It follows from Theorems 1, 2, 3, 7, and 10 and Lemma 5.
Example 3. Let be the set of all negative integers and . Then, forms a -semigroup with the usual multiplication. Define a HFS on by for all . Then,for all and . Hence, it follows from Theorem 13 that is a - of . Since is not an IvFS of , we get that it is not an of .
Lemma 6. Every of is a - of .
Proof. It follows from Lemmas 1 and 3.
By Example 3 and Lemma 6, we see that a - of is a general concept of an of .
Lemma 7. Let be an IvFS of . Then, is an of if and only if is a- of .
Proof. It follows from Lemma 6.
Conversely, assume that is a - of . By Theorem 13, we get and for all and . Thus,for all and . Therefore, is an of .
In Theorem 14, equivalent conditions for an IvFS to be an are given in terms of level sets, FSs, IFSs, IvFSs, and HFSs.
Theorem 14. For , the following are equivalent:(1) is an of (2) is a - of (3) and are of (4) and are of (5) and are of (6) is an of (7) is an of (8) is an of for all (9) and are of for all (10)For all , nonempty subsets and of are of
Proof. It follows from Lemma 7 and Theorem 13.
Theorem 15. For a nonempty subset of , the following are equivalent:(1) is a of (2)The CIvFS is a - of (3)The CHFS is a - of (4) is a - of for all with and
Proof. It follows from Theorems 4, 11, and 12 and Remark 1.
4. Conclusions
In this paper, we have introduced the notions of an - and a -, which are a generalization of an , of a -semigroup and examined their characterizations in terms of sets, FSs, IFSs, IvFSs, and HFSs. Furthermore, we have discussed the relation between a and a generalization of the CHFS and CIvFS. From the study results, we found that the following conditions are all equivalent in a -semigroup: a nonempty subset is a , is an -, is a -, and is a -.
In the future, we will study an - and a - in LA-semigroups and -algebras and examine their characterizations in terms of sets, FSs, IFSs, IvFSs, and HFSs.
Data Availability
No data were used to support this research.
Conflicts of Interest
The authors declare that they have no conflicts of interest.