Hindawi Mathematical Problems in Engineering Volume 2017, Article ID 3634258, 13 pages https://doi.org/10.1155/2017/3634258
Research Article Hesitant Anti-Fuzzy Soft Set in BCK-Algebras Halimah Alshehri1,2 and Noura Alshehri1 1
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Department of Natural and Engineering Science, Faculty of Applied Studies and Community Service, King Saud University, Riyadh, Saudi Arabia
2
Correspondence should be addressed to Halimah Alshehri;
[email protected] Received 6 January 2017; Accepted 30 April 2017; Published 5 June 2017 Academic Editor: Anna M. Gil-Lafuente Copyright ยฉ 2017 Halimah Alshehri and Noura Alshehri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 Zadehโs [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 Morganโs 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 antifuzzy 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 (๐; โ, 0) of type (2, 0) is said to be a BCK-algebra if it satisfies the axioms: for all ๐ฅ, ๐ฆ, ๐ง โ ๐, (BCK-1) ((๐ฅ โ ๐ฆ) โ (๐ฅ โ ๐ง)) โ (๐ง โ ๐ฆ) = 0, (BCK-2) (๐ฅ โ (๐ฅ โ ๐ฆ)) โ ๐ฆ = 0, (BCK-3) ๐ฅ โ ๐ฅ = 0, (BCK-4) 0 โ ๐ฅ = 0, (BCK-5) ๐ฅ โ ๐ฆ = 0 and ๐ฆ โ ๐ฅ = 0 imply that ๐ฅ = ๐ฆ.
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Define a binary relation โค on ๐ by letting ๐ฅ โค ๐ฆ if and only if ๐ฅ โ ๐ฆ = 0. Then (๐; โค) is a partially ordered set with the least element 0. In any BCK-algebra ๐, the following hold: (1) (๐ฅ โ ๐ฆ) โ ๐ง = (๐ฅ โ ๐ง) โ ๐ฆ. (2) ๐ฅ โ ๐ฆ โค ๐ฅ. (3) ๐ฅ โ 0 = ๐ฅ.
โ 0 ๐ ๐ ๐
0 0 ๐ ๐ ๐
๐ 0 0 ๐ ๐
๐ 0 ๐ 0 ๐
๐ 0 0 0 0
Proposition 5 (see [8]). Let ๐ป๐ด fl {(๐ฅ, โ๐ด (๐ฅ)) | ๐ฅ โ ๐} be an ๐ด-hesitant anti-fuzzy ideal of ๐. Then the following hold: for all ๐ฅ, ๐ฆ, ๐ง โ ๐ด,
(4) (๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง) โค ๐ฅ โ ๐ฆ. (5) ๐ฅ โ (๐ฅ โ (๐ฅ โ ๐ฆ)) = ๐ฅ โ ๐ฆ. (6) ๐ฅ โค ๐ฆ implies that ๐ฅ โ ๐ง โค ๐ฆ โ ๐ง and ๐ง โ ๐ฆ โค ๐ง โ ๐ฅ, for all ๐ฅ, ๐ฆ, ๐ง โ ๐. A nonempty subset ๐ผ of ๐ is called a subalgebra of ๐ if, for any ๐ฅ, ๐ฆ โ ๐ผ, ๐ฅโ๐ฆ โ ๐ผ. That is, it is closed under the binary operation โ of ๐. A nonempty subset ๐ผ of ๐ is called an ideal of ๐ if (๐ผ1 ) 0 โ ๐ผ; (๐ผ2 ) ๐ฅ โ ๐ฆ โ ๐ผ and ๐ฆ โ ๐ผ imply that ๐ฅ โ ๐ผ. A nonempty subset ๐ผ of ๐ is called an implicative ideal if it satisfies (๐ผ1 ) and (๐ผ3 ) ๐ฅ โ ๐ผ whenever (๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง โ ๐ผ and ๐ง โ ๐ผ. It is called a commutative ideal if it satisfies (๐ผ1 ) and (๐ผ4 ) ๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)) โ ๐ผ whenever (๐ฅ โ ๐ฆ) โ ๐ง โ ๐ผ and ๐ง โ ๐ผ; and it is called a positive implicative ideal if it satisfies (๐ผ1 ) and (๐ผ5 ) ๐ฅ โ ๐ง โ ๐ผ whenever (๐ฅ โ ๐ฆ) โ ๐ง โ ๐ผ and ๐ฆ โ ๐ง โ ๐ผ. A BCK-algebra ๐ is said to be implicative if it satisfies โ๐ฅ, ๐ฆ โ ๐ : ๐ฅ = ๐ฅ โ (๐ฆ โ ๐ฅ). A BCK-algebra ๐ is said to be positive implicative if it satisfies โ๐ฅ, ๐ฆ, ๐ง โ ๐ : (๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง) = (๐ฅ โ ๐ฆ) โ ๐ง. A BCK-algebra ๐ is said to be commutative if it satisfies โ๐ฅ, ๐ฆ โ ๐ : ๐ฅ โ (๐ฅ โ ๐ฆ) = ๐ฆ โ (๐ฆ โ ๐ฅ). Definition 1 (see [3]). Let ๐ be a set. A fuzzy set in ๐ is a function ๐ : ๐ โ [0, 1] Definition 2 (see [2]). Let ๐ be an initial universe set and let ๐ธ be a set of parameters. Let F(๐) denote the set of all fuzzy ฬ ๐ด) is called a fuzzy soft set over ๐, where sets in ๐. Then (๐น, ฬ ๐ด โ ๐ธ and ๐น is a mapping given by ๐นฬ : ๐ด โ F(๐). Definition 3 (see [4, 7]). Let ๐ธ be a reference set. A hesitant fuzzy set on ๐ธ is defined in terms of a function that when applied to ๐ธ returns a subset of [0, 1] which can be viewed as the following mathematical representation: ๐ป๐ธ fl {(๐, โ๐ธ (๐)) : ๐ โ ๐ธ} ,
(1)
Definition 4 (see [7]). Given a nonempty subset ๐ด of ๐, a hesitant fuzzy set ๐ป๐ fl {(๐ฅ; โ๐ (๐ฅ) : ๐ฅ โ ๐} on ๐ satisfying the condition โ๐ฅ โ ๐ด
(a) if ๐ฅ โค ๐ฆ, then โ๐ด (๐ฅ) โ โ๐ด (๐ฆ), which means that โ๐ด preserves the order, (b) if ๐ฅ โ ๐ฆ โค ๐ง, then โ๐ด (๐ฅ) โ โ๐ด (๐ฆ) โช โ๐ด (๐ง). Proposition 6 (see [9]). In a BCK-algebra ๐, the following hold: for all ๐ฅ, ๐ฆ, ๐ง โ ๐, (i) ((๐ฅ โ ๐ง) โ ๐ง) โ (๐ฆ โ ๐ง) โค (๐ฅ โ ๐ฆ) โ ๐ง, (ii) (๐ฅ โ ๐ง) โ (๐ฅ โ (๐ฅ โ ๐ง)) = (๐ฅ โ ๐ง) โ ๐ง, (iii) (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ (๐ฆ โ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)))) โค ๐ฅ โ ๐ฆ. Definition 7 (see [5, 6]). Denote by ๐ปF(๐) the set of all ฬ ๐ด) is called a hesitant fuzzy hesitant fuzzy sets. A pair (๐ป, ฬ is a mapping given soft set over a reference set ๐, where ๐ป ฬ by ๐ป : ๐ด โ ๐ปF(๐).
3. Hesitant Anti-Fuzzy Soft Subalgebras Definition 8. Given a nonempty subset (subalgebra as much as possible) ๐ด of ๐, let ๐ป๐ด fl {(๐ฅ, โ๐ด (๐ฅ)) | ๐ฅ โ ๐} be an ๐ดhesitant fuzzy set on ๐. Then ๐ป๐ด fl {(๐ฅ, โ๐ด (๐ฅ)) | ๐ฅ โ ๐} is called a hesitant anti-fuzzy subalgebra of ๐ related to ๐ด (briefly, ๐ด-hesitant anti-fuzzy subalgebra of ๐) if it satisfies the following condition: (โ๐ด (๐ฅ โ ๐ฆ) โ โ๐ด (๐ฅ) โช โ๐ด (๐ฆ))
(โ๐ฅ, ๐ฆ โ ๐ด) .
(2)
is called a hesitant fuzzy set related to ๐ด (briefly, ๐ด-hesitant fuzzy set) on ๐ and is represented by ๐ป๐ด fl {(๐ฅ, โ๐ด (๐ฅ)) : ๐ฅ โ ๐}; โ๐ด is a mapping from ๐ to ๐([0, 1]) with โ๐ด (๐ฅ) = ๐, for all ๐ฅ โ ๐ด.
(3)
An ๐ด-hesitant anti-fuzzy subalgebra of ๐ with ๐ด = ๐ is called a hesitant anti-fuzzy subalgebra of ๐. Definition 9. For a subset ๐ด of ๐ธ, a hesitant fuzzy soft set ฬ ๐ด) over ๐ is called a hesitant anti-fuzzy soft subalgebra (๐ป, based on ๐ โ ๐ด (briefly, ๐-hesitant anti-fuzzy soft subalgebra) over ๐ if the hesitant fuzzy set, ฬ [๐] fl {(๐ฅ, โ ฬ (๐ฅ)) | ๐ฅ โ ๐} , ๐ป ๐ป[๐]
where โ๐ธ : ๐ธ โ ๐([0, 1]).
โ๐ (๐ฅ) = ๐;
Table 1
(4)
ฬ ๐ด) is an on ๐ is a hesitant anti-fuzzy subalgebra of ๐. If (๐ป, ๐-hesitant anti-fuzzy soft subalgebra over ๐, for all ๐ โ ๐ด, we ฬ ๐ด) is a hesitant anti-fuzzy soft subalgebra. say that (๐ป, Example 10. Let ๐ = {0, ๐, ๐, ๐} be a BCK-algebra in Table 1 (Cayley). Consider a set of parameters ๐ธ fl {๐1 , ๐2 , ๐3 , ๐4 }. Let ฬ ๐ด) be a hesitant fuzzy soft set over ๐, where ๐ด = (๐ป, {๐1 , ๐2 , ๐3 }, which is given in Table 2. It is routine to verify that
Mathematical Problems in Engineering
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ฬ ๐ด). Table 2: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬ ๐ป ๐1 ๐2 ๐3
0 (0.2, 0.3) {0.3} {0.4}
๐ [0.2, 0.8] {0.3} [0.3, 0.7]
๐ [0.2, 0.8] [0.1, 0, 2) โช (0, 2, 0.8] [0.3, 0.5]
๐ [0.2, 0.8] [0.1, 0.9] [0.3, 0.7]
ฬ 1 ], ๐ป[๐ ฬ 2 ] and ๐ป[๐ ฬ 3 ] are hesitant anti-fuzzy subalgebra ๐ป[๐ ฬ ๐ด) over ๐ based on parameters ๐1 , ๐2 , and ๐3 . Therefore (๐ป, is a hesitant anti-fuzzy soft subalgebra over ๐. ฬ ๐ด) is a hesitant anti-fuzzy soft subalgeProposition 11. If (๐ป, bra over ๐, then โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ) , โ๐ฅ โ ๐,
(5)
where ๐ is any parameter in ๐ด. Proof. For any ๐ฅ โ ๐ and ๐ โ ๐ด, we have โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (๐ฅ โ ๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ) โช โ๐ป[๐] ฬ (๐ฅ) = โ๐ป[๐] ฬ (๐ฅ) .
(6)
This completes the proof. ฬ ๐ด) be a hesitant anti-fuzzy soft subalgeTheorem 12. Let (๐ป, ฬ ๐ต , ๐ต) is a hesitant anti-fuzzy soft bra over ๐. If ๐ต โ ๐ด, then (๐ป| subalgebra over ๐. ฬ ๐ด) is a hesitant anti-fuzzy soft subProof. Suppose that (๐ป, ฬ๐ด is a hesitant anti-fuzzy subalgebra algebra over ๐. Then ๐ป of ๐, โ๐ฅ โ ๐ด. Since ๐ต โ ๐ด, โ๐ต (๐) is a hesitant anti-fuzzy ฬ ๐ต , ๐ต) is a hesitant antisubalgebra of ๐, โ๐ โ ๐ต. Hence, (๐ป| fuzzy soft subalgebra over ๐. The following example shows that there exists a hesitant ฬ ๐ด) over ๐ such that fuzzy soft set (๐ป, ฬ ๐ด) is not a hesitant anti-fuzzy soft subalgebra over (i) (๐ป, ๐, ฬ ๐ต , ๐ต) is a (ii) there exists a subset ๐ต of ๐ด such that (๐ป| hesitant anti-fuzzy soft subalgebra over ๐. Example 13. Let ๐ = {0, ๐, ๐, ๐} be a BCK-algebra in Table 3 (Cayley). ฬ ๐ด) be Consider a set of parameters ๐ด = {๐1 , ๐2 , ๐3 }. Let (๐ป, a hesitant fuzzy soft set over ๐ which is described in Table 4. ฬ 3 ] fl {(๐ฅ, โ ฬ (๐ฅ)) | ๐ฅ โ ๐} on ๐ is not hesitant Then ๐ป[๐ ๐ป[๐3 ] anti-fuzzy subalgebra of ๐, because โ๐ป[๐ ฬ 3 ] (๐ โ ๐) = โ๐ป[๐ ฬ 3 ] (0) = {0.1} โฬธ (0.1, 0.2) = โ๐ป[๐ ฬ 3 ] (๐) โช โ๐ป[๐ ฬ 3 ] (๐) .
(7)
ฬ ๐ด) is not hesitant anti-fuzzy soft subalgebra Therefore (๐ป, ฬ ๐ต , ๐ต) is a hesitant of ๐. But if we take ๐ต = {๐1 , ๐2 }, then (๐ป| anti-fuzzy soft subalgebra over ๐.
Table 3 โ 0 ๐ ๐ ๐
0 0 ๐ ๐ ๐
๐ 0 0 ๐ ๐
๐ 0 ๐ 0 ๐
๐ 0 0 0 0
ฬ ๐ด). Table 4: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬ ๐ป ๐1 ๐2 ๐3
0 (0.3, 0.5) [0.3, 0.6) {0.1}
๐ (0.3, 0.8) [0.3, 0.9) (0.1, 0.2)
๐ [0.3, 0.8] [0.3, 0.8) [0.1, 0.3]
๐ [0.3, 0.5] [0.3, 0.6] [0.1, 0.4]
ฬ ๐ด) be a hesitant fuzzy soft set over ๐. Definition 14. Let (๐ป, ฬ ๐ด)๐ = (๐ป ฬ๐ , ๐ด) is called a For each ๐ โ ๐([0, 1]), the set (๐ป, ฬ ๐ด), where ๐ป ฬ๐ = {๐ฅ โ ๐ : hesitant anti-๐-level soft set of (๐ป, โ๐ป[๐] ฬ (๐ฅ) โ ๐} for all ๐ โ ๐ด. ฬ ๐ด) be a hesitant fuzzy soft set over Theorem 15. Let (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft subalgebra of ๐ if and ๐. (๐ป, ฬ ๐ด)๐ is a subalgebra over ๐ for each ๐ โ ๐([0, 1]). only if (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft subalProof. Suppose that (๐ป, ฬ๐ , gebra of ๐. For each ๐ โ ๐([0, 1]), ๐ โ ๐ด and ๐ฅ1 , ๐ฅ2 โ ๐ป โ๐ป[๐] ฬ (๐ฅ1 ) โ ๐ and โ๐ป[๐] ฬ (๐ฅ2 ) โ ๐. Thus โ๐ป[๐] ฬ (๐ฅ1 โ ๐ฅ2 ) โ โ๐ป[๐] ฬ (๐ฅ1 ) โช โ๐ป[๐] ฬ (๐ฅ) โ ๐.
(8)
ฬ๐ . Hence, (๐ป, ฬ ๐ด)๐ is a This implies that ๐ฅ1 โ ๐ฅ2 โ ๐ป subalgebra over ๐. ฬ ๐ด)๐ is a subalgebra over ๐ for Conversely, assume that (๐ป, each ๐ โ ๐([0, 1]). For each ๐ โ ๐ด and ๐ฅ, ๐ฆ โ ๐, let โ๐ป[๐] ฬ (๐ฅ) = ฬ๐ ๐๐ฅ and let โ๐ป[๐] ฬ (๐ฆ) = ๐๐ฆ . Take ๐ = ๐๐ฅ โช ๐๐ฆ . Then ๐ฅ, ๐ฆ โ ๐ป and ๐ ฬ . Hence, so ๐ฅ โ ๐ฆ โ ๐ป โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ ๐ = ๐๐ฅ โช ๐๐ฆ = โ๐ป[๐] ฬ (๐ฅ) โช โ๐ป[๐] ฬ (๐ฆ) .
(9)
ฬ ๐ด) is a hesitant anti-fuzzy subalgebra over Therefore (๐ป, ฬ ๐ด) is a hesitant ๐. Then, by Definition 8, we conclude that (๐ป, anti-fuzzy soft subalgebra of ๐. This completes the proof.
4. Hesitant Anti-Fuzzy Soft Ideals Definition 16. Given a nonempty subset (subalgebra as much as possible) ๐ด of ๐, let ๐ป๐ด fl {(๐ฅ, โ๐ด (๐ฅ)) | ๐ฅ โ ๐} be an ๐ด-hesitant fuzzy set on ๐. Then ๐ป๐ด fl {(๐ฅ, โ๐ด (๐ฅ)) | ๐ฅ โ ๐} is called a hesitant anti-fuzzy ideal of ๐ related to ๐ด (briefly, ๐ด-hesitant anti-fuzzy ideal of ๐) if it satisfies the following conditions: (HAFI1) โ๐ด (0) โ โ๐ด (๐ฅ). (HAFI2) โ๐ด (๐ฅ) โ โ๐ด (๐ฅ โ ๐ฆ) โช โ๐ด (๐ฆ) for all ๐ฅ, ๐ฆ โ ๐ด. An ๐ด-hesitant anti-fuzzy ideal of ๐ with ๐ด = ๐ is called a hesitant anti-fuzzy ideal of ๐.
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Mathematical Problems in Engineering
ฬ ๐ด). Table 5: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬ ๐ป Apple Cat (0.2, 0.3) Cow [0.3, 0.4) Dog [0.1, 0.3] Horse {0.3}
Banana [0.2, 0.5] [0.3, 0.4) [0.1, 0.7) [0.3, 0.4)
Carrot [0.2, 0.5] [0.3, 0.4] [0.1, 0.8) [0.3, 0.4)
Peach [0.2, 0.5] [0.3, 0.4) [0.1, 0.7) [0.3, 0.9)
Radish [0.2, 0.5] [0.3, 0.4) [0.1, 0.7) [0.3, 0.6)
ฬ ๐ด) be a hesitant fuzzy soft set over ๐, Definition 17. Let (๐ป, ฬ ๐ด) is called a where ๐ด is a subset of ๐ธ. Given ๐ โ ๐ด, (๐ป, hesitant anti-fuzzy soft ideal based on ๐ (briefly, ๐-hesitant anti-fuzzy soft ideal) over ๐, if the hesitant fuzzy set, ฬ [๐] fl {(๐ฅ, โ ฬ (๐ฅ)) | ๐ฅ โ ๐} , ๐ป ๐ป[๐]
(10)
ฬ ๐ด) is an ๐on ๐ is a hesitant anti-fuzzy ideal of ๐. If (๐ป, hesitant anti-fuzzy soft ideal over ๐ for all ๐ โ ๐ด, we say ฬ ๐ด) is a hesitant anti-fuzzy soft ideal over ๐. that (๐ป, Example 18. Let ๐ fl {apple, banana, carrot, peach, radish} be a reference set, and consider a soft machine โ which produces the following products: {apple if ๐ฅ โ {carrot, peach, radish} apple โ๐ฅ = { carrot if ๐ฅ โ {apple, banana} {
(13)
ฬ๐ , and this implies that (๐ป, ฬ ๐ด)๐ ฬ๐ and ๐ฅ โ ๐ป Hence, 0 โ ๐ป is an ideal over ๐. ฬ ๐ด)๐ is an ideal over ๐ for Conversely, assume that (๐ป, each ๐ โ ๐([0, 1]). For each ๐ โ ๐ด and ๐ฅ, ๐ฆ โ ๐, let ฬ๐ ฬ ๐ โ๐ป[๐] ฬ (๐ฅ) = ๐๐ฅ , and then ๐ฅ โ ๐ป . Since (๐ป, ๐ด) is an ideal ฬ๐ and so โ ฬ (0) โ ๐๐ฅ = โ ฬ (๐ฅ). over ๐, we have 0 โ ๐ป ๐ป[๐] ๐ป[๐] Let โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) = ๐๐ฅโ๐ฆ and let โ๐ป[๐] ฬ (๐ฆ) = ๐๐ฆ . If we take ฬ๐ and ๐ฆ โ ๐ป ฬ๐ which imply ๐ = ๐๐ฅโ๐ฆ โช ๐๐ฆ , then ๐ฅ โ ๐ฆ โ ๐ป ฬ๐ . Thus, that ๐ฅ โ ๐ป โ๐ป[๐] ฬ (๐ฅ) โ ๐ = ๐๐ฅโ๐ฆ โช ๐๐ฆ = โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) . (14) ฬ ๐ด) is a hesitant anti-fuzzy soft ideal of ๐. Therefore (๐ป, ฬ ๐ด) Proposition 20. Every hesitant anti-fuzzy soft ideal (๐ป, over a BCK/BCI-algebra ๐ satisfies the following condition for all ๐ โ ๐ด and ๐ฅ, ๐ฆ, ๐ง โ ๐: (b) โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โช โ๐ป[๐] ฬ (๐ง โ ๐ฆ). (c) If ๐ฅ โ ๐ฆ โค ๐ง, then โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฆ) โช โ๐ป[๐] ฬ (๐ง). (d) If โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) = โ๐ป[๐] ฬ (0), then โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฆ).
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{peach if ๐ง โ {carrot, radish} peach โ๐ง = { carrot if ๐ง โ {apple, banana, peach} {
(e) โ๐ป[๐] ฬ (0 โ (0 โ ๐ฅ)) โ โ๐ป[๐] ฬ (๐ฅ). Proof. Let ๐ โ ๐ด and ๐ฅ, ๐ฆ, ๐ง โ ๐. ฬ ๐ด) is a hesitant (a) If ๐ฅ โค ๐ฆ, then ๐ฅ โ ๐ฆ = 0. Since (๐ป, anti-fuzzy soft ideal of ๐,
{radish if ๐ข โ {carrot, peach} radish โ๐ข = { carrot if ๐ข โ {apple, banana, radish} {
โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) = โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ฆ) = โ๐ป[๐] ฬ (๐ฆ) .
carrot โV = carrot โV โ ๐. Then ๐ is a BCK-algebra under the soft machine โ. Consider a set of parameters ๐ด fl {Cat, Cow, Dog, Horse}; let ฬ ๐ด) be a hesitant fuzzy soft set over ๐ which is described (๐ป, in Table 5. ฬ ๐ด) is a hesitant anti-fuzzy It is routine to verify that (๐ป, soft ideal over ๐ based on parameters โcat,โ โcow,โ and โdog.โ ฬ ๐ด) is not a hesitant anti-fuzzy soft ideal of ๐ based on But (๐ป, parameter โhorseโ because ฬ [horse] (peach) = [0.3, 0.9) โฬธ [0.3, 0.4) ๐ป ฬ [horse] (peachโbanana) =๐ป
โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) โ ๐.
(a) If ๐ฅ โค ๐ฆ, then โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฆ).
banana โ๐ฆ {banana if ๐ฆ โ {apple, carrot, peach, radish} ={ carrot if ๐ฆ = banana {
ฬ ๐ด) is a hesitant anti-fuzzy soft ideal Proof. Suppose that (๐ป, ฬ๐ such that of ๐. For each ๐ โ ๐([0, 1]), ๐ โ ๐ด and ๐ฅ, ๐ฆ โ ๐ป ๐ ๐ ฬ and ๐ฆ โ ๐ป ฬ ; then โ ฬ (๐ฅโ๐ฆ) โ ๐ and โ ฬ (๐ฆ) โ ๐. ๐ฅโ๐ฆ โ ๐ป ๐ป[๐] ๐ป[๐] Thus,
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ฬ [horse] (banana) . โช๐ป ฬ ๐ด) be a hesitant fuzzy soft set over Theorem 19. Let (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft ideal of ๐ if and only ๐. (๐ป, ฬ ๐ด)๐ is an ideal over ๐ for each ๐ โ ๐([0, 1]). if (๐ป,
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(b) Since (๐ฅ โ ๐ฆ) โ (๐ฅ โ ๐ง) โค ๐ง โ ๐ฆ, it follows from (a) that โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ (๐ฅ โ ๐ง)) โ โ๐ป[๐] ฬ (๐ง โ ๐ฆ). Hence โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ (๐ฅ โ ๐ง)) โช โ๐ป[๐] ฬ (๐ฅ โ ๐ง)
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โ โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โช โ๐ป[๐] ฬ (๐ง โ ๐ฆ) , (c) If ๐ฅ โ ๐ฆ โค ๐ง, then (๐ฅ โ ๐ฆ) โ ๐ง = 0. Since โ๐ป[๐] ฬ (๐ง) = โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ง) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง)
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โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) , it follows that โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) โ โ๐ป[๐] ฬ (๐ฆ) โช โ๐ป[๐] ฬ (๐ง).
Mathematical Problems in Engineering
5
(d) If โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) = โ๐ป[๐] ฬ (0), then we have โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) = โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ฆ) = โ๐ป[๐] ฬ (๐ฆ) .
Also, (18)
(e) For all ๐ฅ โ ๐,
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= โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ฅ) = โ๐ป[๐] ฬ (๐ฅ) . The proof is complete. ฬ ๐ด) be a hesitant fuzzy soft set over ๐ Theorem 21. Let (๐ป, which satisfies condition (5) and (c) from Proposition 20. Then ฬ ๐ด) is a hesitant anti-fuzzy soft ideal over ๐. (๐ป, Proof. Let ๐ โ ๐ด. Since ๐ฅ โ (๐ฅ โ ๐ฆ) โค ๐ฆ, for all ๐ฅ, ๐ฆ โ ๐, it follows from Proposition 20(c) that โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) .
= โ๐ป[๐ ฬ 2 ] (๐ โ ๐) โช โ๐ป[๐ ฬ 2 ] (๐) .
(23)
We provide a condition for a hesitant anti-fuzzy soft subalgebra over ๐ to be a hesitant anti-fuzzy soft ideal over ๐.
โ๐ป[๐] ฬ (0 โ (0 โ ๐ฅ)) โ โ๐ป[๐] ฬ ((0 โ (0 โ ๐ฅ)) โ ๐ฅ) โช โ๐ป[๐] ฬ (๐ฅ)
โ๐ป[๐ ฬ 2 ] (๐) = [0.3, 0.9) โฬธ [0.3, 0.6]
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ฬ ๐ด) is a hesitant anti-fuzzy soft ideal over ๐. Hence, (๐ป, Theorem 22. Every hesitant anti-fuzzy soft ideal (based on a parameter) over BCK-algebra ๐ is a hesitant anti-fuzzy soft subalgebra (based on the same parameter) over ๐.
ฬ ๐ด) be a hesitant anti-fuzzy soft subalgeTheorem 24. Let (๐ป, ฬ ๐ด) bra over ๐. If the inequality ๐ฅ โ ๐ฆ โค ๐ง holds in ๐, then (๐ป, is a hesitant anti-fuzzy soft ideal over ๐. ฬ ๐ด) is a hesitant anti-fuzzy soft subalProof. Suppose that (๐ป, gebra over ๐. Then, from Proposition 11, we have โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ) , โ๐ฅ โ ๐, ๐ โ ๐ด.
(24)
Assume that ๐ฅ โ ๐ฆ โค ๐ง holds in ๐. Then, by Proposition 20(c), we get โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฆ) โช โ๐ป[๐] ฬ (๐ง)
โ๐ฅ, ๐ฆ, ๐ง โ ๐.
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Since ๐ฅ โ (๐ฅ โ ๐ฆ) โค ๐ฆ, โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ)
โ๐ฅ, ๐ฆ โ ๐.
(26)
ฬ ๐ด) is a hesitant anti-fuzzy soft ideal over ๐. Hence, (๐ป,
ฬ ๐ด) is a hesitant antiProof. For any ๐ โ ๐ด, assume that (๐ป, fuzzy soft ideal over ๐. Then
5. Hesitant Anti-Fuzzy Soft Implicative Ideals
โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฅ) โช โ๐ป[๐] ฬ (๐ฅ) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฅ) โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฅ) = โ๐ป[๐] ฬ (0 โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฅ)
(21)
= โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ฅ)
(HAFII1) โ๐ (0) โ โ๐ (๐ฅ), for all ๐ฅ โ ๐.
โ โ๐ป[๐] ฬ (๐ฅ) โช โ๐ป[๐] ฬ (๐ฆ) , ฬ ๐ด) is a hesitant anti-fuzzy soft for all ๐ฅ, ๐ฆ โ ๐, and so (๐ป, subalgebra over ๐. The following example shows that the converse of Theorem 22 is not true in general. Example 23. Let ๐ = {0, ๐, ๐, ๐} in Example 13 and ๐ต = ฬ ๐ต) is a hesitant anti-fuzzy soft subalge{๐1 , ๐2 }, and then (๐ป, bra over ๐. But it is not a hesitant anti-fuzzy soft ideal over ๐ based on parameters โ๐1 โ and โ๐2 โ since โ๐ป[๐ ฬ 1 ] (๐) = (0.3, 0.8) โฬธ [0.3, 0.5] = (0.3, 0.5) โช [0.3, 0.5] = โ๐ป[๐ ฬ 1 ] (0) โช โ๐ป[๐ ฬ 1 ] (๐) = โ๐ป[๐ ฬ 1 ] (๐ โ ๐) โช โ๐ป[๐ ฬ 1 ] (๐) .
Definition 25. A hesitant fuzzy set ๐ป๐ fl {(๐ฅ, โ๐ (๐ฅ)) | ๐ฅ โ ๐} in a BCK-algebra ๐ is called a hesitant antifuzzy implicative ideal (briefly, HAFII) of ๐ if it satisfies the following conditions:
(22)
(HAFII2) โ๐ (๐ฅ) โ โ๐ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ (๐ง), for all ๐ฅ, ๐ฆ, ๐ง โ ๐. ฬ ๐ด) be a hesitant fuzzy soft set over ๐, Definition 26. Let (๐ป, ฬ ๐ด) is called a where ๐ด is a subset of ๐ธ. Given ๐ โ ๐ด, (๐ป, hesitant anti-fuzzy soft implicative ideal based on ๐ (briefly, ๐-hesitant anti-fuzzy soft implicative ideal) over ๐ if the hesitant fuzzy set, ฬ [๐] fl {(๐ฅ, โ ฬ (๐ฅ)) | ๐ฅ โ ๐} , ๐ป ๐ป[๐]
(27)
ฬ ๐ด) on ๐ is a hesitant anti-fuzzy implicative ideal of ๐. If (๐ป, is an ๐-hesitant anti-fuzzy soft implicative ideal over ๐, for ฬ ๐ด) is a hesitant anti-fuzzy soft all ๐ โ ๐ด, we say that (๐ป, implicative ideal over ๐. Proposition 27. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra ๐ is order-preserving.
6
Mathematical Problems in Engineering
ฬ ๐ด) be a hesitant anti-fuzzy soft implicative Proof. Let (๐ป, ideal over ๐. Let ๐ โ ๐ด and ๐ฅ, ๐ฆ โ ๐ be such that ๐ฅ โค ๐ฆ; then โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ง โ ๐ฅ)) โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ (๐ง โ ๐ฅ)) โช โ๐ป[๐] ฬ (๐ฆ) = โ๐ป[๐] ฬ (0 โ (๐ง โ ๐ฅ)) โช โ๐ป[๐] ฬ (๐ฆ)
(28)
Table 6 โ 0 ๐ ๐ ๐
0 0 ๐ ๐ ๐
Hence, โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฆ), and this completes the proof.
0 1 1 ( , ) 4 2 1 (0, ) 4
๐1 ๐2
Proposition 28. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra ๐ is a hesitant anti-fuzzy soft ideal of ๐. ฬ ๐ด) be a hesitant anti-fuzzy soft implicative Proof. Let (๐ป, ideal over ๐. Let ๐ โ ๐ด and ๐ฅ, ๐ฆ, ๐ง โ ๐; then โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) .
(29)
Replace ๐ฆ = ๐ฅ, and using ๐ฅ โ ๐ฅ = 0 we get โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฅ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) = โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) .
(30)
๐ 0 0 0 ๐
๐ 0 ๐ ๐ 0
ฬ ๐ด). Table 7: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬ ๐ป
= โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ฆ) = โ๐ป[๐] ฬ (๐ฆ) .
๐ 0 0 ๐ ๐
๐ 1 [0, ) 2 1 (0, ) 2
๐ 1 [0, ) 2 1 [0, ] 2
๐ 3 [0, ] 4 3 [0, ) 4
Also, 1 โ๐ป[๐ ฬ 2 ] (๐) = (0, ) 2 โฬธ โ๐ป[๐ ฬ 2 ] ((๐ โ (๐ โ ๐)) โ 0) โช โ๐ป[๐ ฬ 2 ] (0)
(32)
1 = โ๐ป[๐ ฬ 2 ] (0) โช โ๐ป[๐ ฬ 2 ] (0) = (0, ) . 4 Now we give a condition for a hesitant anti-fuzzy soft ideal over ๐ to be a hesitant anti-fuzzy soft implicative ideal over ๐.
ฬ ๐ด) is a hesitant anti-fuzzy soft ideal. For all ๐ฅ, ๐ง โ ๐, (๐ป,
Theorem 31. If ๐ is an implicative BCK-algebra, then every hesitant anti-fuzzy soft ideal over ๐ is a hesitant anti-fuzzy soft implicative ideal over ๐.
Combining Proposition 28 and Theorem 22 yields the following result.
Proof. Let ๐ be an implicative BCK-algebra; it follows that ฬ ๐ด) be hesitant anti๐ฅ = ๐ฅ โ (๐ฆ โ ๐ฅ), โ๐ฅ, ๐ฆ โ ๐. And let (๐ป, fuzzy soft ideal over ๐. Then, for any ๐ โ ๐ด, we have
Corollary 29. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra ๐ is a hesitant anti-fuzzy soft subalgebra of ๐. The following example shows that the converse of Proposition 28 may not be true in general. Example 30. Let ๐ = {0, ๐, ๐, ๐} be a BCK-algebra in Table 6 (Cayley). ฬ ๐ด) be a Consider a set of parameters ๐ด = {๐1 , ๐2 }. Let (๐ป, hesitant fuzzy soft set over ๐ which is described in Table 7. ฬ ๐ด) is a hesitant anti-fuzzy soft ideal (subalgebra) Then (๐ป, ฬ 1 ] fl {(๐ฅ, โ ฬ (๐ฅ)) | ๐ฅ โ ๐} and ๐ป[๐ ฬ 2 ] fl of ๐. But ๐ป[๐ ๐ป[๐1 ] {(๐ฅ, โ๐ป[๐ ฬ 2 ] (๐ฅ)) | ๐ฅ โ ๐} on ๐ is not hesitant anti-fuzzy soft implicative ideal of ๐, because 1 โ๐ป[๐ ฬ 1 ] (๐) = [0, ) 2 โฬธ โ๐ป[๐ ฬ 1 ] ((๐ โ (๐ โ ๐)) โ 0) โช โ๐ป[๐ ฬ 1 ] (0) 1 1 = โ๐ป[๐ ฬ 1 ] (0) โช โ๐ป[๐ ฬ 1 ] (0) = ( , ) . 4 2
(31)
โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) = โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง)
(33)
for all ๐ฅ, ๐ฆ, ๐ง โ ๐. Hence, it is a hesitant anti-fuzzy impliฬ ๐ด) is hesitant anti-fuzzy soft cative ideal of ๐. That is, (๐ป, implicative ideal of ๐. ฬ ๐ด) be a hesitant fuzzy soft set in a BCKTheorem 32. Let (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft implicative algebra ๐. Then (๐ป, ideal of ๐ if and only if for each ๐ โ ๐([0, 1]) the hesitant antiฬ ๐ด)๐ of (๐ป, ฬ ๐ด) is empty or an implicative fuzzy soft ๐-level set (๐ป, ideal of ๐. ฬ ๐ด) be a hesitant anti-fuzzy soft implicative Proof. Let (๐ป, ฬ๐ =ฬธ โ for every ๐ โ ๐([0, 1]). ideal of ๐ and assume that ๐ป Note that โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ), for all ๐ โ ๐ด and ๐ฅ โ ๐. In ฬ๐ , โ ฬ (0) โ โ ฬ (๐ฅ) โ ๐ and so particular, for each ๐ฅ โ ๐ป ๐ป[๐] ๐ป[๐] ฬ๐ . Let (๐ฅโ(๐ฆโ๐ฅ))โ๐ง โ ๐ป ฬ๐ and ๐ง โ ๐ป ฬ๐ , and since (๐ป, ฬ ๐ด) 0โ๐ป is a hesitant anti-fuzzy implicative ideal over ๐, it follows that โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) โ ๐. (34) ฬ๐ is an implicative ideal of ๐. ฬ๐ , and ๐ป Hence ๐ฅ โ ๐ป
Mathematical Problems in Engineering
7
ฬ๐ be an implicative ideal of ๐; we first Conversely, let ๐ป show โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ), for all ๐ โ ๐ด and ๐ฅ โ ๐. If not, then there exists ๐ฅ0 โ ๐ such that โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ0 ). Let (0) + โ (๐ฅ )}; then 0 โค โ ๐0 = (1/2){ โ๐ป[๐] ฬ ฬ ฬ (๐ฅ0 ) โ ๐0 โ 0 ๐ป[๐] ๐ป[๐] ๐0 ๐0 ฬ ฬ ฬ๐0 is an โ๐ป[๐] =ฬธ โ. As ๐ป ฬ (0) โค 1. Thus, ๐ฅ0 โ ๐ป and ๐ป ๐ ฬ 0 , which implies that โ ฬ (0) โ implicative ideal of ๐, 0 โ ๐ป ๐ป[๐] ๐0 . This is a contradiction and so โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ) for all ๐ โ ๐ด and ๐ฅ โ ๐. Now assume that โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ))โ๐ง)โชโ๐ป[๐] ฬ (๐ง) does not hold. Then, โ๐ป[๐] ฬ (๐ฅ0 ) โ โ๐ป[๐] ฬ ((๐ฅ0 โ (๐ฆ0 โ ๐ฅ0 )) โ ๐ง0 ) โช โ๐ป[๐] ฬ (๐ง0 ) for some ๐ฅ0 , ๐ฆ0 , ๐ง0 โ ๐. If we take ๐0 =
1 {โ ฬ (๐ฅ ) 2 ๐ป[๐] 0
(35)
+ {โ๐ป[๐] ฬ ((๐ฅ0 โ (๐ฆ0 โ ๐ฅ0 )) โ ๐ง0 ) โช โ๐ป[๐] ฬ (๐ง0 )}} , then 0 โค โ๐ป[๐] ฬ ((๐ฅ0 โ (๐ฆ0 โ ๐ฅ0 )) โ ๐ง0 ) โช โ๐ป[๐] ฬ (๐ง0 ) โ ๐0 โ โ๐ป[๐] ฬ (๐ฅ0 ) โค 1. It follows that โ๐ป[๐] ฬ ((๐ฅ0 โ (๐ฆ0 โ ๐ฅ0 )) โ ๐ง0 ) โ ๐0 ฬ๐0 and โ๐ป[๐] ฬ (๐ง0 ) โ ๐0 , which imply that (๐ฅ0 โ(๐ฆ0 โ๐ฅ0 ))โ๐ง0 โ ๐ป ๐ ๐ ฬ 0 . But ๐ป ฬ 0 is an implicative ideal of ๐, so ๐ฅ0 โ and ๐ง0 โ ๐ป ๐0 ฬ ๐ป or โ๐ป[๐] ฬ (๐ฅ0 ) โ ๐0 , and this is a contradiction. Therefore ฬ ๐ด) is a hesitant anti-fuzzy soft implicative ideal of ๐. (๐ป, ฬ ๐ด) be a hesitant anti-fuzzy soft ideal of Theorem 33. Let (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft a BCK-algebra ๐. Then (๐ป, implicative ideal of ๐ if and only if it satisfies the condition โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ))
(36)
for all ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด. ฬ ๐ด) is a hesitant anti-fuzzy soft Proof. Assume that (๐ป, implicative ideal of ๐. Take ๐ง = 0 in โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) = โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ 0) โช โ๐ป[๐] ฬ (0)
(37)
= โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) . ฬ ๐ด) satisfies the condition. As Conversely, suppose that (๐ป, ฬ (๐ป, ๐ด) is a hesitant anti-fuzzy soft ideal of ๐, we have โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) .
(38)
ฬ ๐ด) is a hesitant anti-fuzzy soft implicative ideal Then (๐ป, of ๐ and the proof is completed. Now we give characterizations of hesitant anti-fuzzy soft implicative ideals. ฬ ๐ด) be a hesitant anti-fuzzy soft ideal of Theorem 34. Let (๐ป, a BCK-algebra ๐. Then the following are equivalent: ฬ ๐ด) is a hesitant anti-fuzzy soft implicative ideal of (i) (๐ป, ๐. (ii) โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅโ(๐ฆโ๐ฅ)) for all ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด. (iii) โ๐ป[๐] ฬ (๐ฅ) = โ๐ป[๐] ฬ (๐ฅโ(๐ฆโ๐ฅ)) for all ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด.
ฬ ๐ด) be a hesitant anti-fuzzy soft Proof. (i) โ (ii) Let (๐ป, implicative ideal of ๐. Then by Theorem 33 we have โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ 0) โช โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) โช โ๐ป[๐] ฬ (0)
(39)
= โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) , for all ๐ฅ, ๐ฆ โ ๐. Hence condition (ii) holds. (ii) โ (iii) Observe that, in BCK-algebra ๐, ๐ฅโ(๐ฆโ๐ฅ) โค ๐ฅ. Applying Proposition 20(a), we obtain โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) โ โ๐ป[๐] ฬ (๐ฅ). Since (ii) holds, โ๐ป[๐] ฬ (๐ฅ) = โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) for all ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด. So, condition (iii) holds. ฬ ๐ด) (iii) โ (i) Suppose that condition (iii) holds. Since (๐ป, is a hesitant anti-fuzzy soft ideal of ๐, by Definition 16, we have โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) .
(40)
Combining (iii), we obtain โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) .
(41)
ฬ ๐ด) is a hesitant anti-fuzzy soft implicative ideal Thus (๐ป, of ๐. So, condition (i) holds. ฬ ๐ด) be a hesitant anti-fuzzy soft implicaTheorem 35. Let (๐ป, tive ideal of a BCK-algebra ๐; then the set ๐ปโ๐ป[๐] fl {๐ฅ โ ๐ | โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (๐ฅ)} ฬ
(42)
is an implicative ideal of ๐. . Let ๐ฅ, ๐ฆ, ๐ง โ ๐ปโ๐ป[๐] be such that Proof. Clearly, 0 โ ๐ปโ๐ป[๐] ฬ ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง โ ๐ปโ๐ป[๐] and ๐ง โ ๐ปโ๐ป[๐] , for all ๐ โ ๐ด; then ฬ ฬ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) = โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (๐ง) .
(43)
ฬ ๐ด) is a hesitant anti-fuzzy soft implicative ideal Since (๐ป, of ๐, it follows that โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) = โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (0) .
(44)
Combining Definition 25(HAFII1), we get โ๐ป[๐] ฬ (๐ฅ) = . Therefore ๐ปโ๐ป[๐] is an โ๐ป[๐] ฬ (0) and hence ๐ฅ โ ๐ปโ๐ป[๐] ฬ ฬ implicative ideal of ๐. Theorem 36. A hesitant anti-fuzzy soft subalgebra โ๐ป[๐] of ๐ ฬ is a hesitant anti-fuzzy soft implicative ideal if and only if it satisfies the condition (๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง โค ๐ข implying that โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ง) โช โ๐ป[๐] ฬ (๐ข) for all ๐ฅ, ๐ฆ, ๐ง, ๐ข โ ๐ and โ๐ โ ๐ด. is a hesitant anti-fuzzy soft implicaProof. Assume that โ๐ป[๐] ฬ tive ideal of ๐ and let ๐ฅ, ๐ฆ, ๐ง, ๐ข โ ๐ be such that (๐ฅ โ (๐ฆ โ
8
Mathematical Problems in Engineering
๐ฅ)) โ ๐ง โค ๐ข. Since โ๐ป[๐] is also hesitant anti-fuzzy soft ideal of ฬ ๐, by Proposition 28, it follows from Proposition 20(c) that โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) โ โ๐ป[๐] ฬ (๐ง) โช โ๐ป[๐] ฬ (๐ข) .
(45)
Making use of Theorem 34, we obtain โ๐ป[๐] โ ฬ (๐ฅ) โ๐ป[๐] ฬ (๐ง) โช โ๐ป[๐] ฬ (๐ข). satisfies the condition in Conversely, suppose that โ๐ป[๐] ฬ satisfies (5) from Proposition 11. theorem. Obviously, โ๐ป[๐] ฬ Since (๐ฅ โ (๐ฆ โ ๐ฅ)) โ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โค ๐ง, it follows from Proposition 20(c) and Theorem 34 that โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ฅ)) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) ,
(46)
which shows that โ๐ป[๐] satisfies conditions of a hesitant antiฬ is a hesitant antifuzzy implicative ideal of ๐ and so โ๐ป[๐] ฬ fuzzy soft implicative ideal of ๐. The proof is complete.
Definition 37. A hesitant fuzzy set ๐ป๐ fl {(๐ฅ, โ๐ (๐ฅ)) | ๐ฅ โ ๐} in a BCK-algebra ๐ is called a hesitant anti-fuzzy positive implicative ideal (briefly, HAFPII) of ๐ if it satisfies the following conditions: (HAFPII1) โ๐ (0) โ โ๐ (๐ฅ), for all ๐ฅ โ ๐. (HAFPII2) โ๐ (๐ฅ โ ๐ง) โ โ๐ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ (๐ฆ โ ๐ง), for all ๐ฅ, ๐ฆ, ๐ง โ ๐. ฬ ๐ด) be a hesitant fuzzy soft set over ๐, Definition 38. Let (๐ป, ฬ ๐ด) is called a where ๐ด is a subset of ๐ธ. Given ๐ โ ๐ด, (๐ป, hesitant anti-fuzzy soft positive implicative ideal based on ๐ (briefly, ๐-hesitant anti-fuzzy soft positive implicative ideal) over ๐ if the hesitant fuzzy set, (47)
on ๐ is a hesitant anti-fuzzy positive implicative ideal of ๐. ฬ ๐ด) is an ๐-hesitant anti-fuzzy soft positive implicative If (๐ป, ฬ ๐ด) is a hesitant ideal over ๐, for all ๐ โ ๐ด, we say that (๐ป, anti-fuzzy soft positive implicative ideal over ๐. Proposition 39. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra ๐ is order-preserving. Proof. Let ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด be such that ๐ฅ โค ๐ฆ. Since ฬ ๐ด) is a hesitant anti-fuzzy soft positive implicative ideal (๐ป, of ๐, โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) = โ๐ป[๐] ฬ (0 โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) = โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง)
(48)
(50)
Putting ๐ง = 0, โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) .
(51)
ฬ ๐ด) is a hesitant anti-fuzzy soft ideal of ๐. Therefore (๐ป,
Proposition 41. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra ๐ is a hesitant anti-fuzzy soft subalgebra of ๐. Remark 42. A hesitant anti-fuzzy soft ideal (subalgebra) of a BCK-algebra ๐ may not be a hesitant anti-fuzzy soft positive implicative ideal of ๐ as shown in the following example. Example 43. Let ๐ be the BCK-algebra in Example 30. ฬ ๐ด) is a hesitant anti-fuzzy Routine calculations give that (๐ป, soft ideal (subalgebra) of ๐, but it is not a hesitant anti-fuzzy soft positive implicative ideal of ๐, because 1 โ๐ป[๐ ฬ 1 ] (๐ โ ๐) = โ๐ป[๐ ฬ 1 ] (๐) = [0, ) 2 โฬธ โ๐ป[๐ ฬ 1 ] ((๐ โ ๐) โ ๐) โช โ๐ป[๐ ฬ 1 ] (๐ โ ๐)
The converse of Proposition 40 is valid if the following condition holds. Proposition 44. If ๐ is a positive implicative BCK-algebra, then every hesitant anti-fuzzy soft ideal of ๐ is a hesitant antifuzzy soft positive implicative ideal of ๐. ฬ ๐ด) is a hesitant anti-fuzzy soft ideal Proof. Assume that (๐ป, of a positive implicative BCK-algebra ๐, for all ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด, and then (53)
By replacing ๐ฅ by ๐ฅ โ ๐ง and ๐ฆ by ๐ฆ โ ๐ง, we get
โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) . (49)
(52)
1 1 = โ๐ป[๐ ฬ 1 ] (0) โช โ๐ป[๐ ฬ 1 ] (0) = ( , ) . 4 2
โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง))
Putting ๐ง = 0,
The proof is complete.
โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) .
โ๐ป[๐] ฬ (๐ฅ) โ โH[๐] ฬ (๐ฅ โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ) .
= โ๐ป[๐] ฬ (๐ฆ โ ๐ง) . โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฆ) .
ฬ ๐ด) be a hesitant anti-fuzzy soft positive Proof. Let (๐ป, implicative ideal of a BCK-algebra ๐, so for all ๐ฅ, ๐ฆ, ๐ง โ ๐ and ๐ โ ๐ด we have
Combining Theorem 21 and Proposition 39 yields the following result.
6. Hesitant Anti-Fuzzy Soft Positive Implicative Ideals
ฬ [๐] fl {(๐ฅ, โ ฬ (๐ฅ)) | ๐ฅ โ ๐} , ๐ป ๐ป[๐]
Proposition 40. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra ๐ is a hesitant anti-fuzzy soft ideal of ๐.
(54)
Since ๐ is a positive implicative BCK-algebra, (๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง) = (๐ฅ โ ๐ฆ) โ ๐ง for all ๐ฅ, ๐ฆ, ๐ง โ ๐. Hence, โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) .
(55)
Mathematical Problems in Engineering
9
ฬ ๐ด) is a hesitant anti-fuzzy soft positive This shows that (๐ป, implicative ideal of ๐. ฬ ๐ด) be a hesitant anti-fuzzy soft ideal over Theorem 45. Let (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft positive implicative ๐; then (๐ป, ideal of ๐ if and only if โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง)) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) ,
(56)
ฬ ๐ด) is a hesitant anti-fuzzy soft ideal Proof. Suppose that (๐ป, over ๐ and (57)
for all ๐ โ ๐ด and ๐ฅ, ๐ฆ, ๐ง โ ๐.
(58)
ฬ ๐ด) is a hesitant anti-fuzzy soft positive implicaThus (๐ป, tive ideal of ๐. ฬ ๐ด) is a hesitant anti-fuzzy Conversely, assume that (๐ป, ฬ ๐ด) is soft positive implicative ideal of ๐ implying that (๐ป, a hesitant anti-fuzzy soft ideal of ๐ by Proposition 40. Let ๐ = ๐ฅ โ (๐ฆ โ ๐ง) and let ๐ = ๐ฅ โ ๐ฆ; since (59)
fl {๐ฅ โ ๐ | โ๐ป[๐] ๐ปโ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (๐ฅ)} ฬ
(64)
. Let ๐ฅ, ๐ฆ, ๐ง โ ๐ปโ๐ป[๐] be Proof. Let ๐ โ ๐ด. Clearly, 0 โ ๐ปโ๐ป[๐] ฬ ฬ such that (๐ฅ โ ๐ฆ) โ ๐ง โ ๐ปโ๐ป[๐] and ๐ฆ โ ๐ง โ ๐ป , and then โ ฬ ฬ ๐ป[๐] (65)
It follows that โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) = โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (0) ,
(66)
ฬ ๐ด) is a hesitant anti-fuzzy positive implicative ideal since (๐ป, of ๐, and then โ๐ป[๐] ฬ (๐ฅ โ ๐ง) = โ๐ป[๐] ฬ (0) and hence ๐ฅ โ ๐ง โ .Therefore ๐ป is a positive implicative ideal of ๐. ๐ปโ๐ป[๐] โ๐ป[๐] ฬ ฬ ฬ ๐ด) be a hesitant anti-fuzzy soft ideal over Theorem 48. Let (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft positive implicative ๐; then (๐ป, ideal of ๐ if and only if it satisfies the inequalities
using Proposition 20(a), we have โ๐ป[๐] ฬ ((๐ โ ๐) โ ๐ง) = โ๐ป[๐] ฬ (((๐ฅ โ (๐ฆ โ ๐ง)) โ (๐ฅ โ ๐ฆ)) โ ๐ง)
(63)
ฬ ๐ด) be a hesitant anti-fuzzy soft positive Theorem 47. Let (๐ป, implicative ideal of a BCK-algebra ๐, for all ๐ โ ๐ด. Then the set
โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) = โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (๐ฆ โ ๐ง) .
โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) .
((๐ฅ โ (๐ฆ โ ๐ง)) โ (๐ฅ โ ๐ฆ)) โค ๐ฆ โ (๐ฆ โ ๐ง) ,
โ โ๐ป[๐] ฬ (๐) โช โ๐ป[๐] ฬ (๐) .
is a positive implicative ideal of ๐.
โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง)) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง)
โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง)) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง)
This completes the proof.
for all ๐ โ ๐ด and ๐ฅ, ๐ฆ, ๐ง โ ๐.
โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง)) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) ,
Proof. Let ๐ฅ, ๐ฆ, ๐ง, ๐, ๐ โ ๐ be such that ((๐ฅ โ ๐ฆ) โ ๐ง) โ ๐ โค ๐. ฬ ๐ด) is a hesitant anti-fuzzy soft positive implicative Since (๐ป, ideal of ๐, it follows from Theorem 45 and Proposition 20(c) that
(60)
โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ)
โ โ๐ป[๐] ฬ ((๐ฆ โ (๐ฆ โ ๐ง)) โ ๐ง) = โ๐ป[๐] ฬ (0) .
โ๐ฅ, ๐ฆ โ ๐, ๐ โ ๐ด,
(67)
for all ๐ โ ๐ด and ๐ฅ, ๐ฆ, ๐ง โ ๐.
And so โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง)) = โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ ๐ง)) โ ๐ง) = โ๐ป[๐] ฬ (๐ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ โ ๐) โ ๐ง) โช โ๐ป[๐] ฬ (๐ โ ๐ง) โ โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ โ ๐ง) = โ๐ป[๐] ฬ (๐ โ ๐ง)
(61)
ฬ ๐ด) Proof. Suppose that the hesitant anti-fuzzy soft ideal (๐ป, of a BCK-algebra ๐ is a hesitant anti-fuzzy soft filter positive implicative ideal of ๐. So โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง)
= โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) .
โ๐ฅ, ๐ฆ, ๐ง โ ๐, ๐ โ ๐ด.
Therefore โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง)) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง), for all ๐ โ ๐ด and ๐ฅ, ๐ฆ, ๐ง โ ๐. ฬ ๐ด) be a hesitant anti-fuzzy soft positive Theorem 46. Let (๐ป, implicative ideal of ๐. Then, for all ๐ฅ, ๐ฆ, ๐ง, ๐, ๐ โ ๐ and ๐ โ ๐ด, ((๐ฅ โ ๐ฆ) โ ๐ง) โ ๐ โค ๐ ๓ณจโ โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง)) โ โ๐ป[๐] ฬ (๐) โช โ๐ป[๐] ฬ (๐) .
(62)
(68)
Substituting ๐ง = ๐ฆ, we have โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ฆ) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ) โช โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ) , for all ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด.
(69)
10
Mathematical Problems in Engineering
ฬ ๐ด) is a hesitant anti-fuzzy Conversely, suppose that (๐ป, soft ideal over ๐ and satisfies inequality (67). โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ) โ๐ฅ, ๐ฆ โ ๐, ๐ โ ๐ด,
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since โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ). Now we can prove that โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) for all ๐ฅ, ๐ฆ, ๐ง โ ๐ and ๐ โ ๐ด. In contrast, there exist ๐ฅ,ฬ ๐ฆฬ โ ๐ such that ฬ โ โ๐ป[๐] ฬ โ ๐ฆ)ฬ โช โ๐ป[๐] ฬ โ๐ป[๐] ฬ (๐ฅฬ โ ๐ฆ) ฬ ((๐ฅฬ โ ๐ฆ) ฬ (๐ฆฬ โ ๐ฆ) ฬ โ ๐ฆ)ฬ โช โ๐ป[๐] = โ๐ป[๐] ฬ ((๐ฅฬ โ ๐ฆ) ฬ (0)
(71)
ฬ โ ๐ฆ)ฬ , = โ๐ป[๐] ฬ ((๐ฅฬ โ ๐ฆ) which is a contradiction. Therefore โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) for all ๐ฅ, ๐ฆ, ๐ง โ ๐ and ๐ โ ๐ด. Thus ฬ ๐ด) is a hesitant anti-fuzzy soft positive implicative ideal (๐ป, of ๐. Now we give the condition that makes equality in Theorem 48 hold. Theorem 49. If ๐ is positive implicative, then a hesitant antifuzzy soft ideal of ๐ is a hesitant anti-fuzzy soft positive implicative ideal of ๐ if and only if it satisfies โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ) โ๐ฅ, ๐ฆ โ ๐, ๐ โ ๐ด.
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ฬ ๐ด) Proof. Suppose that the hesitant anti-fuzzy soft ideal (๐ป, of a BCK-algebra ๐ is a hesitant anti-fuzzy soft positive implicative ideal of ๐. So by Theorem 48 we have โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ) โ๐ฅ, ๐ฆ โ ๐, ๐ โ ๐ด.
(73)
โ๐ฅ, ๐ฆ โ ๐, ๐ โ ๐ด.
(74)
Conversely, assume that โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ) for all ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด. We want to prove that a hesitant ฬ ๐ด) of a BCK-algebra ๐ is a hesitant anti-fuzzy soft ideal (๐ป, anti-fuzzy soft positive implicative ideal of ๐. It is clear that ฬ โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ) for all ๐ฅ โ ๐ and ๐ โ ๐ด. Since (๐ป, ๐ด) is a hesitant anti-fuzzy soft ideal and ๐ is positive implicative, โ๐ป[๐] ฬ (๐ฅ โ ๐ง) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ (๐ฆ โ ๐ง)) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) ,
ฬ ๐ด) be a hesitant anti-fuzzy soft positive Theorem 50. Let (๐ป, implicative ideal of ๐ if and only if for each ๐ โ ๐([0, 1]) the ฬ ๐ด)๐ of (๐ป, ฬ ๐ด) is a positive hesitant anti-fuzzy soft ๐-level set (๐ป, implicative ideal of ๐. ฬ ๐ด) is a hesitant anti-fuzzy soft posiProof. Suppose that (๐ป, tive implicative ideal of ๐ and for every ๐ โ ๐([0, 1]) define the sets ๐
ฬ ๐ด) fl {๐ฅ โ ๐ | โ ฬ (๐ฅ) โ ๐} (๐ป, ๐ป[๐]
(75)
โ๐ โ ๐ด.
(76)
ฬ ๐ด)๐ โ โ ฬ (๐ฅ) โ ๐. By ฬ ๐ด)๐ =ฬธ โ, let ๐ฅ โ (๐ป, Since (๐ป, ๐ป[๐] Definition 37, we have โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ), for all ๐ โ ๐ด and ฬ ๐ด)๐ . Let ๐ฅ, ๐ฆ, ๐ง โ ๐ be such ๐ฅ โ ๐, implying that 0 โ (๐ป, ๐ ฬ ๐ด) and ๐ฆ โ ๐ง โ (๐ป, ฬ ๐ด)๐ , implying that that (๐ฅ โ ๐ฆ) โ ๐ง โ (๐ป, ฬ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โ ๐ and โ๐ป[๐] ฬ (๐ฆ โ ๐ง) โ ๐. Since (๐ป, ๐ด) is a hesitant anti-fuzzy soft positive implicative ideal over ๐, it follows that โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) โ ๐.
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ฬ ๐ด)๐ , and (๐ป, ฬ ๐ด)๐ is a positive Hence ๐ฅ โ ๐ง โ (๐ป, implicative ideal of ๐. ฬ ๐ด)๐ is a positive implicative Conversely, suppose that (๐ป, ideal of ๐ for all ๐ โ ๐([0, 1]). Put โ๐ป[๐] ฬ (๐ฅ) = ๐ for any ๐ฅ โ ๐. ฬ ๐ด)๐ โ โ ฬ (0) โ ๐ = โ ฬ (๐ฅ), for all ๐ โ ๐ด Hence 0 โ (๐ป, ๐ป[๐] ๐ป[๐] and ๐ฅ โ ๐. Now we prove that โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง). In contrast, there exists ๐ฅ0 , ๐ฆ0 , ๐ง0 โ ๐ such that โ๐ป[๐] ฬ (๐ฅ0 โ ๐ง0 ) โ โ๐ป[๐] ฬ ((๐ฅ0 โ ๐ฆ0 ) โ ๐ง0 ) โช โ๐ป[๐] ฬ (๐ฆ0 โ ๐ง0 ). Taking ๐0 =
On the other hand, since (๐ฅ โ ๐ฆ) โ ๐ฆ โค ๐ฅ โ ๐ฆ, it follows from Proposition 20(a) that โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ). Thus we have โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) = โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ฆ)
for all ๐ฅ, ๐ฆ, ๐ง โ ๐ and ๐ โ ๐ด. Therefore a hesitant anti-fuzzy ฬ ๐ด) of ๐ is a hesitant anti-fuzzy soft positive soft ideal (๐ป, implicative ideal of ๐.
1 {โ ฬ (๐ฅ โ ๐ง0 ) 2 ๐ป[๐] 0
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+ {โ๐ป[๐] ฬ ((๐ฅ0 โ ๐ฆ0 ) โ ๐ง0 ) โช โ๐ป[๐] ฬ (๐ฆ0 โ ๐ง0 )}} , it follows that โ๐ป[๐] ฬ ((๐ฅ0 โ๐ฆ0 )โ๐ง0 ) โ ๐0 and โ๐ป[๐] ฬ (๐ฆ0 โ๐ง0 ) โ ๐0 , ๐0 ฬ which imply that (๐ฅ0 โ ๐ฆ0 ) โ ๐ง0 โ (๐ป, ๐ด) and ๐ฆ0 โ ๐ง0 โ ฬ ๐ด)๐0 . But (๐ป, ฬ ๐ด)๐0 is a positive implicative ideal of ๐, and (๐ป, ฬ ๐ด)๐0 or โ ฬ (๐ฅ0 โ ๐ง0 ) โ ๐0 , and this is a thus ๐ฅ0 โ ๐ง0 โ (๐ป, ๐ป[๐] contradiction. Therefore โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง) โ๐ฅ, ๐ฆ, ๐ง โ ๐, ๐ โ ๐ด.
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ฬ ๐ด) is a hesitant anti-fuzzy soft positive implicative Hence (๐ป, ideal of ๐.
7. Hesitant Anti-Fuzzy Soft Commutative Ideals Definition 51. A hesitant fuzzy set ๐ป๐ fl {(๐ฅ, โ๐ (๐ฅ)) | ๐ฅ โ ๐} in a BCK-algebra ๐ is called a hesitant anti-fuzzy
Mathematical Problems in Engineering
11
commutative ideal (briefly, HAFCI) of ๐ if it satisfies the following conditions: (HAFCI1) โ๐ (0) โ โ๐ (๐ฅ), for all ๐ฅ โ ๐. (HAFCI2) โ๐ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ โ๐ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ (๐ง), for all ๐ฅ, ๐ฆ, ๐ง โ ๐. ฬ ๐ด) be a hesitant fuzzy soft set over ๐, Definition 52. Let (๐ป, ฬ ๐ด) is called a where ๐ด is a subset of ๐ธ. Given ๐ โ ๐ด, (๐ป, hesitant anti-fuzzy soft commutative ideal based on ๐ (briefly, ๐-hesitant anti-fuzzy soft commutative ideal) over ๐ if the hesitant fuzzy set, ฬ [๐] fl {(๐ฅ, โ ฬ (๐ฅ)) | ๐ฅ โ ๐} , ๐ป ๐ป[๐]
Proposition 53. Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebra ๐ is order-preserving. Proof. Let ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด be such that ๐ฅ โค ๐ฆ. Since ฬ ๐ด) is a hesitant anti-fuzzy soft commutative ideal of ๐, (๐ป, โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)))
= โ๐ป[๐] ฬ (0 โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) = โ๐ป[๐] ฬ (0) โช โ๐ป[๐] ฬ (๐ง)
โ 0 ๐ ๐
0 0 ๐ ๐
๐ 0 0 ๐
ฬ ๐ด). Table 9: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬ ๐ป ๐1 ๐2
0 (0.2, 0.4) (0, 0.2)
๐ [0.2, 0.7) (0, 0.5)
๐ [0, 1) [0, 0.7)
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Remark 56. A hesitant anti-fuzzy soft ideal (subalgebra) of a BCK-algebra ๐ may not be a hesitant anti-fuzzy soft commutative ideal of ๐ as shown in the following example. Example 57. Let ๐ = {0, ๐, ๐} be the BCK-algebra with in Table 8 (Cayley). ฬ ๐ด) be a Consider a set of parameters ๐ด = {๐1 , ๐2 }. Let (๐ป, hesitant fuzzy soft set over ๐ which is described in Table 9. ฬ ๐ด) is a hesitant anti-fuzzy soft ideal (subThen (๐ป, ฬ ๐ด) is a algebra) of ๐. Routine calculations give that (๐ป, ฬ 1 ] fl hesitant anti-fuzzy soft ideal (subalgebra) of ๐, but ๐ป[๐ ฬ {(๐ฅ, โ๐ป[๐ (๐ฅ)) | ๐ฅ โ ๐} and ๐ป[๐ ] fl {(๐ฅ, โ (๐ฅ)) | ๐ฅ โ ๐} ฬ 1] ฬ 2] 2 ๐ป[๐ on ๐ is not hesitant anti-fuzzy soft commutative ideal of ๐, because โ๐ป[๐ ฬ 1 ] (๐ โ (๐ โ (๐ โ ๐))) = [0.2, 0.7) โฬธ (0.2, 0.4) = โ๐ป[๐ ฬ 1 ] ((๐ โ ๐) โ 0) โช โ๐ป[๐ ฬ 1 ] (0) .
= โ๐ป[๐] ฬ (๐ง) .
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Also,
Putting ๐ฆ = 0 and replacing ๐ง = ๐ฆ, we obtain โ๐ป[๐] ฬ (๐ฅ) โ โ๐ป[๐] ฬ (๐ฆ) .
๐ 0 0 0
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ฬ ๐ด) on ๐ is a hesitant anti-fuzzy commutative ideal of ๐. If (๐ป, is an ๐-hesitant anti-fuzzy soft commutative ideal over ๐, for ฬ ๐ด) is a hesitant anti-fuzzy soft all ๐ โ ๐ด, we say that (๐ป, commutative ideal over ๐.
โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง)
Table 8
โ๐ป[๐ ฬ 2 ] (๐ โ (๐ โ (๐ โ ๐))) = (0, 0.5) โฬธ (0, 0.2)
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= โ๐ป[๐ ฬ 2 ] ((๐ โ ๐) โ 0) โช โ๐ป[๐ ฬ 2 ] (0)
The proof is complete.
(85)
Theorem 54. Any hesitant anti-fuzzy soft commutative ideal of BCK-algebra ๐ is hesitant anti-fuzzy soft ideal of ๐.
The condition in the following theorem makes converse Theorem 54 valid.
ฬ ๐ด) be a Proof. Let ๐ฅ, ๐ฆ, ๐ง โ ๐ and let ๐ โ ๐ด. And let (๐ป, hesitant anti-fuzzy soft commutative ideal of a BCK-algebra ๐, so we have
Theorem 58. In a commutative BCK-algebra ๐. Every hesitant anti-fuzzy soft ideal of ๐ is a hesitant anti-fuzzy soft commutative ideal of ๐.
โ๐ป[๐] ฬ (๐ฅ) = โ๐ป[๐] ฬ (๐ฅ โ (0 โ (0 โ ๐ฅ))) โ โ๐ป[๐] ฬ ((๐ฅ โ 0) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง)
(83)
= โ๐ป[๐] ฬ (๐ฅ โ ๐ง) โช โ๐ (๐ง) ,
ฬ ๐ด) is a hesitant anti-fuzzy soft ideal of Proof. Suppose that (๐ป, ฬ ๐ด) satisfies a BCK-algebra ๐. It is sufficient to show that (๐ป, condition (HAFCI2). Let ๐ฅ, ๐ฆ, ๐ง โ ๐. Then ((๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ ((๐ฅ โ ๐ฆ) โ ๐ง)) โ ๐ง
ฬ ๐ด) is a hesitant antifor all ๐ฅ, ๐ง โ ๐ and ๐ โ ๐ด. Hence (๐ป, fuzzy soft ideal of ๐.
= ((๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ ๐ง) โ ((๐ฅ โ ๐ฆ) โ ๐ง) โค (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ (๐ฅ โ ๐ฆ)
Combining Theorems 22 and 54 yields the following result. Corollary 55. Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebra ๐ is a hesitant anti-fuzzy soft subalgebra of ๐.
(86)
= (๐ฅ โ (๐ฅ โ ๐ฆ)) โ (๐ฆ โ (๐ฆ โ ๐ฅ)) = 0; that is, (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ ((๐ฅ โ ๐ฆ) โ ๐ง) โค ๐ง.
(87)
12
Mathematical Problems in Engineering
It follows from Proposition 20(c) that โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) ,
(88)
ฬ ๐ด) is a hesitant antifor all ๐ฅ, ๐ฆ, ๐ง โ ๐ and ๐ โ ๐ด. Thus (๐ป, fuzzy soft commutative ideal of a BCK-algebra ๐, and the proof is complete. ฬ ๐ด) be a hesitant anti-fuzzy soft ideal of Theorem 59. Let (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft a BCK-algebra ๐. Then (๐ป, commutative ideal of ๐ if and only if it satisfies the condition โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ๐ฅ, ๐ฆ โ ๐, ๐ โ ๐ด.
โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ 0) โช โ๐ป[๐] ฬ (0) = โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) .
โ๐ฅ, ๐ฆ, ๐ง โ ๐, ๐ โ ๐ด.
(90)
โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) ,
(91)
(92)
Observing ๐ฅ โ ๐ฆ โค ๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)) and using Proposition 20(a), we have โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) for all ๐ฅ, ๐ฆ โ ๐ and ๐ โ ๐ด. Hence, Theorem 59 can be improved as follows. ฬ ๐ด) of a BCKTheorem 60. A hesitant anti-fuzzy soft ideal (๐ป, algebra ๐ is a hesitant anti-fuzzy soft commutative ideal of ๐ if and only if it satisfies the identity
โ๐ฅ, ๐ฆ โ ๐, ๐ โ ๐ด.
= โ๐ป[๐] ฬ ((๐ฅ โ ๐ง))
(94)
โ (๐ฅ โ (๐ฅ โ ๐ง)) , [by Proposition 6 (ii)]
ฬ ๐ด) is a hesitant anti-fuzzy soft positive implicaHence (๐ป, tive ideal of ๐. Also, By Propositions 20 and 6(iii) and Theorem 34(iii), we have โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) (95)
โ (๐ฆ โ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)))) โ โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) . ฬ ๐ด) is hesitant antiIt follows from Theorem 59 that (๐ป, fuzzy soft commutative ideal of ๐. ฬ ๐ด) is both hesitant antiConversely, suppose that (๐ป, fuzzy soft positive implicative ideal of ๐ and hesitant antifuzzy soft commutative ideal of ๐. Since (๐ฆโ(๐ฆโ๐ฅ))โ(๐ฆโ๐ฅ) โค ๐ฅ โ (๐ฆ โ ๐ฅ), it follows from Proposition 20(a) that โ๐ป[๐] ฬ ((๐ฆ โ (๐ฆ โ ๐ฅ)) โ (๐ฆ โ ๐ฅ)) โ โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) (96)
ฬ ๐ด) is hesitant antifor all ๐ฅ, ๐ฆ, ๐ง โ ๐ and ๐ โ ๐ด. Hence (๐ป, fuzzy soft commutative ideal of ๐, which completes the proof.
โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) = โ๐ป[e] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)))
โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ง) โ ๐ง)
= โ๐ป[๐] ฬ ((๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))))
Combining (91) and (89), we obtain โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)))
โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ฆ โ ๐ง)
= โ๐ป[๐] ฬ (๐ฅ โ ๐ง) , [by Theorem 34 (iii)] .
ฬ ๐ด) satisfies condition (89) as Conversely, assume that (๐ป, ฬ ๐ด) is a hesitant anti-fuzzy soft ideal of a BCK-algebra ๐. (๐ป, Hence โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง)
ฬ ๐ด) is a hesitant anti-fuzzy soft Proof. Assume that (๐ป, implicative ideal. By Propositions 6(i) and 20(c), for all ๐ฅ, ๐ฆ, ๐ง โ ๐ and ๐ โ ๐ด, we have
(89)
ฬ ๐ด) Proof. Let ๐ฅ, ๐ฆ, ๐ง โ ๐ and let ๐ โ ๐ด. And suppose that (๐ป, is a hesitant anti-fuzzy soft commutative ideal of ๐. Taking ๐ง = 0, we have โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)))
ฬ ๐ด) of a BCKTheorem 61. A hesitant anti-fuzzy soft ideal (๐ป, algebra ๐ is a hesitant anti-fuzzy soft implicative ideal if and ฬ ๐ด) is both hesitant anti-fuzzy soft commutative ideal only if (๐ป, and hesitant anti-fuzzy soft positive implicative ideal.
Using Theorem 49, we have โ๐ป[๐] ฬ ((๐ฆ โ (๐ฆ โ ๐ฅ)) โ (๐ฆ โ ๐ฅ)) = โ๐ป[๐] ฬ (๐ฆ โ (๐ฆ โ ๐ฅ)) and so โ๐ป[๐] ฬ (๐ฆ โ (๐ฆ โ ๐ฅ)) โ โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) .
(97)
On the other hand, ๐ฅโ๐ฆ โค ๐ฅโ(๐ฆโ๐ฅ) implies that โ๐ป[๐] ฬ (๐ฅโ ฬ ๐ฆ) โ โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)). Since (๐ป, ๐ด) is a hesitant anti-fuzzy soft commutative ideal of ๐, By Theorem 60, we have โ๐ป[๐] ฬ (๐ฅ โ ๐ฆ) = โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) .
(98)
Hence, โ๐ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ โ๐ (๐ฅ โ (๐ฆ โ ๐ฅ)) .
(99)
Combining (97), we obtain โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ ๐ฅ)) โ โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)))
(93)
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.
โช โ๐ป[๐] ฬ (๐ฆ โ (๐ฆ โ ๐ฅ))
(100)
โ โ๐ป[๐] ฬ (๐ฅ) . ฬ ๐ด) is a hesitant anti-fuzzy soft implicative ideal of So, (๐ป, ๐ by Theorem 34. The proof is complete.
Mathematical Problems in Engineering
13
ฬ ๐ด) be hesitant anti-fuzzy soft commutaTheorem 62. Let (๐ป, tive ideal of ๐ if and only if for each ๐ โ ๐([0, 1]) the hesitant ฬ ๐ด)๐ of (๐ป, ฬ ๐ด) is a commutative anti-fuzzy soft ๐-level set (๐ป, ideal of ๐. ฬ ๐ด) is a hesitant anti-fuzzy soft comProof. Suppose that (๐ป, mutative ideal of ๐ and for every ๐ โ ๐([0, 1]) define the sets ๐
ฬ ๐ด) fl {๐ฅ โ ๐ | โ ฬ (๐ฅ) โ ๐} (๐ป, ๐ป[๐]
โ๐ โ ๐ด.
(101)
ฬ ๐ด)๐ =ฬธ โ, let ๐ฅ โ (๐ป, ฬ ๐ด)๐ โ โ ฬ (๐ฅ) โ ๐. By Since (๐ป, ๐ป[๐] definition, we have โ๐ป[๐] ฬ (0) โ โ๐ป[๐] ฬ (๐ฅ), for all ๐ โ ๐ด and ฬ ๐ด)๐ . Let ๐ฅ, ๐ฆ, ๐ง โ ๐ be such ๐ฅ โ ๐, implying that 0 โ (๐ป, ฬ ๐ด)๐ and ๐ง โ (๐ป, ฬ ๐ด)๐ , implying that that (๐ฅ โ ๐ฆ) โ ๐ง โ (๐ป, ฬ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โ ๐ and โ๐ป[๐] ฬ (๐ง) โ ๐. Since (๐ป, ๐ด) is a hesitant anti-fuzzy soft commutative ideal over ๐, it follows that โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง) โ ๐.
(102)
ฬ ๐ด)๐ , and (๐ป, ฬ ๐ด)๐ is a Namely, ๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ)) โ (๐ป, commutative ideal of ๐. ฬ ๐ด)๐ is a commutative ideal Conversely, suppose that (๐ป, of ๐ for all ๐ โ ๐([0, 1]). Put โ๐ป[๐] ฬ (๐ฅ) = ๐ for any ๐ฅ โ ๐ and ๐ ฬ ๐ โ ๐ด. Hence 0 โ (๐ป, ๐ด) โ โ๐ป[๐] ฬ (0) โ ๐ = โ๐ป[๐] ฬ (๐ฅ), for all ๐ โ ๐ด and ๐ฅ โ ๐. Now we prove that โ๐ป[๐] ฬ (๐ฅ โ (๐ฆ โ (๐ฆ โ ๐ฅ))) โ โ๐ป[๐] ฬ ((๐ฅ โ ๐ฆ) โ ๐ง) โช โ๐ป[๐] ฬ (๐ง). In contrast, there exists ๐ฅ0 , ๐ฆ0 , ๐ง0 โ ๐ such that โ๐ป[๐] ฬ (๐ฅ0 โ (๐ฆ0 โ (๐ฆ0 โ ๐ฅ0 ))) โ โ๐ป[๐] ฬ ((๐ฅ0 โ ๐ฆ0 ) โ ๐ง0 ) โช โ๐ป[๐] ฬ (๐ง0 ) .
(103)
Taking ๐0 =
1 {โ ฬ (๐ฅ โ (๐ฆ0 โ (๐ฆ0 โ ๐ฅ0 ))) 2 ๐ป[๐] 0
(104)
+ {โ๐ป[๐] ฬ ((๐ฅ0 โ ๐ฆ0 ) โ ๐ง0 ) โช โ๐ป[๐] ฬ (๐ง0 )}} , it follows that โ๐ป[๐] ฬ ((๐ฅ0 โ ๐ฆ0 ) โ ๐ง0 ) โ ๐0 and โ๐ป[๐] ฬ (๐ง0 ) โ ๐0 , ๐0 ฬ ฬ ๐ด)๐0 . which imply that ((๐ฅ0 โ๐ฆ0 )โ๐ง0 ) โ (๐ป, ๐ด) and ๐ง0 โ (๐ป, ๐ 0 ฬ ๐ด) is a commutative ideal of ๐, and thus ๐ฅ0 โ (๐ฆ0 โ But (๐ป, ฬ ๐ด)๐0 or โ ฬ (๐ฅ0 โ (๐ฆ0 โ (๐ฆ0 โ ๐ฅ0 ))) โ ๐0 , (๐ฆ0 โ ๐ฅ0 )) โ (๐ป, ๐ป[๐] ฬ ๐ด) is a hesitant antiand this is a contradiction. Therefore (๐ป, fuzzy soft commutative ideal of ๐.
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 BCKalgebras. 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.
References [1] D. Molodtsov, โSoft set theory- first results,โ Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19โ31, 1999. [2] P. K. Maji, R. Biswas, and A. R. Roy, โFuzzy soft sets,โ Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589โ602, 2001. [3] L. A. Zadeh, โFuzzy sets,โ Information and Control, vol. 8, no. 3, pp. 338โ353, 1965. [4] V. Torra, โHesitant fuzzy sets,โ International Journal of Intelligent Systems, vol. 25, no. 6, pp. 529โ539, 2010. [5] K. V. Babitha and S. J. John, โHesitant fuzzy soft sets,โ Journal of New Results in Science, vol. 3, pp. 98โ107, 2013. [6] Y. B. Jun, S. S. Ahn, and G. Muhiuddin, โHesitant fuzzy soft subalgebras and ideals in BCK/BCI -algebras,โ Scientific World Journal, vol. 2014, Article ID 763929, 2014. [7] Y. B. Jun and S. S. Ahn, โHesitant fuzzy sets theory applied to in BCK/BCI- algebras,โ J. Computational Analysis and Applications, vol. 20, no. 4, pp. 635โ646, 2016. [8] K. Isยดeki, โAn algebraic related with a propositional calclus,โ Proc. Japan Academy, vol. 42, pp. 26โ29, 1996. [9] J. Meng, Y. B. Jun, and H. S. Kim, โFuzzy implicative ideals of BCK-algebras,โ Fuzzy Sets and Systems, vol. 89, pp. 243โ248, 1997.
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