Abstract
As a link between classical soft sets and hesitant fuzzy sets, the notion of hesitant fuzzy soft sets is introduced and applied to a decision making problem in the papers by Babitha and John (2013) and Wang et al. (2014). The aim of this paper is to apply hesitant fuzzy soft set for dealing with several kinds of theories in BCK/BCI-algebras. The notions of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals are introduced, and related properties are investigated. Relations between a hesitant fuzzy soft subalgebra and a (closed) hesitant fuzzy soft ideal are discussed. Conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra are given, and conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal are provided. Characterizations of a (closed) hesitant fuzzy soft ideal are considered.
1. Introduction
Nowadays, even if Molodtsov’s soft set theory (see [1]) is a newly emerging mathematical tool to handle uncertainty, the classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In order to overcome this situation, Maji et al. [2] introduced the concept of fuzzy soft sets as a generalization of the standard soft sets and presented an application of fuzzy soft sets in a decision making problem. The notions of Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets, fuzzy multisets, and so forth are a generalization of fuzzy sets. As another generalization of fuzzy sets, Torra [3] introduced the notion of hesitant fuzzy sets which are very useful to express peoples hesitancy in daily life. The hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. Xu and Xia [4] proposed a variety of distance measures for hesitant fuzzy sets, based on which the corresponding similarity measures can be obtained. They investigated the connections of the aforementioned distance measures and further develop a number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures. Also, hesitant fuzzy set theory is used in decision making problem, and so forth (see [5–9]), and is applied to residuated lattices and MTL-algebras (see [10, 11]). In soft set theory membership is decided by adequate parameters, and hesitant fuzzy set employs all possible values for the membership of an element. The soft set model has been combined with other mathematical models, for example, fuzzy soft sets by combining fuzzy sets and soft sets (see [2]), intuitionistic fuzzy soft sets which are based on a combination of intuitionistic fuzzy sets and soft sets (see [12, 13]), interval-valued fuzzy soft sets (see [14]), trapezoidal fuzzy soft sets (see [15]), and forth. As a link between classical soft sets and hesitant fuzzy sets, the notion of hesitant fuzzy soft sets is introduced and applied to a decision making problem in [6, 16].
In this paper, we apply the notion of hesitant fuzzy soft sets to subalgebras and ideals in BCK/BCI-algebras. We introduce the notion of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals and investigate related properties. We consider relations between a hesitant fuzzy soft subalgebra and a hesitant fuzzy soft ideal. We provide conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra. We also give conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal. We discuss characterizations of a (closed) hesitant fuzzy soft ideal.
2. Preliminaries
A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki and was extensively investigated by several researchers.
An algebra of type is called a BCI-algebra if it satisfies the following conditions:(I) , (II) , (III) , (IV) . If a BCI-algebra satisfies the following identity:(V) , then is called a BCK-algebra; any BCK-algebra satisfies the following axioms:(a1) , (a2) , (a3) , (a4) , where if and only if ; Any BCI-algebra satisfies the following axioms:(a5) .
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A nonempty subset of a BCK/BCI-algebra is called a subalgebra of if for all .
A subset of a BCK/BCI-algebra is called an ideal of if it satisfies
An ideal of a BCI-algebra is said to be closed if for every .
We refer the reader to the books [17, 18] for further information regarding BCK/BCI-algebras.
Definition 1 (see [3]). A hesitant fuzzy set on a reference set (or an initial universe set) is defined in terms of a function that when applied to returns a subset of , which can be viewed as the following mathematical representation: where .
Denote by the set of all hesitant fuzzy sets on a reference set (or an initial universe set) .
Definition 2 (see [19]). Let be a BCK/BCI-algebra. A hesitant fuzzy set, on is called a hesitant fuzzy subalgebra of if it satisfies
Definition 3 (see [19]). Let be a BCK/BCI-algebra. A hesitant fuzzy set, on is called a hesitant fuzzy ideal of if it satisfies
Definition 4 (see [6, 16]). A pair is called a hesitant fuzzy soft set over a reference set , where is a mapping given by
3. Hesitant Fuzzy Soft Subalgebras
In what follows let be a set of parameters and we take a BCK/BCI-algebra as a reference set unless otherwise specified.
Definition 5. For a subset of , a hesitant fuzzy soft set over is called a hesitant fuzzy soft subalgebra based on (briefly, -hesitant fuzzy soft subalgebra) over if the hesitant fuzzy set, on is a hesitant fuzzy subalgebra of . If is an -hesitant fuzzy soft subalgebra over for all , we say that is a hesitant fuzzy soft subalgebra.
Example 6. Let be a BCK-algebra with the following Cayley table.
Consider a set of parameters .
(1) Let be a hesitant fuzzy soft set over where which is given in Table 1.
It is routine to verify that , , and are hesitant fuzzy subalgebras of ; that is, is a hesitant fuzzy soft subalgebra over based on parameters “,” “,” and “.” Therefore is a hesitant fuzzy soft subalgebra over .
(2) Let be a hesitant fuzzy soft set over where which is defined in Table 2.
It is easily checked that is a hesitant fuzzy soft subalgebra based on parameters “” and “” over . But it is not an -hesitant fuzzy soft subalgebra over since the hesitant fuzzy set,
is not a hesitant fuzzy subalgebra of . In fact,
Example 7. Let be a BCI-algebra with the following Cayley table.
Given a set of parameters, let be a hesitant fuzzy soft set over which is described in Table 3.
Then is a hesitant fuzzy soft subalgebra over .
Proposition 8. If is a hesitant fuzzy soft subalgebra over , then where is any parameter in .
Proof. For any and , we have This completes the proof.
Theorem 9. Let be a hesitant fuzzy soft subalgebra over . If is a subset of , then is a hesitant fuzzy soft subalgebra over .
Proof. Straightforward.
The following example shows that there exists a hesitant fuzzy soft set over such that(i) is not a hesitant fuzzy soft subalgebra over ,(ii)there exists a subset of such that is a hesitant fuzzy soft subalgebra over .
Example 10. Let be a BCK-algebra as in Example 6. Consider a set of parameters. Let be a hesitant fuzzy soft set over which is described in Table 4.
Then two hesitant fuzzy sets,
on are not hesitant fuzzy subalgebras of since
respectively. Therefore is not a hesitant fuzzy soft subalgebra over . But if we take , then is a hesitant fuzzy soft subalgebra over .
4. Hesitant Fuzzy Soft Ideals
Definition 11. Let be a hesitant fuzzy soft set over where is a subset of . Given , is called a hesitant fuzzy soft ideal based on (briefly, -hesitant fuzzy soft ideal) over if the hesitant fuzzy set, on is a hesitant fuzzy ideal of . If is an -hesitant fuzzy soft ideal over for all , we say that is a hesitant fuzzy soft ideal over .
Example 12. Let be a reference set, and consider an operation which produces the following products:
Then is a BCK-algebra. Consider a set of parameters
Let be a hesitant fuzzy soft set over which is described in Table 5.
Then is a hesitant fuzzy soft ideal over based on parameters “cat,” “cow,” “dog,” “duck,” and “pig.” But is not a hesitant fuzzy soft ideal of based on parameter “horse” since
In general, we know that horses like grapes best of all. In the above example, we know that is not a hesitant fuzzy soft ideal of based on parameter “horse.” This means that if a horse likes a grape better than the others, then cannot be a hesitant fuzzy ideal over .
Proposition 13. Every hesitant fuzzy soft ideal over satisfies the following condition:
Proof. Let and be such that . Then , and so It follows that . This completes the proof.
Theorem 14. Let be a hesitant fuzzy soft set over which satisfies the condition (13) and (21). Then is a hesitant fuzzy soft ideal over .
Proof. Let . Since for all , it follows from (21) that Hence is an -hesitant fuzzy soft ideal over . Since is arbitrary, we know that is a hesitant fuzzy soft ideal over .
Theorem 15. In a BCK-algebra , every hesitant fuzzy soft ideal (based on a parameter) over is a hesitant fuzzy soft subalgebra (based on the same parameter) over .
Proof. For any , let be an -hesitant fuzzy soft ideal over . Then for all , and so is an -hesitant fuzzy soft subalgebra over .
The following example shows that the converse of Theorem 15 is not true in general.
Example 16. Let be a universe, and consider an operation which produces the following products:
Then is a BCK-algebra. Consider a set of parameters:
Let be a hesitant fuzzy soft set over which is given in Table 6.
Then is a hesitant fuzzy soft subalgebra over , but it is not a hesitant fuzzy soft ideal over based on parameter “horse” since
We provide a condition for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal.
Theorem 17. Let be a hesitant fuzzy soft subalgebra over . Then is a hesitant fuzzy soft ideal over if and only if the condition (21) is valid.
Proof. Necessity is by Proposition 13.
Conversely, assume that the condition (21) is valid. Since for all , it follows that for all and . Combining this and (13), we know that is a hesitant fuzzy soft ideal over .
Proposition 18. Every hesitant fuzzy soft ideal over a BCI-algebra satisfies the following inequality:
Proof. Let be a hesitant fuzzy soft ideal of a BCI-algebra . Then for all and .
The following example shows that a hesitant fuzzy soft ideal over a BCI-algebra (based on a parameter) may not be a hesitant fuzzy soft subalgebra (based on the same parameter).
Example 19. Let be a reference set which consists of all nonzero rational numbers. Let be a binary operation which is defined as division in general. Then is a BCI-algebra. For a subset of , let be a hesitant fuzzy soft set over in which is defined as follows: for all and , where is the set of all nonzero integers. Then is a hesitant fuzzy soft ideal over , but it is not a hesitant fuzzy soft subalgebra over since
Definition 20. A hesitant fuzzy ideal of a -algebra is said to be closed if for all .
Definition 21. A hesitant fuzzy soft ideal over a -algebra based on a parameter is said to be closed if the hesitant fuzzy set, on is a closed hesitant fuzzy ideal of .
Example 22. The hesitant fuzzy soft subalgebra over which is described in Example 7 is a closed hesitant fuzzy soft ideal over .
Theorem 23. In a -algebra , every closed hesitant fuzzy soft ideal over based on a parameter is a hesitant fuzzy soft subalgerba over based on the same parameter.
Proof. Let be a closed hesitant fuzzy soft ideal over based on a parameter . Then for all . It follows that for all . Therefore is a hesitant fuzzy soft subalgebra over based on the parameter .
Theorem 24. Let be a hesitant fuzzy soft ideal over a -algebra based on a parameter . Then it is closed if and only if it satisfies
Proof. Assume that is a closed hesitant fuzzy soft ideal over a BCI-algebra based on a parameter . Since for all , it follows from (21) that
for all .
Conversely, let be a hesitant fuzzy soft ideal over a BCI-algebra based on a parameter that satisfies the condition (34). Since for all , we have
for all . Therefore is a closed hesitant fuzzy soft ideal over a BCI-algebra based on a parameter .
5. Conclusion
As another generalization of the standard soft sets, the concept of hesitant fuzzy soft sets has been introduced in [6, 16]. They have presented an application of hesitant fuzzy soft soft sets in a decision making problem. The first author [20] has applied the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. Also, Jun et al. [21] have discussed soft set theory which is applied to ideals in -algebras. In this paper, we applied the notion of hesitant fuzzy soft sets to the theory of BCK/BCI-algebras. We introduced the concepts of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals, and then we investigated related properties. We provided relations between a hesitant fuzzy soft subalgebra and a hesitant fuzzy soft ideal. We gave conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra. We also provided conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal. We discussed characterizations of a (closed) hesitant fuzzy soft ideal. On the basis of these results, we will study applications of hesitant fuzzy soft sets to several ideals and filters of BCK/BCI-algebras, MTL-algebras, -algebras, -algebras, and so forth.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors wish to thank the anonymous reviewers for their valuable suggestions.