Abstract
We introduce the notions of hesitant anti-fuzzy soft set (subalgebras and ideals) and provide relation between them. However, we study new types of hesitant anti-fuzzy soft ideals (implicative, positive implicative, and commutative). Also, we stated and proved some theorems which determine the relationship between these notions.
1. Introduction
In the real world, there are many complicated problems in economic science, engineering, environment, social science, and management science. They are characterized by uncertainty, imprecision, and vagueness. We cannot successfully utilize the classical methods to deal with these problems because there are various types of uncertainties involved in these problems. Moreover, although there are many theories, such as theory of probability, theory of fuzzy sets, theory of interval mathematics, and theory of rough sets, to be considered as mathematical tools to deal with uncertainties, Molodtsov [1] pointed out that all these theories had their own limitations. Also, in order to overcome these difficulties, Molodtsov [1] firstly proposed a new mathematical tool named soft set theory to deal with uncertainty and imprecision. This theory has been demonstrated to be a useful tool in many applications such as decision-making, measurement theory, and game theory.
The soft set model can be combined with other mathematical models. Maji et al. [2] firstly presented the concept of fuzzy soft set by combining the theories of fuzzy set and soft set together. The hesitant fuzzy set, as one of the extensions of Zadeh’s [3] (1965) fuzzy set, allows the membership degree of an element to a set presented by several possible values, and it can express the hesitant information more comprehensively than other extensions of fuzzy set. In [4], Torra introduced the concept of hesitant fuzzy set and Babitha and John (2013) [5] defined another important soft set, hesitant fuzzy soft set. They introduced basic operations such as intersection, union, and compliment, and De Morgan’s law was proven. In 2014, Jun et al. [6] applied the notion of hesitant fuzzy soft sets to subalgebras and ideals in BCK/BCI-algebras. In this paper, in Section 3, we introduce the concepts of hesitant anti-fuzzy soft set of subalgebra. In Section 4, we define the hesitant anti-fuzzy soft ideal in BCK-algebras and give some basic relations. In Section 5, we discuss notion of hesitant anti-fuzzy soft implicative ideals and provide some properties. In Section 6, we investigate concept of hesitant anti-fuzzy soft positive implicative ideals and give some relations. In Section 7, we introduce the notion of hesitant anti-fuzzy soft commutative ideals in BCK-algebras and related properties are investigated. Finally, conclusions are presented in the last section.
2. Preliminaries
An algebra of type (2, 0) is said to be a BCK-algebra if it satisfies the axioms: for all , (BCK-1) , (BCK-2) , (BCK-3) , (BCK-4) , (BCK-5) and imply that .
Define a binary relation on by letting if and only if
Then is a partially ordered set with the least element . In any BCK-algebra , the following hold:(1).(2).(3).(4).(5).(6) implies that and , for all .
A nonempty subset of is called a subalgebra of if, for any , That is, it is closed under the binary operation of
A nonempty subset of is called an ideal of if () and imply that .
A nonempty subset of is called an implicative ideal if it satisfies and whenever and .
It is called a commutative ideal if it satisfies and whenever and ; and it is called a positive implicative ideal if it satisfies and whenever and .
A BCK-algebra is said to be implicative if it satisfies
A BCK-algebra is said to be positive implicative if it satisfies
A BCK-algebra is said to be commutative if it satisfies
Definition 1 (see [3]). Let be a set. A fuzzy set in is a function
Definition 2 (see [2]). Let be an initial universe set and let be a set of parameters. Let denote the set of all fuzzy sets in . Then is called a fuzzy soft set over , where and is a mapping given by
Definition 3 (see [4, 7]). Let be a reference set. A hesitant fuzzy set on 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
Definition 4 (see [7]). Given a nonempty subset of , a hesitant fuzzy set on satisfying the conditionis called a hesitant fuzzy set related to (briefly, -hesitant fuzzy set) on and is represented by ; is a mapping from to with , for all
Proposition 5 (see [8]). Let be an -hesitant anti-fuzzy ideal of . Then the following hold: for all ,(a)if , then , which means that preserves the order,(b)if , then
Proposition 6 (see [9]). In a BCK-algebra , the following hold: for all ,(i),(ii),(iii)
Definition 7 (see [5, 6]). Denote by the set of all hesitant fuzzy sets. A pair is called a hesitant fuzzy soft set over a reference set , where is a mapping given by
3. Hesitant Anti-Fuzzy Soft Subalgebras
Definition 8. Given a nonempty subset (subalgebra as much as possible) of , let be an -hesitant fuzzy set on . Then is called a hesitant anti-fuzzy subalgebra of related to (briefly, -hesitant anti-fuzzy subalgebra of ) if it satisfies the following condition:An -hesitant anti-fuzzy subalgebra of with is called a hesitant anti-fuzzy subalgebra of
Definition 9. For a subset of , a hesitant fuzzy soft set over is called a hesitant anti-fuzzy soft subalgebra based on (briefly, -hesitant anti-fuzzy soft subalgebra) over if the hesitant fuzzy set,on is a hesitant anti-fuzzy subalgebra of . If is an -hesitant anti-fuzzy soft subalgebra over , for all , we say that is a hesitant anti-fuzzy soft subalgebra.
Example 10. Let be a BCK-algebra in Table 1 (Cayley).
Consider a set of parameters Let be a hesitant fuzzy soft set over , where , which is given in Table 2. It is routine to verify that and are hesitant anti-fuzzy subalgebra over based on parameters , and Therefore is a hesitant anti-fuzzy soft subalgebra over .
Proposition 11. If is a hesitant anti-fuzzy soft subalgebra over , thenwhere is any parameter in
Proof. For any and , we haveThis completes the proof.
Theorem 12. Let be a hesitant anti-fuzzy soft subalgebra over If , then is a hesitant anti-fuzzy soft subalgebra over
Proof. Suppose that is a hesitant anti-fuzzy soft subalgebra over Then is a hesitant anti-fuzzy subalgebra of , Since , is a hesitant anti-fuzzy subalgebra of , Hence, is a hesitant anti-fuzzy soft subalgebra over .
The following example shows that there exists a hesitant fuzzy soft set over such that(i) is not a hesitant anti-fuzzy soft subalgebra over ,(ii)there exists a subset of such that is a hesitant anti-fuzzy soft subalgebra over
Example 13. Let be a BCK-algebra in Table 3 (Cayley).
Consider a set of parameters . Let be a hesitant fuzzy soft set over which is described in Table 4.
Then on is not hesitant anti-fuzzy subalgebra of , because Therefore is not hesitant anti-fuzzy soft subalgebra of . But if we take , then is a hesitant anti-fuzzy soft subalgebra over
Definition 14. Let be a hesitant fuzzy soft set over For each , the set is called a hesitant anti--level soft set of , where for all
Theorem 15. Let be a hesitant fuzzy soft set over is a hesitant anti-fuzzy soft subalgebra of if and only if is a subalgebra over for each
Proof. Suppose that is a hesitant anti-fuzzy soft subalgebra of For each and , and ThusThis implies that Hence, is a subalgebra over .
Conversely, assume that is a subalgebra over for each For each and , let and let . Take Then and so Hence,Therefore is a hesitant anti-fuzzy subalgebra over Then, by Definition 8, we conclude that is a hesitant anti-fuzzy soft subalgebra of This completes the proof.
4. Hesitant Anti-Fuzzy Soft Ideals
Definition 16. Given a nonempty subset (subalgebra as much as possible) of , let be an -hesitant fuzzy set on . Then is called a hesitant anti-fuzzy ideal of related to (briefly, -hesitant anti-fuzzy ideal of ) if it satisfies the following conditions: . for all An -hesitant anti-fuzzy ideal of with is called a hesitant anti-fuzzy ideal of .
Definition 17. Let be a hesitant fuzzy soft set over , where is a subset of . Given is called a hesitant anti-fuzzy soft ideal based on (briefly, -hesitant anti-fuzzy soft ideal) over , if the hesitant fuzzy set,on is a hesitant anti-fuzzy ideal of . If is an -hesitant anti-fuzzy soft ideal over for all , we say that is a hesitant anti-fuzzy soft ideal over .
Example 18. Let ≔ banana, carrot, peach, be a reference set, and consider a soft machine which produces the following products:Then is a BCK-algebra under the soft machine . Consider a set of parameters Cow, Dog, ; let be a hesitant fuzzy soft set over which is described in Table 5.
It is routine to verify that is a hesitant anti-fuzzy soft ideal over based on parameters “cat,” “cow,” and “dog.” But is not a hesitant anti-fuzzy soft ideal of based on parameter “horse” because
Theorem 19. Let be a hesitant fuzzy soft set over is a hesitant anti-fuzzy soft ideal of if and only if is an ideal over for each
Proof. Suppose that is a hesitant anti-fuzzy soft ideal of For each and such that and ; then and . Thus,Hence, and , and this implies that is an ideal over .
Conversely, assume that is an ideal over for each For each and , let , and then Since is an ideal over , we have and so Let and let If we take , then and which imply that Thus,Therefore is a hesitant anti-fuzzy soft ideal of .
Proposition 20. Every hesitant anti-fuzzy soft ideal over a BCK/BCI-algebra satisfies the following condition for all and :(a)If , then (b)(c)If , then (d)If , then (e)
Proof. Let and
(a) If , then Since is a hesitant anti-fuzzy soft ideal of ,(b) Since , it follows from (a) that Hence (c) If , then Since it follows that
(d) If , then we have (e) For all , The proof is complete.
Theorem 21. Let be a hesitant fuzzy soft set over which satisfies condition (5) and (c) from Proposition 20. Then is a hesitant anti-fuzzy soft ideal over .
Proof. Let . Since , for all , it follows from Proposition 20(c) thatHence, is a hesitant anti-fuzzy soft ideal over .
Theorem 22. Every hesitant anti-fuzzy soft ideal (based on a parameter) over BCK-algebra is a hesitant anti-fuzzy soft subalgebra (based on the same parameter) over
Proof. For any , assume that is a hesitant anti-fuzzy soft ideal over . Then for all , and so is a hesitant anti-fuzzy soft subalgebra over .
The following example shows that the converse of Theorem 22 is not true in general.
Example 23. Let in Example 13 and , and then is a hesitant anti-fuzzy soft subalgebra over But it is not a hesitant anti-fuzzy soft ideal over based on parameters “” and “” since Also, We provide a condition for a hesitant anti-fuzzy soft subalgebra over to be a hesitant anti-fuzzy soft ideal over
Theorem 24. Let be a hesitant anti-fuzzy soft subalgebra over . If the inequality holds in , then is a hesitant anti-fuzzy soft ideal over .
Proof. Suppose that is a hesitant anti-fuzzy soft subalgebra over Then, from Proposition 11, we haveAssume that holds in Then, by Proposition 20(c), we getSince ,Hence, is a hesitant anti-fuzzy soft ideal over .
5. Hesitant Anti-Fuzzy Soft Implicative Ideals
Definition 25. A hesitant fuzzy set in a BCK-algebra is called a hesitant anti-fuzzy implicative ideal (briefly, HAFII) of if it satisfies the following conditions: , for all , for all
Definition 26. Let be a hesitant fuzzy soft set over , where is a subset of Given is called a hesitant anti-fuzzy soft implicative ideal based on (briefly, -hesitant anti-fuzzy soft implicative ideal) over if the hesitant fuzzy set,on is a hesitant anti-fuzzy implicative ideal of . If is an -hesitant anti-fuzzy soft implicative ideal over , for all , we say that is a hesitant anti-fuzzy soft implicative ideal over .
Proposition 27. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra is order-preserving.
Proof. Let be a hesitant anti-fuzzy soft implicative ideal over . Let and be such that ; thenHence, , and this completes the proof.
Proposition 28. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra is a hesitant anti-fuzzy soft ideal of .
Proof. Let be a hesitant anti-fuzzy soft implicative ideal over . Let and ; thenReplace , and using we get For all , is a hesitant anti-fuzzy soft ideal.
Combining Proposition 28 and Theorem 22 yields the following result.
Corollary 29. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra is a hesitant anti-fuzzy soft subalgebra of .
The following example shows that the converse of Proposition 28 may not be true in general.
Example 30. Let be a BCK-algebra in Table 6 (Cayley).
Consider a set of parameters . Let be a hesitant fuzzy soft set over which is described in Table 7.
Then is a hesitant anti-fuzzy soft ideal (subalgebra) of But and on is not hesitant anti-fuzzy soft implicative ideal of , because Also, Now we give a condition for a hesitant anti-fuzzy soft ideal over to be a hesitant anti-fuzzy soft implicative ideal over .
Theorem 31. If is an implicative BCK-algebra, then every hesitant anti-fuzzy soft ideal over is a hesitant anti-fuzzy soft implicative ideal over .
Proof. Let be an implicative BCK-algebra; it follows that And let be hesitant anti-fuzzy soft ideal over . Then, for any , we have for all Hence, it is a hesitant anti-fuzzy implicative ideal of . That is, is hesitant anti-fuzzy soft implicative ideal of .
Theorem 32. Let be a hesitant fuzzy soft set in a BCK-algebra Then is a hesitant anti-fuzzy soft implicative ideal of if and only if for each the hesitant anti-fuzzy soft -level set of is empty or an implicative ideal of .
Proof. Let be a hesitant anti-fuzzy soft implicative ideal of and assume that for every Note that , for all and In particular, for each and so Let and , and since is a hesitant anti-fuzzy implicative ideal over , it follows thatHence , and is an implicative ideal of .
Conversely, let be an implicative ideal of ; we first show , for all and If not, then there exists such that Let ; then Thus, and As is an implicative ideal of , which implies that This is a contradiction and so for all and Now assume that does not hold. Then, for some If we takethen It follows that and , which imply that and But is an implicative ideal of , so or , and this is a contradiction. Therefore is a hesitant anti-fuzzy soft implicative ideal of .
Theorem 33. Let be a hesitant anti-fuzzy soft ideal of a BCK-algebra . Then is a hesitant anti-fuzzy soft implicative ideal of if and only if it satisfies the conditionfor all and
Proof. Assume that is a hesitant anti-fuzzy soft implicative ideal of . Take in Conversely, suppose that satisfies the condition. As is a hesitant anti-fuzzy soft ideal of , we haveThen is a hesitant anti-fuzzy soft implicative ideal of and the proof is completed.
Now we give characterizations of hesitant anti-fuzzy soft implicative ideals.
Theorem 34. Let be a hesitant anti-fuzzy soft ideal of a BCK-algebra . Then the following are equivalent:(i) is a hesitant anti-fuzzy soft implicative ideal of .(ii) for all and .(iii) for all and .
Proof. Let be a hesitant anti-fuzzy soft implicative ideal of . Then by Theorem 33 we have for all Hence condition (ii) holds.
Observe that, in BCK-algebra Applying Proposition 20(a), we obtain Since (ii) holds, for all and So, condition holds.
(iii) ⇒ (i) Suppose that condition (iii) holds. Since is a hesitant anti-fuzzy soft ideal of , by Definition 16, we haveCombining (iii), we obtainThus is a hesitant anti-fuzzy soft implicative ideal of . So, condition (i) holds.
Theorem 35. Let be a hesitant anti-fuzzy soft implicative ideal of a BCK-algebra ; then the set is an implicative ideal of .
Proof. Clearly, Let be such that and , for all ; thenSince is a hesitant anti-fuzzy soft implicative ideal of , it follows that Combining Definition 25(HAFII1), we get and hence Therefore is an implicative ideal of .
Theorem 36. A hesitant anti-fuzzy soft subalgebra of is a hesitant anti-fuzzy soft implicative ideal if and only if it satisfies the condition implying that for all and
Proof. Assume that is a hesitant anti-fuzzy soft implicative ideal of and let be such that . Since is also hesitant anti-fuzzy soft ideal of , by Proposition 28, it follows from Proposition 20(c) thatMaking use of Theorem 34, we obtain
Conversely, suppose that satisfies the condition in theorem. Obviously, satisfies (5) from Proposition 11. Since , it follows from Proposition 20(c) and Theorem 34 thatwhich shows that satisfies conditions of a hesitant anti-fuzzy implicative ideal of and so is a hesitant anti-fuzzy soft implicative ideal of . The proof is complete.
6. Hesitant Anti-Fuzzy Soft Positive Implicative Ideals
Definition 37. A hesitant fuzzy set in a BCK-algebra is called a hesitant anti-fuzzy positive implicative ideal (briefly, HAFPII) of if it satisfies the following conditions: , for all , for all
Definition 38. Let be a hesitant fuzzy soft set over , where is a subset of Given is called a hesitant anti-fuzzy soft positive implicative ideal based on (briefly, -hesitant anti-fuzzy soft positive implicative ideal) over if the hesitant fuzzy set,on is a hesitant anti-fuzzy positive implicative ideal of . If is an -hesitant anti-fuzzy soft positive implicative ideal over , for all , we say that is a hesitant anti-fuzzy soft positive implicative ideal over .
Proposition 39. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra is order-preserving.
Proof. Let and be such that . Since is a hesitant anti-fuzzy soft positive implicative ideal of ,Putting ,The proof is complete.
Proposition 40. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra is a hesitant anti-fuzzy soft ideal of .
Proof. Let be a hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra , so for all and we havePutting ,Therefore is a hesitant anti-fuzzy soft ideal of .
Combining Theorem 21 and Proposition 39 yields the following result.
Proposition 41. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra is a hesitant anti-fuzzy soft subalgebra of .
Remark 42. A hesitant anti-fuzzy soft ideal (subalgebra) of a BCK-algebra may not be a hesitant anti-fuzzy soft positive implicative ideal of as shown in the following example.
Example 43. Let be the BCK-algebra in Example 30. Routine calculations give that is a hesitant anti-fuzzy soft ideal (subalgebra) of , but it is not a hesitant anti-fuzzy soft positive implicative ideal of , because The converse of Proposition 40 is valid if the following condition holds.
Proposition 44. If is a positive implicative BCK-algebra, then every hesitant anti-fuzzy soft ideal of is a hesitant anti-fuzzy soft positive implicative ideal of .
Proof. Assume that is a hesitant anti-fuzzy soft ideal of a positive implicative BCK-algebra , for all and , and thenBy replacing by and by , we getSince is a positive implicative BCK-algebra, for all Hence,This shows that is a hesitant anti-fuzzy soft positive implicative ideal of .
Theorem 45. Let be a hesitant anti-fuzzy soft ideal over ; then is a hesitant anti-fuzzy soft positive implicative ideal of if and only iffor all and
Proof. Suppose that is a hesitant anti-fuzzy soft ideal over andfor all and . Thus is a hesitant anti-fuzzy soft positive implicative ideal of .
Conversely, assume that is a hesitant anti-fuzzy soft positive implicative ideal of implying that is a hesitant anti-fuzzy soft ideal of by Proposition 40. Let and let ; sinceusing Proposition 20(a), we haveAnd so Therefore , for all and
Theorem 46. Let be a hesitant anti-fuzzy soft positive implicative ideal of . Then, for all and ,
Proof. Let be such that Since is a hesitant anti-fuzzy soft positive implicative ideal of , it follows from Theorem 45 and Proposition 20(c) thatThis completes the proof.
Theorem 47. Let be a hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra , for all Then the setis a positive implicative ideal of .
Proof. Let Clearly, Let be such that and , and thenIt follows that since is a hesitant anti-fuzzy positive implicative ideal of , and then and hence Therefore is a positive implicative ideal of .
Theorem 48. Let be a hesitant anti-fuzzy soft ideal over ; then is a hesitant anti-fuzzy soft positive implicative ideal of if and only if it satisfies the inequalitiesfor all and
Proof. Suppose that the hesitant anti-fuzzy soft ideal of a BCK-algebra is a hesitant anti-fuzzy soft filter positive implicative ideal of . SoSubstituting , we have for all and
Conversely, suppose that is a hesitant anti-fuzzy soft ideal over and satisfies inequality (67).since Now we can prove that for all and In contrast, there exist such that which is a contradiction. Therefore for all and Thus is a hesitant anti-fuzzy soft positive implicative ideal of .
Now we give the condition that makes equality in Theorem 48 hold.
Theorem 49. If is positive implicative, then a hesitant anti-fuzzy soft ideal of is a hesitant anti-fuzzy soft positive implicative ideal of if and only if it satisfies
Proof. Suppose that the hesitant anti-fuzzy soft ideal of a BCK-algebra is a hesitant anti-fuzzy soft positive implicative ideal of . So by Theorem 48 we haveOn the other hand, since , it follows from Proposition 20(a) that Thus we haveConversely, assume that for all and We want to prove that a hesitant anti-fuzzy soft ideal of a BCK-algebra is a hesitant anti-fuzzy soft positive implicative ideal of . It is clear that for all and Since is a hesitant anti-fuzzy soft ideal and is positive implicative,for all and Therefore a hesitant anti-fuzzy soft ideal of is a hesitant anti-fuzzy soft positive implicative ideal of .
Theorem 50. Let be a hesitant anti-fuzzy soft positive implicative ideal of if and only if for each the hesitant anti-fuzzy soft -level set of is a positive implicative ideal of .
Proof. Suppose that is a hesitant anti-fuzzy soft positive implicative ideal of and for every define the setsSince , let By Definition 37, we have , for all and , implying that Let be such that and , implying that and . Since is a hesitant anti-fuzzy soft positive implicative ideal over , it follows thatHence , and is a positive implicative ideal of .
Conversely, suppose that is a positive implicative ideal of for all Put for any Hence , for all and Now we prove that In contrast, there exists such that Takingit follows that and , which imply that and But is a positive implicative ideal of , and thus or , and this is a contradiction. ThereforeHence is a hesitant anti-fuzzy soft positive implicative ideal of .
7. Hesitant Anti-Fuzzy Soft Commutative Ideals
Definition 51. A hesitant fuzzy set in a BCK-algebra is called a hesitant anti-fuzzy commutative ideal (briefly, HAFCI) of if it satisfies the following conditions: , for all , for all
Definition 52. Let be a hesitant fuzzy soft set over , where is a subset of Given is called a hesitant anti-fuzzy soft commutative ideal based on (briefly, -hesitant anti-fuzzy soft commutative ideal) over if the hesitant fuzzy set,on is a hesitant anti-fuzzy commutative ideal of . If is an -hesitant anti-fuzzy soft commutative ideal over , for all , we say that is a hesitant anti-fuzzy soft commutative ideal over .
Proposition 53. Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebra is order-preserving.
Proof. Let and be such that . Since is a hesitant anti-fuzzy soft commutative ideal of ,Putting and replacing , we obtainThe proof is complete.
Theorem 54. Any hesitant anti-fuzzy soft commutative ideal of BCK-algebra is hesitant anti-fuzzy soft ideal of
Proof. Let and let And let be a hesitant anti-fuzzy soft commutative ideal of a BCK-algebra , so we have for all and Hence is a hesitant anti-fuzzy soft ideal of .
Combining Theorems 22 and 54 yields the following result.
Corollary 55. Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebra is a hesitant anti-fuzzy soft subalgebra of .
Remark 56. A hesitant anti-fuzzy soft ideal (subalgebra) of a BCK-algebra may not be a hesitant anti-fuzzy soft commutative ideal of as shown in the following example.
Example 57. Let be the BCK-algebra with in Table 8 (Cayley).
Consider a set of parameters . Let be a hesitant fuzzy soft set over which is described in Table 9.
Then is a hesitant anti-fuzzy soft ideal (subalgebra) of Routine calculations give that is a hesitant anti-fuzzy soft ideal (subalgebra) of , but and on is not hesitant anti-fuzzy soft commutative ideal of , because Also, The condition in the following theorem makes converse Theorem 54 valid.
Theorem 58. In a commutative BCK-algebra . Every hesitant anti-fuzzy soft ideal of is a hesitant anti-fuzzy soft commutative ideal of
Proof. Suppose that is a hesitant anti-fuzzy soft ideal of a BCK-algebra . It is sufficient to show that satisfies condition (HAFCI2). Let . Thenthat is,It follows from Proposition 20(c) thatfor all and Thus is a hesitant anti-fuzzy soft commutative ideal of a BCK-algebra , and the proof is complete.
Theorem 59. Let be a hesitant anti-fuzzy soft ideal of a BCK-algebra . Then is a hesitant anti-fuzzy soft commutative ideal of if and only if it satisfies the condition
Proof. Let and let And suppose that is a hesitant anti-fuzzy soft commutative ideal of Taking , we have Conversely, assume that satisfies condition (89) as is a hesitant anti-fuzzy soft ideal of a BCK-algebra . HenceCombining (91) and (89), we obtainfor all and Hence is hesitant anti-fuzzy soft commutative ideal of , which completes the proof.
Observing and using Proposition 20(a), we have for all and Hence, Theorem 59 can be improved as follows.
Theorem 60. A hesitant anti-fuzzy soft ideal of a BCK-algebra is a hesitant anti-fuzzy soft commutative ideal of if and only if it satisfies the identity
Now we describe a relation between hesitant anti-fuzzy soft implicative ideals, hesitant anti-fuzzy soft commutative ideals, and hesitant anti-fuzzy soft positive implicative ideals.
Theorem 61. A hesitant anti-fuzzy soft ideal of a BCK-algebra is a hesitant anti-fuzzy soft implicative ideal if and only if is both hesitant anti-fuzzy soft commutative ideal and hesitant anti-fuzzy soft positive implicative ideal.
Proof. Assume that is a hesitant anti-fuzzy soft implicative ideal. By Propositions 6(i) and 20(c), for all and , we have Hence is a hesitant anti-fuzzy soft positive implicative ideal of .
Also, By Propositions 20 and 6(iii) and Theorem 34(iii), we haveIt follows from Theorem 59 that is hesitant anti-fuzzy soft commutative ideal of
Conversely, suppose that is both hesitant anti-fuzzy soft positive implicative ideal of and hesitant anti-fuzzy soft commutative ideal of . Since , it follows from Proposition 20(a) thatUsing Theorem 49, we have and soOn the other hand, implies that Since is a hesitant anti-fuzzy soft commutative ideal of , By Theorem 60, we haveHence,Combining (97), we obtainSo, is a hesitant anti-fuzzy soft implicative ideal of by Theorem 34. The proof is complete.
Theorem 62. Let be hesitant anti-fuzzy soft commutative ideal of if and only if for each the hesitant anti-fuzzy soft -level set of is a commutative ideal of
Proof. Suppose that is a hesitant anti-fuzzy soft commutative ideal of and for every define the setsSince , let By definition, we have , for all and , implying that Let be such that and , implying that and . Since is a hesitant anti-fuzzy soft commutative ideal over , it follows thatNamely, , and is a commutative ideal of
Conversely, suppose that is a commutative ideal of for all Put for any and Hence , for all and Now we prove that In contrast, there exists such thatTakingit follows that and , which imply that and But is a commutative ideal of , and thus or , and this is a contradiction. Therefore is a hesitant anti-fuzzy soft commutative ideal of .
8. Conclusions
When people make a decision, they are usually hesitant and irresolute for one thing or another which makes it difficult to reach a final agreement. So, we can see that hesitant fuzzy set is a very useful tool to deal with uncertainty. Thus, it is very necessary to develop some concepts about hesitant fuzzy set. Therefore, we introduce the hesitant anti-fuzzy soft set for basic notions in BCK-algebras. In this paper, we introduced the notions of hesitant anti-fuzzy soft subalgebras and hesitant anti-fuzzy soft ideals of BCK-algebras and investigated their relations and properties. However, we defined the notions of hesitant anti-fuzzy soft ideals (implicative, positive implicative, and commutative) in BCK-algebras. Also, we presented the relationship between them and we gave conditions for a hesitant anti-fuzzy soft ideal to be a hesitant anti-fuzzy soft (implicative, positive implicative, and commutative) ideal in BCK-algebras.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. D-078-363-1437. The authors, therefore, gratefully acknowledge the DSR technical and financial support.