Hesitant Anti-Fuzzy Soft Set in BCK-Algebras

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Apr 30, 2017 - study new types of hesitant anti-fuzzy soft ideals (implicative, positive implicative, ... fuzzy soft ideal in BCK-algebras and give some basic rela-.
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

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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 Zadehโ€™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 Morganโ€™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 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‘‹; โˆ—, 0) of type (2, 0) is said to be a BCK-algebra if it satisfies the axioms: for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹, (BCK-1) ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— (๐‘ฅ โˆ— ๐‘ง)) โˆ— (๐‘ง โˆ— ๐‘ฆ) = 0, (BCK-2) (๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ฆ)) โˆ— ๐‘ฆ = 0, (BCK-3) ๐‘ฅ โˆ— ๐‘ฅ = 0, (BCK-4) 0 โˆ— ๐‘ฅ = 0, (BCK-5) ๐‘ฅ โˆ— ๐‘ฆ = 0 and ๐‘ฆ โˆ— ๐‘ฅ = 0 imply that ๐‘ฅ = ๐‘ฆ.

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Mathematical Problems in Engineering

Define a binary relation โ‰ค on ๐‘‹ by letting ๐‘ฅ โ‰ค ๐‘ฆ if and only if ๐‘ฅ โˆ— ๐‘ฆ = 0. Then (๐‘‹; โ‰ค) is a partially ordered set with the least element 0. In any BCK-algebra ๐‘‹, the following hold: (1) (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง = (๐‘ฅ โˆ— ๐‘ง) โˆ— ๐‘ฆ. (2) ๐‘ฅ โˆ— ๐‘ฆ โ‰ค ๐‘ฅ. (3) ๐‘ฅ โˆ— 0 = ๐‘ฅ.

โˆ— 0 ๐‘Ž ๐‘ ๐‘

0 0 ๐‘Ž ๐‘ ๐‘

๐‘Ž 0 0 ๐‘ ๐‘

๐‘ 0 ๐‘Ž 0 ๐‘Ž

๐‘ 0 0 0 0

Proposition 5 (see [8]). Let ๐ป๐ด fl {(๐‘ฅ, โ„Ž๐ด (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} be an ๐ด-hesitant anti-fuzzy ideal of ๐‘‹. Then the following hold: for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐ด,

(4) (๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง) โ‰ค ๐‘ฅ โˆ— ๐‘ฆ. (5) ๐‘ฅ โˆ— (๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ฆ)) = ๐‘ฅ โˆ— ๐‘ฆ. (6) ๐‘ฅ โ‰ค ๐‘ฆ implies that ๐‘ฅ โˆ— ๐‘ง โ‰ค ๐‘ฆ โˆ— ๐‘ง and ๐‘ง โˆ— ๐‘ฆ โ‰ค ๐‘ง โˆ— ๐‘ฅ, for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. A nonempty subset ๐ผ of ๐‘‹ is called a subalgebra of ๐‘‹ if, for any ๐‘ฅ, ๐‘ฆ โˆˆ ๐ผ, ๐‘ฅโˆ—๐‘ฆ โˆˆ ๐ผ. That is, it is closed under the binary operation โˆ— of ๐‘‹. A nonempty subset ๐ผ of ๐‘‹ is called an ideal of ๐‘‹ if (๐ผ1 ) 0 โˆˆ ๐ผ; (๐ผ2 ) ๐‘ฅ โˆ— ๐‘ฆ โˆˆ ๐ผ and ๐‘ฆ โˆˆ ๐ผ imply that ๐‘ฅ โˆˆ ๐ผ. A nonempty subset ๐ผ of ๐‘‹ is called an implicative ideal if it satisfies (๐ผ1 ) and (๐ผ3 ) ๐‘ฅ โˆˆ ๐ผ whenever (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง โˆˆ ๐ผ and ๐‘ง โˆˆ ๐ผ. It is called a commutative ideal if it satisfies (๐ผ1 ) and (๐ผ4 ) ๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆˆ ๐ผ whenever (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง โˆˆ ๐ผ and ๐‘ง โˆˆ ๐ผ; and it is called a positive implicative ideal if it satisfies (๐ผ1 ) and (๐ผ5 ) ๐‘ฅ โˆ— ๐‘ง โˆˆ ๐ผ whenever (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง โˆˆ ๐ผ and ๐‘ฆ โˆ— ๐‘ง โˆˆ ๐ผ. A BCK-algebra ๐‘‹ is said to be implicative if it satisfies โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ : ๐‘ฅ = ๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ). A BCK-algebra ๐‘‹ is said to be positive implicative if it satisfies โˆ€๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ : (๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง) = (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง. A BCK-algebra ๐‘‹ is said to be commutative if it satisfies โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ : ๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ฆ) = ๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ). Definition 1 (see [3]). Let ๐‘† be a set. A fuzzy set in ๐‘† is a function ๐œ‡ : ๐‘† โ†’ [0, 1] Definition 2 (see [2]). Let ๐‘ˆ be an initial universe set and let ๐ธ be a set of parameters. Let F(๐‘ˆ) denote the set of all fuzzy ฬƒ ๐ด) is called a fuzzy soft set over ๐‘ˆ, where sets in ๐‘ˆ. Then (๐น, ฬƒ ๐ด โŠ† ๐ธ and ๐น is a mapping given by ๐นฬƒ : ๐ด โ†’ F(๐‘ˆ). Definition 3 (see [4, 7]). Let ๐ธ be a reference set. A hesitant fuzzy set on ๐ธ is defined in terms of a function that when applied to ๐ธ returns a subset of [0, 1] which can be viewed as the following mathematical representation: ๐ป๐ธ fl {(๐‘’, โ„Ž๐ธ (๐‘’)) : ๐‘’ โˆˆ ๐ธ} ,

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Definition 4 (see [7]). Given a nonempty subset ๐ด of ๐‘‹, a hesitant fuzzy set ๐ป๐‘‹ fl {(๐‘ฅ; โ„Ž๐‘‹ (๐‘ฅ) : ๐‘ฅ โˆˆ ๐‘‹} on ๐‘‹ satisfying the condition โˆ€๐‘ฅ โˆ‰ ๐ด

(a) if ๐‘ฅ โ‰ค ๐‘ฆ, then โ„Ž๐ด (๐‘ฅ) โŠ† โ„Ž๐ด (๐‘ฆ), which means that โ„Ž๐ด preserves the order, (b) if ๐‘ฅ โˆ— ๐‘ฆ โ‰ค ๐‘ง, then โ„Ž๐ด (๐‘ฅ) โŠ† โ„Ž๐ด (๐‘ฆ) โˆช โ„Ž๐ด (๐‘ง). Proposition 6 (see [9]). In a BCK-algebra ๐‘‹, the following hold: for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹, (i) ((๐‘ฅ โˆ— ๐‘ง) โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง) โ‰ค (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง, (ii) (๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ง)) = (๐‘ฅ โˆ— ๐‘ง) โˆ— ๐‘ง, (iii) (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โˆ— (๐‘ฆ โˆ— (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)))) โ‰ค ๐‘ฅ โˆ— ๐‘ฆ. Definition 7 (see [5, 6]). Denote by ๐ปF(๐‘ˆ) the set of all ฬƒ ๐ด) is called a hesitant fuzzy hesitant fuzzy sets. A pair (๐ป, ฬƒ is a mapping given soft set over a reference set ๐‘ˆ, where ๐ป ฬƒ by ๐ป : ๐ด โ†’ ๐ปF(๐‘ˆ).

3. Hesitant Anti-Fuzzy Soft Subalgebras Definition 8. Given a nonempty subset (subalgebra as much as possible) ๐ด of ๐‘‹, let ๐ป๐ด fl {(๐‘ฅ, โ„Ž๐ด (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} be an ๐ดhesitant fuzzy set on ๐‘‹. Then ๐ป๐ด fl {(๐‘ฅ, โ„Ž๐ด (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} is called a hesitant anti-fuzzy subalgebra of ๐‘‹ related to ๐ด (briefly, ๐ด-hesitant anti-fuzzy subalgebra of ๐‘‹) if it satisfies the following condition: (โ„Ž๐ด (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ด (๐‘ฅ) โˆช โ„Ž๐ด (๐‘ฆ))

(โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐ด) .

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is called a hesitant fuzzy set related to ๐ด (briefly, ๐ด-hesitant fuzzy set) on ๐‘‹ and is represented by ๐ป๐ด fl {(๐‘ฅ, โ„Ž๐ด (๐‘ฅ)) : ๐‘ฅ โˆˆ ๐‘‹}; โ„Ž๐ด is a mapping from ๐‘‹ to ๐‘([0, 1]) with โ„Ž๐ด (๐‘ฅ) = ๐œ™, for all ๐‘ฅ โˆ‰ ๐ด.

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An ๐ด-hesitant anti-fuzzy subalgebra of ๐‘‹ with ๐ด = ๐‘‹ is called a hesitant anti-fuzzy subalgebra of ๐‘‹. Definition 9. For a subset ๐ด of ๐ธ, a hesitant fuzzy soft set ฬƒ ๐ด) over ๐‘‹ is called a hesitant anti-fuzzy soft subalgebra (๐ป, based on ๐‘’ โˆˆ ๐ด (briefly, ๐‘’-hesitant anti-fuzzy soft subalgebra) over ๐‘‹ if the hesitant fuzzy set, ฬƒ [๐‘’] fl {(๐‘ฅ, โ„Ž ฬƒ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} , ๐ป ๐ป[๐‘’]

where โ„Ž๐ธ : ๐ธ โ†’ ๐‘([0, 1]).

โ„Ž๐‘‹ (๐‘ฅ) = ๐œ™;

Table 1

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ฬƒ ๐ด) is an on ๐‘‹ is a hesitant anti-fuzzy subalgebra of ๐‘‹. If (๐ป, ๐‘’-hesitant anti-fuzzy soft subalgebra over ๐‘‹, for all ๐‘’ โˆˆ ๐ด, we ฬƒ ๐ด) is a hesitant anti-fuzzy soft subalgebra. say that (๐ป, Example 10. Let ๐‘‹ = {0, ๐‘Ž, ๐‘, ๐‘} be a BCK-algebra in Table 1 (Cayley). Consider a set of parameters ๐ธ fl {๐‘’1 , ๐‘’2 , ๐‘’3 , ๐‘’4 }. Let ฬƒ ๐ด) be a hesitant fuzzy soft set over ๐‘‹, where ๐ด = (๐ป, {๐‘’1 , ๐‘’2 , ๐‘’3 }, which is given in Table 2. It is routine to verify that

Mathematical Problems in Engineering

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ฬƒ ๐ด). Table 2: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬƒ ๐ป ๐‘’1 ๐‘’2 ๐‘’3

0 (0.2, 0.3) {0.3} {0.4}

๐‘Ž [0.2, 0.8] {0.3} [0.3, 0.7]

๐‘ [0.2, 0.8] [0.1, 0, 2) โˆช (0, 2, 0.8] [0.3, 0.5]

๐‘ [0.2, 0.8] [0.1, 0.9] [0.3, 0.7]

ฬƒ 1 ], ๐ป[๐‘’ ฬƒ 2 ] and ๐ป[๐‘’ ฬƒ 3 ] are hesitant anti-fuzzy subalgebra ๐ป[๐‘’ ฬƒ ๐ด) over ๐‘‹ based on parameters ๐‘’1 , ๐‘’2 , and ๐‘’3 . Therefore (๐ป, is a hesitant anti-fuzzy soft subalgebra over ๐‘‹. ฬƒ ๐ด) is a hesitant anti-fuzzy soft subalgeProposition 11. If (๐ป, bra over ๐‘‹, then โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) , โˆ€๐‘ฅ โˆˆ ๐‘‹,

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where ๐‘’ is any parameter in ๐ด. Proof. For any ๐‘ฅ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด, we have โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) .

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This completes the proof. ฬƒ ๐ด) be a hesitant anti-fuzzy soft subalgeTheorem 12. Let (๐ป, ฬƒ ๐ต , ๐ต) is a hesitant anti-fuzzy soft bra over ๐‘‹. If ๐ต โŠ† ๐ด, then (๐ป| subalgebra over ๐‘‹. ฬƒ ๐ด) is a hesitant anti-fuzzy soft subProof. Suppose that (๐ป, ฬƒ๐ด is a hesitant anti-fuzzy subalgebra algebra over ๐‘‹. Then ๐ป of ๐‘‹, โˆ€๐‘ฅ โˆˆ ๐ด. Since ๐ต โŠ† ๐ด, โ„Ž๐ต (๐‘) is a hesitant anti-fuzzy ฬƒ ๐ต , ๐ต) is a hesitant antisubalgebra of ๐‘‹, โˆ€๐‘ โˆˆ ๐ต. Hence, (๐ป| fuzzy soft subalgebra over ๐‘‹. The following example shows that there exists a hesitant ฬƒ ๐ด) over ๐‘‹ such that fuzzy soft set (๐ป, ฬƒ ๐ด) is not a hesitant anti-fuzzy soft subalgebra over (i) (๐ป, ๐‘‹, ฬƒ ๐ต , ๐ต) is a (ii) there exists a subset ๐ต of ๐ด such that (๐ป| hesitant anti-fuzzy soft subalgebra over ๐‘‹. Example 13. Let ๐‘‹ = {0, ๐‘Ž, ๐‘, ๐‘} be a BCK-algebra in Table 3 (Cayley). ฬƒ ๐ด) be Consider a set of parameters ๐ด = {๐‘’1 , ๐‘’2 , ๐‘’3 }. Let (๐ป, a hesitant fuzzy soft set over ๐‘‹ which is described in Table 4. ฬƒ 3 ] fl {(๐‘ฅ, โ„Ž ฬƒ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} on ๐‘‹ is not hesitant Then ๐ป[๐‘’ ๐ป[๐‘’3 ] anti-fuzzy subalgebra of ๐‘‹, because โ„Ž๐ป[๐‘’ ฬƒ 3 ] (๐‘Ž โˆ— ๐‘Ž) = โ„Ž๐ป[๐‘’ ฬƒ 3 ] (0) = {0.1} โŠ†ฬธ (0.1, 0.2) = โ„Ž๐ป[๐‘’ ฬƒ 3 ] (๐‘Ž) โˆช โ„Ž๐ป[๐‘’ ฬƒ 3 ] (๐‘Ž) .

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ฬƒ ๐ด) is not hesitant anti-fuzzy soft subalgebra Therefore (๐ป, ฬƒ ๐ต , ๐ต) is a hesitant of ๐‘‹. But if we take ๐ต = {๐‘’1 , ๐‘’2 }, then (๐ป| anti-fuzzy soft subalgebra over ๐‘‹.

Table 3 โˆ— 0 ๐‘Ž ๐‘ ๐‘

0 0 ๐‘Ž ๐‘ ๐‘

๐‘Ž 0 0 ๐‘ ๐‘

๐‘ 0 ๐‘Ž 0 ๐‘

๐‘ 0 0 0 0

ฬƒ ๐ด). Table 4: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬƒ ๐ป ๐‘’1 ๐‘’2 ๐‘’3

0 (0.3, 0.5) [0.3, 0.6) {0.1}

๐‘Ž (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]

ฬƒ ๐ด) be a hesitant fuzzy soft set over ๐‘‹. Definition 14. Let (๐ป, ฬƒ ๐ด)๐œ€ = (๐ป ฬƒ๐œ– , ๐ด) is called a For each ๐œ€ โˆˆ ๐‘ƒ([0, 1]), the set (๐ป, ฬƒ ๐ด), where ๐ป ฬƒ๐œ– = {๐‘ฅ โˆˆ ๐‘‹ : hesitant anti-๐œ€-level soft set of (๐ป, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† ๐œ€} for all ๐‘’ โˆˆ ๐ด. ฬƒ ๐ด) be a hesitant fuzzy soft set over Theorem 15. Let (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft subalgebra of ๐‘‹ if and ๐‘‹. (๐ป, ฬƒ ๐ด)๐œ€ is a subalgebra over ๐‘‹ for each ๐œ€ โˆˆ ๐‘ƒ([0, 1]). only if (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft subalProof. Suppose that (๐ป, ฬƒ๐œ– , gebra of ๐‘‹. For each ๐œ€ โˆˆ ๐‘ƒ([0, 1]), ๐‘’ โˆˆ ๐ด and ๐‘ฅ1 , ๐‘ฅ2 โˆˆ ๐ป โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ1 ) โŠ† ๐œ€ and โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ2 ) โŠ† ๐œ€. Thus โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ1 โˆ— ๐‘ฅ2 ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ1 ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† ๐œ€.

(8)

ฬƒ๐œ– . Hence, (๐ป, ฬƒ ๐ด)๐œ€ is a This implies that ๐‘ฅ1 โˆ— ๐‘ฅ2 โˆˆ ๐ป subalgebra over ๐‘‹. ฬƒ ๐ด)๐œ€ is a subalgebra over ๐‘‹ for Conversely, assume that (๐ป, each ๐œ€ โˆˆ ๐‘ƒ([0, 1]). For each ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, let โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = ฬƒ๐œ– ๐œ€๐‘ฅ and let โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = ๐œ€๐‘ฆ . Take ๐œ€ = ๐œ€๐‘ฅ โˆช ๐œ€๐‘ฆ . Then ๐‘ฅ, ๐‘ฆ โˆˆ ๐ป and ๐œ– ฬƒ . Hence, so ๐‘ฅ โˆ— ๐‘ฆ โˆˆ ๐ป โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† ๐œ€ = ๐œ€๐‘ฅ โˆช ๐œ€๐‘ฆ = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

(9)

ฬƒ ๐ด) is a hesitant anti-fuzzy subalgebra over Therefore (๐ป, ฬƒ ๐ด) is a hesitant ๐‘‹. Then, by Definition 8, we conclude that (๐ป, 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) ๐ด of ๐‘‹, let ๐ป๐ด fl {(๐‘ฅ, โ„Ž๐ด (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} be an ๐ด-hesitant fuzzy set on ๐‘‹. Then ๐ป๐ด fl {(๐‘ฅ, โ„Ž๐ด (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} is called a hesitant anti-fuzzy ideal of ๐‘‹ related to ๐ด (briefly, ๐ด-hesitant anti-fuzzy ideal of ๐‘‹) if it satisfies the following conditions: (HAFI1) โ„Ž๐ด (0) โŠ† โ„Ž๐ด (๐‘ฅ). (HAFI2) โ„Ž๐ด (๐‘ฅ) โŠ† โ„Ž๐ด (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ด (๐‘ฆ) for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐ด. An ๐ด-hesitant anti-fuzzy ideal of ๐‘‹ with ๐ด = ๐‘‹ is called a hesitant anti-fuzzy ideal of ๐‘‹.

4

Mathematical Problems in Engineering

ฬƒ ๐ด). Table 5: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬƒ ๐ป 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)

ฬƒ ๐ด) be a hesitant fuzzy soft set over ๐‘‹, Definition 17. Let (๐ป, ฬƒ ๐ด) is called a where ๐ด is a subset of ๐ธ. Given ๐‘’ โˆˆ ๐ด, (๐ป, hesitant anti-fuzzy soft ideal based on ๐‘’ (briefly, ๐‘’-hesitant anti-fuzzy soft ideal) over ๐‘‹, if the hesitant fuzzy set, ฬƒ [๐‘’] fl {(๐‘ฅ, โ„Ž ฬƒ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} , ๐ป ๐ป[๐‘’]

(10)

ฬƒ ๐ด) is an ๐‘’on ๐‘‹ is a hesitant anti-fuzzy ideal of ๐‘‹. If (๐ป, hesitant anti-fuzzy soft ideal over ๐‘‹ for all ๐‘’ โˆˆ ๐ด, we say ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal over ๐‘‹. that (๐ป, Example 18. Let ๐‘‹ fl {apple, banana, carrot, peach, radish} be a reference set, and consider a soft machine โ—Š which produces the following products: {apple if ๐‘ฅ โˆˆ {carrot, peach, radish} apple โ—Š๐‘ฅ = { carrot if ๐‘ฅ โˆˆ {apple, banana} {

(13)

ฬƒ๐œ– , and this implies that (๐ป, ฬƒ ๐ด)๐œ€ ฬƒ๐œ– and ๐‘ฅ โˆˆ ๐ป Hence, 0 โˆˆ ๐ป is an ideal over ๐‘‹. ฬƒ ๐ด)๐œ€ is an ideal over ๐‘‹ for Conversely, assume that (๐ป, each ๐œ€ โˆˆ ๐‘ƒ([0, 1]). For each ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, let ฬƒ๐œ– ฬƒ ๐œ€ โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = ๐œ€๐‘ฅ , and then ๐‘ฅ โˆˆ ๐ป . Since (๐ป, ๐ด) is an ideal ฬƒ๐œ– and so โ„Ž ฬƒ (0) โŠ† ๐œ€๐‘ฅ = โ„Ž ฬƒ (๐‘ฅ). over ๐‘‹, we have 0 โˆˆ ๐ป ๐ป[๐‘’] ๐ป[๐‘’] Let โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) = ๐œ€๐‘ฅโˆ—๐‘ฆ and let โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = ๐œ€๐‘ฆ . If we take ฬƒ๐œ– and ๐‘ฆ โˆˆ ๐ป ฬƒ๐œ– which imply ๐œ€ = ๐œ€๐‘ฅโˆ—๐‘ฆ โˆช ๐œ€๐‘ฆ , then ๐‘ฅ โˆ— ๐‘ฆ โˆˆ ๐ป ฬƒ๐œ– . Thus, that ๐‘ฅ โˆˆ ๐ป โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† ๐œ€ = ๐œ€๐‘ฅโˆ—๐‘ฆ โˆช ๐œ€๐‘ฆ = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) . (14) ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal of ๐‘‹. Therefore (๐ป, ฬƒ ๐ด) Proposition 20. Every hesitant anti-fuzzy soft ideal (๐ป, over a BCK/BCI-algebra ๐‘‹ satisfies the following condition for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹: (b) โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง โˆ— ๐‘ฆ). (c) If ๐‘ฅ โˆ— ๐‘ฆ โ‰ค ๐‘ง, then โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง). (d) If โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (0), then โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ).

(11)

{peach if ๐‘ง โˆˆ {carrot, radish} peach โ—Š๐‘ง = { carrot if ๐‘ง โˆˆ {apple, banana, peach} {

(e) โ„Ž๐ป[๐‘’] ฬƒ (0 โˆ— (0 โˆ— ๐‘ฅ)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ). Proof. Let ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. ฬƒ ๐ด) is a hesitant (a) If ๐‘ฅ โ‰ค ๐‘ฆ, then ๐‘ฅ โˆ— ๐‘ฆ = 0. Since (๐ป, anti-fuzzy soft ideal of ๐‘‹,

{radish if ๐‘ข โˆˆ {carrot, peach} radish โ—Š๐‘ข = { carrot if ๐‘ข โˆˆ {apple, banana, radish} {

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

carrot โ—ŠV = carrot โˆ€V โˆˆ ๐‘‹. Then ๐‘‹ is a BCK-algebra under the soft machine โ—Š. Consider a set of parameters ๐ด fl {Cat, Cow, Dog, Horse}; let ฬƒ ๐ด) be a hesitant fuzzy soft set over ๐‘‹ which is described (๐ป, in Table 5. ฬƒ ๐ด) is a hesitant anti-fuzzy It is routine to verify that (๐ป, soft ideal over ๐‘‹ based on parameters โ€œcat,โ€ โ€œcow,โ€ and โ€œdog.โ€ ฬƒ ๐ด) is not a hesitant anti-fuzzy soft ideal of ๐‘‹ based on But (๐ป, parameter โ€œhorseโ€ because ฬƒ [horse] (peach) = [0.3, 0.9) โŠ†ฬธ [0.3, 0.4) ๐ป ฬƒ [horse] (peachโ—Šbanana) =๐ป

โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) โŠ† ๐œ€.

(a) If ๐‘ฅ โ‰ค ๐‘ฆ, then โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ).

banana โ—Š๐‘ฆ {banana if ๐‘ฆ โˆˆ {apple, carrot, peach, radish} ={ carrot if ๐‘ฆ = banana {

ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal Proof. Suppose that (๐ป, ฬƒ๐œ– such that of ๐‘‹. For each ๐œ€ โˆˆ ๐‘ƒ([0, 1]), ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ โˆˆ ๐ป ๐œ– ๐œ– ฬƒ and ๐‘ฆ โˆˆ ๐ป ฬƒ ; then โ„Ž ฬƒ (๐‘ฅโˆ—๐‘ฆ) โŠ† ๐œ€ and โ„Ž ฬƒ (๐‘ฆ) โŠ† ๐œ€. ๐‘ฅโˆ—๐‘ฆ โˆˆ ๐ป ๐ป[๐‘’] ๐ป[๐‘’] Thus,

(12)

ฬƒ [horse] (banana) . โˆช๐ป ฬƒ ๐ด) be a hesitant fuzzy soft set over Theorem 19. Let (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal of ๐‘‹ if and only ๐‘‹. (๐ป, ฬƒ ๐ด)๐œ€ is an ideal over ๐‘‹ for each ๐œ€ โˆˆ ๐‘ƒ([0, 1]). if (๐ป,

(15)

(b) Since (๐‘ฅ โˆ— ๐‘ฆ) โˆ— (๐‘ฅ โˆ— ๐‘ง) โ‰ค ๐‘ง โˆ— ๐‘ฆ, it follows from (a) that โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— (๐‘ฅ โˆ— ๐‘ง)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง โˆ— ๐‘ฆ). Hence โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— (๐‘ฅ โˆ— ๐‘ง)) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง)

(16)

โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง โˆ— ๐‘ฆ) , (c) If ๐‘ฅ โˆ— ๐‘ฆ โ‰ค ๐‘ง, then (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง = 0. Since โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง)

(17)

โŠ‡ โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) , it follows that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง).

Mathematical Problems in Engineering

5

(d) If โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (0), then we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

Also, (18)

(e) For all ๐‘ฅ โˆˆ ๐‘‹,

(19)

= โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) . The proof is complete. ฬƒ ๐ด) be a hesitant fuzzy soft set over ๐‘‹ Theorem 21. Let (๐ป, which satisfies condition (5) and (c) from Proposition 20. Then ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal over ๐‘‹. (๐ป, Proof. Let ๐‘’ โˆˆ ๐ด. Since ๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ฆ) โ‰ค ๐‘ฆ, for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, it follows from Proposition 20(c) that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

= โ„Ž๐ป[๐‘’ ฬƒ 2 ] (๐‘Ž โˆ— ๐‘) โˆช โ„Ž๐ป[๐‘’ ฬƒ 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‘‹.

โ„Ž๐ป[๐‘’] ฬƒ (0 โˆ— (0 โˆ— ๐‘ฅ)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((0 โˆ— (0 โˆ— ๐‘ฅ)) โˆ— ๐‘ฅ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ)

โ„Ž๐ป[๐‘’ ฬƒ 2 ] (๐‘Ž) = [0.3, 0.9) โŠ†ฬธ [0.3, 0.6]

(20)

ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal over ๐‘‹. Hence, (๐ป, 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‘‹.

ฬƒ ๐ด) be a hesitant anti-fuzzy soft subalgeTheorem 24. Let (๐ป, ฬƒ ๐ด) bra over ๐‘‹. If the inequality ๐‘ฅ โˆ— ๐‘ฆ โ‰ค ๐‘ง holds in ๐‘‹, then (๐ป, is a hesitant anti-fuzzy soft ideal over ๐‘‹. ฬƒ ๐ด) is a hesitant anti-fuzzy soft subalProof. Suppose that (๐ป, gebra over ๐‘‹. Then, from Proposition 11, we have โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) , โˆ€๐‘ฅ โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

(24)

Assume that ๐‘ฅ โˆ— ๐‘ฆ โ‰ค ๐‘ง holds in ๐‘‹. Then, by Proposition 20(c), we get โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง)

โˆ€๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹.

(25)

Since ๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ฆ) โ‰ค ๐‘ฆ, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ)

โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹.

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ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal over ๐‘‹. Hence, (๐ป,

ฬƒ ๐ด) is a hesitant antiProof. For any ๐‘’ โˆˆ ๐ด, assume that (๐ป, fuzzy soft ideal over ๐‘‹. Then

5. Hesitant Anti-Fuzzy Soft Implicative Ideals

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฅ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฅ) โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = โ„Ž๐ป[๐‘’] ฬƒ (0 โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ)

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= โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ)

(HAFII1) โ„Ž๐‘‹ (0) โŠ† โ„Ž๐‘‹ (๐‘ฅ), for all ๐‘ฅ โˆˆ ๐‘‹.

โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) , ฬƒ ๐ด) is a hesitant anti-fuzzy soft for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, and so (๐ป, subalgebra over ๐‘‹. The following example shows that the converse of Theorem 22 is not true in general. Example 23. Let ๐‘‹ = {0, ๐‘Ž, ๐‘, ๐‘} in Example 13 and ๐ต = ฬƒ ๐ต) is a hesitant anti-fuzzy soft subalge{๐‘’1 , ๐‘’2 }, and then (๐ป, bra over ๐‘‹. But it is not a hesitant anti-fuzzy soft ideal over ๐‘‹ based on parameters โ€œ๐‘’1 โ€ and โ€œ๐‘’2 โ€ since โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘Ž) = (0.3, 0.8) โŠ†ฬธ [0.3, 0.5] = (0.3, 0.5) โˆช [0.3, 0.5] = โ„Ž๐ป[๐‘’ ฬƒ 1 ] (0) โˆช โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘) = โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘Ž โˆ— ๐‘) โˆช โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘) .

Definition 25. A hesitant fuzzy set ๐ป๐‘‹ fl {(๐‘ฅ, โ„Ž๐‘‹ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} in a BCK-algebra ๐‘‹ is called a hesitant antifuzzy implicative ideal (briefly, HAFII) of ๐‘‹ if it satisfies the following conditions:

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(HAFII2) โ„Ž๐‘‹ (๐‘ฅ) โŠ† โ„Ž๐‘‹ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐‘‹ (๐‘ง), for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. ฬƒ ๐ด) be a hesitant fuzzy soft set over ๐‘‹, Definition 26. Let (๐ป, ฬƒ ๐ด) is called a where ๐ด is a subset of ๐ธ. Given ๐‘’ โˆˆ ๐ด, (๐ป, hesitant anti-fuzzy soft implicative ideal based on ๐‘’ (briefly, ๐‘’-hesitant anti-fuzzy soft implicative ideal) over ๐‘‹ if the hesitant fuzzy set, ฬƒ [๐‘’] fl {(๐‘ฅ, โ„Ž ฬƒ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} , ๐ป ๐ป[๐‘’]

(27)

ฬƒ ๐ด) on ๐‘‹ is a hesitant anti-fuzzy implicative ideal of ๐‘‹. If (๐ป, is an ๐‘’-hesitant anti-fuzzy soft implicative ideal over ๐‘‹, for ฬƒ ๐ด) is a hesitant anti-fuzzy soft all ๐‘’ โˆˆ ๐ด, we say that (๐ป, 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

ฬƒ ๐ด) be a hesitant anti-fuzzy soft implicative Proof. Let (๐ป, ideal over ๐‘‹. Let ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ be such that ๐‘ฅ โ‰ค ๐‘ฆ; then โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ง โˆ— ๐‘ฅ)) โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— (๐‘ง โˆ— ๐‘ฅ)) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (0 โˆ— (๐‘ง โˆ— ๐‘ฅ)) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ)

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Table 6 โˆ— 0 ๐‘Ž ๐‘ ๐‘

0 0 ๐‘Ž ๐‘ ๐‘

Hence, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ), and this completes the proof.

0 1 1 ( , ) 4 2 1 (0, ) 4

๐‘’1 ๐‘’2

Proposition 28. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra ๐‘‹ is a hesitant anti-fuzzy soft ideal of ๐‘‹. ฬƒ ๐ด) be a hesitant anti-fuzzy soft implicative Proof. Let (๐ป, ideal over ๐‘‹. Let ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹; then โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) .

(29)

Replace ๐‘ฆ = ๐‘ฅ, and using ๐‘ฅ โˆ— ๐‘ฅ = 0 we get โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) .

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๐‘ 0 0 0 ๐‘

๐‘ 0 ๐‘Ž ๐‘ 0

ฬƒ ๐ด). Table 7: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬƒ ๐ป

= โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

๐‘Ž 0 0 ๐‘Ž ๐‘

๐‘Ž 1 [0, ) 2 1 (0, ) 2

๐‘ 1 [0, ) 2 1 [0, ] 2

๐‘ 3 [0, ] 4 3 [0, ) 4

Also, 1 โ„Ž๐ป[๐‘’ ฬƒ 2 ] (๐‘Ž) = (0, ) 2 โŠ†ฬธ โ„Ž๐ป[๐‘’ ฬƒ 2 ] ((๐‘Ž โˆ— (๐‘ โˆ— ๐‘Ž)) โˆ— 0) โˆช โ„Ž๐ป[๐‘’ ฬƒ 2 ] (0)

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1 = โ„Ž๐ป[๐‘’ ฬƒ 2 ] (0) โˆช โ„Ž๐ป[๐‘’ ฬƒ 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‘‹.

ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal. For all ๐‘ฅ, ๐‘ง โˆˆ ๐‘‹, (๐ป,

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 ฬƒ ๐ด) be hesitant anti๐‘ฅ = ๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ), โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹. And let (๐ป, fuzzy soft ideal over ๐‘‹. Then, for any ๐‘’ โˆˆ ๐ด, 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‘Ž, ๐‘, ๐‘} be a BCK-algebra in Table 6 (Cayley). ฬƒ ๐ด) be a Consider a set of parameters ๐ด = {๐‘’1 , ๐‘’2 }. Let (๐ป, hesitant fuzzy soft set over ๐‘‹ which is described in Table 7. ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal (subalgebra) Then (๐ป, ฬƒ 1 ] fl {(๐‘ฅ, โ„Ž ฬƒ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} and ๐ป[๐‘’ ฬƒ 2 ] fl of ๐‘‹. But ๐ป[๐‘’ ๐ป[๐‘’1 ] {(๐‘ฅ, โ„Ž๐ป[๐‘’ ฬƒ 2 ] (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} on ๐‘‹ is not hesitant anti-fuzzy soft implicative ideal of ๐‘‹, because 1 โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘Ž) = [0, ) 2 โŠ†ฬธ โ„Ž๐ป[๐‘’ ฬƒ 1 ] ((๐‘Ž โˆ— (๐‘ โˆ— ๐‘Ž)) โˆ— 0) โˆช โ„Ž๐ป[๐‘’ ฬƒ 1 ] (0) 1 1 = โ„Ž๐ป[๐‘’ ฬƒ 1 ] (0) โˆช โ„Ž๐ป[๐‘’ ฬƒ 1 ] (0) = ( , ) . 4 2

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โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง)

(33)

for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. Hence, it is a hesitant anti-fuzzy impliฬƒ ๐ด) is hesitant anti-fuzzy soft cative ideal of ๐‘‹. That is, (๐ป, implicative ideal of ๐‘‹. ฬƒ ๐ด) be a hesitant fuzzy soft set in a BCKTheorem 32. Let (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft implicative algebra ๐‘‹. Then (๐ป, ideal of ๐‘‹ if and only if for each ๐œ€ โˆˆ ๐‘ƒ([0, 1]) the hesitant antiฬƒ ๐ด)๐œ€ of (๐ป, ฬƒ ๐ด) is empty or an implicative fuzzy soft ๐œ€-level set (๐ป, ideal of ๐‘‹. ฬƒ ๐ด) be a hesitant anti-fuzzy soft implicative Proof. Let (๐ป, ฬƒ๐œ€ =ฬธ โŒ€ for every ๐œ€ โˆˆ ๐‘ƒ([0, 1]). ideal of ๐‘‹ and assume that ๐ป Note that โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ), for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ โˆˆ ๐‘‹. In ฬƒ๐œ€ , โ„Ž ฬƒ (0) โŠ† โ„Ž ฬƒ (๐‘ฅ) โŠ† ๐œ€ and so particular, for each ๐‘ฅ โˆˆ ๐ป ๐ป[๐‘’] ๐ป[๐‘’] ฬƒ๐œ€ . Let (๐‘ฅโˆ—(๐‘ฆโˆ—๐‘ฅ))โˆ—๐‘ง โˆˆ ๐ป ฬƒ๐œ€ and ๐‘ง โˆˆ ๐ป ฬƒ๐œ€ , and since (๐ป, ฬƒ ๐ด) 0โˆˆ๐ป is a hesitant anti-fuzzy implicative ideal over ๐‘‹, it follows that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) โŠ† ๐œ€. (34) ฬƒ๐œ€ is an implicative ideal of ๐‘‹. ฬƒ๐œ€ , and ๐ป Hence ๐‘ฅ โˆˆ ๐ป

Mathematical Problems in Engineering

7

ฬƒ๐œ€ be an implicative ideal of ๐‘‹; we first Conversely, let ๐ป show โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ), for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ โˆˆ ๐‘‹. If not, then there exists ๐‘ฅ0 โˆˆ ๐‘‹ such that โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ‡ โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ0 ). Let (0) + โ„Ž (๐‘ฅ )}; then 0 โ‰ค โ„Ž ๐œ€0 = (1/2){ โ„Ž๐ป[๐‘’] ฬƒ ฬƒ ฬƒ (๐‘ฅ0 ) โŠ‚ ๐œ€0 โŠ‚ 0 ๐ป[๐‘’] ๐ป[๐‘’] ๐œ€0 ๐œ€0 ฬƒ ฬƒ ฬƒ๐œ€0 is an โ„Ž๐ป[๐‘’] =ฬธ โŒ€. As ๐ป ฬƒ (0) โ‰ค 1. Thus, ๐‘ฅ0 โˆˆ ๐ป and ๐ป ๐œ€ ฬƒ 0 , which implies that โ„Ž ฬƒ (0) โŠ† implicative ideal of ๐‘‹, 0 โˆˆ ๐ป ๐ป[๐‘’] ๐œ€0 . This is a contradiction and so โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ โˆˆ ๐‘‹. Now assume that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))โˆ—๐‘ง)โˆชโ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) does not hold. Then, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ0 ) โŠ‡ โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— (๐‘ฆ0 โˆ— ๐‘ฅ0 )) โˆ— ๐‘ง0 ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง0 ) for some ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ง0 โˆˆ ๐‘‹. If we take ๐œ€0 =

1 {โ„Ž ฬƒ (๐‘ฅ ) 2 ๐ป[๐‘’] 0

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+ {โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— (๐‘ฆ0 โˆ— ๐‘ฅ0 )) โˆ— ๐‘ง0 ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง0 )}} , then 0 โ‰ค โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— (๐‘ฆ0 โˆ— ๐‘ฅ0 )) โˆ— ๐‘ง0 ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง0 ) โŠ† ๐œ€0 โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ0 ) โ‰ค 1. It follows that โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— (๐‘ฆ0 โˆ— ๐‘ฅ0 )) โˆ— ๐‘ง0 ) โŠ† ๐œ€0 ฬƒ๐œ€0 and โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง0 ) โŠ† ๐œ€0 , which imply that (๐‘ฅ0 โˆ—(๐‘ฆ0 โˆ—๐‘ฅ0 ))โˆ—๐‘ง0 โˆˆ ๐ป ๐œ€ ๐œ€ ฬƒ 0 . But ๐ป ฬƒ 0 is an implicative ideal of ๐‘‹, so ๐‘ฅ0 โˆˆ and ๐‘ง0 โˆˆ ๐ป ๐œ€0 ฬƒ ๐ป or โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ0 ) โŠ† ๐œ€0 , and this is a contradiction. Therefore ฬƒ ๐ด) is a hesitant anti-fuzzy soft implicative ideal of ๐‘‹. (๐ป, ฬƒ ๐ด) be a hesitant anti-fuzzy soft ideal of Theorem 33. Let (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft a BCK-algebra ๐‘‹. Then (๐ป, implicative ideal of ๐‘‹ if and only if it satisfies the condition โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))

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for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. ฬƒ ๐ด) is a hesitant anti-fuzzy soft Proof. Assume that (๐ป, implicative ideal of ๐‘‹. Take ๐‘ง = 0 in โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— 0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (0)

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= โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) . ฬƒ ๐ด) satisfies the condition. As Conversely, suppose that (๐ป, ฬƒ (๐ป, ๐ด) is a hesitant anti-fuzzy soft ideal of ๐‘‹, we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) .

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ฬƒ ๐ด) is a hesitant anti-fuzzy soft implicative ideal Then (๐ป, of ๐‘‹ and the proof is completed. Now we give characterizations of hesitant anti-fuzzy soft implicative ideals. ฬƒ ๐ด) be a hesitant anti-fuzzy soft ideal of Theorem 34. Let (๐ป, a BCK-algebra ๐‘‹. Then the following are equivalent: ฬƒ ๐ด) is a hesitant anti-fuzzy soft implicative ideal of (i) (๐ป, ๐‘‹. (ii) โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅโˆ—(๐‘ฆโˆ—๐‘ฅ)) for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. (iii) โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅโˆ—(๐‘ฆโˆ—๐‘ฅ)) for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด.

ฬƒ ๐ด) be a hesitant anti-fuzzy soft Proof. (i) โ‡’ (ii) Let (๐ป, implicative ideal of ๐‘‹. Then by Theorem 33 we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— 0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆช โ„Ž๐ป[๐‘’] ฬƒ (0)

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= โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) , for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹. Hence condition (ii) holds. (ii) โ‡’ (iii) Observe that, in BCK-algebra ๐‘‹, ๐‘ฅโˆ—(๐‘ฆโˆ—๐‘ฅ) โ‰ค ๐‘ฅ. Applying Proposition 20(a), we obtain โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ). Since (ii) holds, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. So, condition (iii) holds. ฬƒ ๐ด) (iii) โ‡’ (i) Suppose that condition (iii) holds. Since (๐ป, is a hesitant anti-fuzzy soft ideal of ๐‘‹, by Definition 16, we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) .

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Combining (iii), we obtain โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) .

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ฬƒ ๐ด) is a hesitant anti-fuzzy soft implicative ideal Thus (๐ป, of ๐‘‹. So, condition (i) holds. ฬƒ ๐ด) be a hesitant anti-fuzzy soft implicaTheorem 35. Let (๐ป, tive ideal of a BCK-algebra ๐‘‹; then the set ๐ปโ„Ž๐ป[๐‘’] fl {๐‘ฅ โˆˆ ๐‘‹ | โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ)} ฬƒ

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is an implicative ideal of ๐‘‹. . Let ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐ปโ„Ž๐ป[๐‘’] be such that Proof. Clearly, 0 โˆˆ ๐ปโ„Ž๐ป[๐‘’] ฬƒ ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง โˆˆ ๐ปโ„Ž๐ป[๐‘’] and ๐‘ง โˆˆ ๐ปโ„Ž๐ป[๐‘’] , for all ๐‘’ โˆˆ ๐ด; then ฬƒ ฬƒ โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) .

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ฬƒ ๐ด) is a hesitant anti-fuzzy soft implicative ideal Since (๐ป, of ๐‘‹, it follows that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (0) .

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Combining Definition 25(HAFII1), we get โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = . Therefore ๐ปโ„Ž๐ป[๐‘’] is an โ„Ž๐ป[๐‘’] ฬƒ (0) and hence ๐‘ฅ โˆˆ ๐ปโ„Ž๐ป[๐‘’] ฬƒ ฬƒ implicative ideal of ๐‘‹. Theorem 36. A hesitant anti-fuzzy soft subalgebra โ„Ž๐ป[๐‘’] of ๐‘‹ ฬƒ is a hesitant anti-fuzzy soft implicative ideal if and only if it satisfies the condition (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง โ‰ค ๐‘ข implying that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ข) for all ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ข โˆˆ ๐‘‹ and โˆ€๐‘’ โˆˆ ๐ด. is a hesitant anti-fuzzy soft implicaProof. Assume that โ„Ž๐ป[๐‘’] ฬƒ tive ideal of ๐‘‹ and let ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ข โˆˆ ๐‘‹ be such that (๐‘ฅ โˆ— (๐‘ฆ โˆ—

8

Mathematical Problems in Engineering

๐‘ฅ)) โˆ— ๐‘ง โ‰ค ๐‘ข. Since โ„Ž๐ป[๐‘’] is also hesitant anti-fuzzy soft ideal of ฬƒ ๐‘‹, by Proposition 28, it follows from Proposition 20(c) that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ข) .

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Making use of Theorem 34, we obtain โ„Ž๐ป[๐‘’] โŠ† ฬƒ (๐‘ฅ) โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ข). satisfies the condition in Conversely, suppose that โ„Ž๐ป[๐‘’] ฬƒ satisfies (5) from Proposition 11. theorem. Obviously, โ„Ž๐ป[๐‘’] ฬƒ Since (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โ‰ค ๐‘ง, it follows from Proposition 20(c) and Theorem 34 that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) ,

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which shows that โ„Ž๐ป[๐‘’] satisfies conditions of a hesitant antiฬƒ is a hesitant antifuzzy implicative ideal of ๐‘‹ and so โ„Ž๐ป[๐‘’] ฬƒ fuzzy soft implicative ideal of ๐‘‹. The proof is complete.

Definition 37. A hesitant fuzzy set ๐ป๐‘‹ fl {(๐‘ฅ, โ„Ž๐‘‹ (๐‘ฅ)) | ๐‘ฅ โˆˆ 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) โ„Ž๐‘‹ (0) โŠ† โ„Ž๐‘‹ (๐‘ฅ), for all ๐‘ฅ โˆˆ ๐‘‹. (HAFPII2) โ„Ž๐‘‹ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐‘‹ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐‘‹ (๐‘ฆ โˆ— ๐‘ง), for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. ฬƒ ๐ด) be a hesitant fuzzy soft set over ๐‘‹, Definition 38. Let (๐ป, ฬƒ ๐ด) is called a where ๐ด is a subset of ๐ธ. Given ๐‘’ โˆˆ ๐ด, (๐ป, 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‘‹. ฬƒ ๐ด) is an ๐‘’-hesitant anti-fuzzy soft positive implicative If (๐ป, ฬƒ ๐ด) is a hesitant ideal over ๐‘‹, for all ๐‘’ โˆˆ ๐ด, we say that (๐ป, 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 ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด be such that ๐‘ฅ โ‰ค ๐‘ฆ. Since ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive implicative ideal (๐ป, of ๐‘‹, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0 โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง)

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Putting ๐‘ง = 0, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

(51)

ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal of ๐‘‹. Therefore (๐ป,

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. ฬƒ ๐ด) is a hesitant anti-fuzzy Routine calculations give that (๐ป, soft ideal (subalgebra) of ๐‘‹, but it is not a hesitant anti-fuzzy soft positive implicative ideal of ๐‘‹, because 1 โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘ โˆ— ๐‘Ž) = โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘Ž) = [0, ) 2 โŠ†ฬธ โ„Ž๐ป[๐‘’ ฬƒ 1 ] ((๐‘ โˆ— ๐‘Ž) โˆ— ๐‘Ž) โˆช โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘Ž โˆ— ๐‘Ž)

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‘‹. ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal Proof. Assume that (๐ป, of a positive implicative BCK-algebra ๐‘‹, for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด, and then (53)

By replacing ๐‘ฅ by ๐‘ฅ โˆ— ๐‘ง and ๐‘ฆ by ๐‘ฆ โˆ— ๐‘ง, we get

โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) . (49)

(52)

1 1 = โ„Ž๐ป[๐‘’ ฬƒ 1 ] (0) โˆช โ„Ž๐ป[๐‘’ ฬƒ 1 ] (0) = ( , ) . 4 2

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง))

Putting ๐‘ง = 0,

The proof is complete.

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) .

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„ŽH[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

= โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) . โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

ฬƒ ๐ด) be a hesitant anti-fuzzy soft positive Proof. Let (๐ป, implicative ideal of a BCK-algebra ๐‘‹, so for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด we have

Combining Theorem 21 and Proposition 39 yields the following result.

6. Hesitant Anti-Fuzzy Soft Positive Implicative Ideals

ฬƒ [๐‘’] fl {(๐‘ฅ, โ„Ž ฬƒ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} , ๐ป ๐ป[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, (๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง) = (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. Hence, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) .

(55)

Mathematical Problems in Engineering

9

ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive This shows that (๐ป, implicative ideal of ๐‘‹. ฬƒ ๐ด) be a hesitant anti-fuzzy soft ideal over Theorem 45. Let (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive implicative ๐‘‹; then (๐ป, ideal of ๐‘‹ if and only if โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) ,

(56)

ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal Proof. Suppose that (๐ป, over ๐‘‹ and (57)

for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹.

(58)

ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive implicaThus (๐ป, tive ideal of ๐‘‹. ฬƒ ๐ด) is a hesitant anti-fuzzy Conversely, assume that (๐ป, ฬƒ ๐ด) is soft positive implicative ideal of ๐‘‹ implying that (๐ป, a hesitant anti-fuzzy soft ideal of ๐‘‹ by Proposition 40. Let ๐‘Ž = ๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ง) and let ๐‘ = ๐‘ฅ โˆ— ๐‘ฆ; since (59)

fl {๐‘ฅ โˆˆ ๐‘‹ | โ„Ž๐ป[๐‘’] ๐ปโ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ)} ฬƒ

(64)

. Let ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐ปโ„Ž๐ป[๐‘’] be Proof. Let ๐‘’ โˆˆ ๐ด. Clearly, 0 โˆˆ ๐ปโ„Ž๐ป[๐‘’] ฬƒ ฬƒ such that (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง โˆˆ ๐ปโ„Ž๐ป[๐‘’] and ๐‘ฆ โˆ— ๐‘ง โˆˆ ๐ป , and then โ„Ž ฬƒ ฬƒ ๐ป[๐‘’] (65)

It follows that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (0) ,

(66)

ฬƒ ๐ด) is a hesitant anti-fuzzy positive implicative ideal since (๐ป, of ๐‘‹, and then โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) and hence ๐‘ฅ โˆ— ๐‘ง โˆˆ .Therefore ๐ป is a positive implicative ideal of ๐‘‹. ๐ปโ„Ž๐ป[๐‘’] โ„Ž๐ป[๐‘’] ฬƒ ฬƒ ฬƒ ๐ด) be a hesitant anti-fuzzy soft ideal over Theorem 48. Let (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive implicative ๐‘‹; then (๐ป, ideal of ๐‘‹ if and only if it satisfies the inequalities

using Proposition 20(a), we have โ„Ž๐ป[๐‘’] ฬƒ ((๐‘Ž โˆ— ๐‘) โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ง)) โˆ— (๐‘ฅ โˆ— ๐‘ฆ)) โˆ— ๐‘ง)

(63)

ฬƒ ๐ด) be a hesitant anti-fuzzy soft positive Theorem 47. Let (๐ป, implicative ideal of a BCK-algebra ๐‘‹, for all ๐‘’ โˆˆ ๐ด. Then the set

โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) .

โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) .

((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ง)) โˆ— (๐‘ฅ โˆ— ๐‘ฆ)) โ‰ค ๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ง) ,

โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘Ž) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘) .

is a positive implicative ideal of ๐‘‹.

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง)) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง)

โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง)

This completes the proof.

for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹.

โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) ,

Proof. Let ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘Ž, ๐‘ โˆˆ ๐‘‹ be such that ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆ— ๐‘Ž โ‰ค ๐‘. ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive implicative Since (๐ป, ideal of ๐‘‹, it follows from Theorem 45 and Proposition 20(c) that

(60)

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ)

โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ง)) โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) .

โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด,

(67)

for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹.

And so โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง)) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ง)) โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘Ž โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘Ž โˆ— ๐‘) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ โˆ— ๐‘ง)

(61)

ฬƒ ๐ด) Proof. Suppose that the hesitant anti-fuzzy soft ideal (๐ป, of a BCK-algebra ๐‘‹ is a hesitant anti-fuzzy soft filter positive implicative ideal of ๐‘‹. So โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง)

= โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) .

โˆ€๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

Therefore โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง), for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. ฬƒ ๐ด) be a hesitant anti-fuzzy soft positive Theorem 46. Let (๐ป, implicative ideal of ๐‘‹. Then, for all ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘Ž, ๐‘ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด, ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆ— ๐‘Ž โ‰ค ๐‘ ๓ณจโ‡’ โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘Ž) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘) .

(62)

(68)

Substituting ๐‘ง = ๐‘ฆ, we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ) , for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด.

(69)

10

Mathematical Problems in Engineering

ฬƒ ๐ด) is a hesitant anti-fuzzy Conversely, suppose that (๐ป, soft ideal over ๐‘‹ and satisfies inequality (67). โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ) โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด,

(70)

since โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ). Now we can prove that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. In contrast, there exist ๐‘ฅ,ฬ€ ๐‘ฆฬ€ โˆˆ ๐‘‹ such that ฬ€ โŠ‡ โ„Ž๐ป[๐‘’] ฬ€ โˆ— ๐‘ฆ)ฬ€ โˆช โ„Ž๐ป[๐‘’] ฬ€ โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅฬ€ โˆ— ๐‘ฆ) ฬƒ ((๐‘ฅฬ€ โˆ— ๐‘ฆ) ฬƒ (๐‘ฆฬ€ โˆ— ๐‘ฆ) ฬ€ โˆ— ๐‘ฆ)ฬ€ โˆช โ„Ž๐ป[๐‘’] = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅฬ€ โˆ— ๐‘ฆ) ฬƒ (0)

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ฬ€ โˆ— ๐‘ฆ)ฬ€ , = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅฬ€ โˆ— ๐‘ฆ) which is a contradiction. Therefore โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. Thus ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive implicative ideal (๐ป, 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 โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ) โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

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ฬƒ ๐ด) Proof. Suppose that the hesitant anti-fuzzy soft ideal (๐ป, of a BCK-algebra ๐‘‹ is a hesitant anti-fuzzy soft positive implicative ideal of ๐‘‹. So by Theorem 48 we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ) โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

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โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

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Conversely, assume that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ) for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. We want to prove that a hesitant ฬƒ ๐ด) of a BCK-algebra ๐‘‹ is a hesitant anti-fuzzy soft ideal (๐ป, anti-fuzzy soft positive implicative ideal of ๐‘‹. It is clear that ฬƒ โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) for all ๐‘ฅ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. Since (๐ป, ๐ด) is a hesitant anti-fuzzy soft ideal and ๐‘‹ is positive implicative, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— (๐‘ฆ โˆ— ๐‘ง)) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) ,

ฬƒ ๐ด) be a hesitant anti-fuzzy soft positive Theorem 50. Let (๐ป, implicative ideal of ๐‘‹ if and only if for each ๐œ€ โˆˆ ๐‘ƒ([0, 1]) the ฬƒ ๐ด)๐œ€ of (๐ป, ฬƒ ๐ด) is a positive hesitant anti-fuzzy soft ๐œ€-level set (๐ป, implicative ideal of ๐‘‹. ฬƒ ๐ด) is a hesitant anti-fuzzy soft posiProof. Suppose that (๐ป, tive implicative ideal of ๐‘‹ and for every ๐œ€ โˆˆ ๐‘ƒ([0, 1]) define the sets ๐œ€

ฬƒ ๐ด) fl {๐‘ฅ โˆˆ ๐‘‹ | โ„Ž ฬƒ (๐‘ฅ) โŠ† ๐œ€} (๐ป, ๐ป[๐‘’]

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โˆ€๐‘’ โˆˆ ๐ด.

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ฬƒ ๐ด)๐œ€ โ‡’ โ„Ž ฬƒ (๐‘ฅ) โŠ† ๐œ€. By ฬƒ ๐ด)๐œ€ =ฬธ โŒ€, let ๐‘ฅ โˆˆ (๐ป, Since (๐ป, ๐ป[๐‘’] Definition 37, we have โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ), for all ๐‘’ โˆˆ ๐ด and ฬƒ ๐ด)๐œ€ . Let ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ be such ๐‘ฅ โˆˆ ๐‘‹, implying that 0 โˆˆ (๐ป, ๐œ€ ฬƒ ๐ด) and ๐‘ฆ โˆ— ๐‘ง โˆˆ (๐ป, ฬƒ ๐ด)๐œ€ , implying that that (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง โˆˆ (๐ป, ฬƒ โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โŠ† ๐œ€ and โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) โŠ† ๐œ€. Since (๐ป, ๐ด) is a hesitant anti-fuzzy soft positive implicative ideal over ๐‘‹, it follows that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) โŠ† ๐œ€.

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ฬƒ ๐ด)๐œ€ , and (๐ป, ฬƒ ๐ด)๐œ€ is a positive Hence ๐‘ฅ โˆ— ๐‘ง โˆˆ (๐ป, implicative ideal of ๐‘‹. ฬƒ ๐ด)๐œ€ is a positive implicative Conversely, suppose that (๐ป, ideal of ๐‘‹ for all ๐œ€ โˆˆ ๐‘ƒ([0, 1]). Put โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = ๐œ€ for any ๐‘ฅ โˆˆ ๐‘‹. ฬƒ ๐ด)๐œ€ โ‡’ โ„Ž ฬƒ (0) โŠ† ๐œ€ = โ„Ž ฬƒ (๐‘ฅ), for all ๐‘’ โˆˆ ๐ด Hence 0 โˆˆ (๐ป, ๐ป[๐‘’] ๐ป[๐‘’] and ๐‘ฅ โˆˆ ๐‘‹. Now we prove that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง). In contrast, there exists ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ง0 โˆˆ ๐‘‹ such that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ0 โˆ— ๐‘ง0 ) โŠ‡ โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— ๐‘ฆ0 ) โˆ— ๐‘ง0 ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ0 โˆ— ๐‘ง0 ). Taking ๐œ€0 =

On the other hand, since (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ โ‰ค ๐‘ฅ โˆ— ๐‘ฆ, it follows from Proposition 20(a) that โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ). Thus we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ฆ)

for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. Therefore a hesitant anti-fuzzy ฬƒ ๐ด) of ๐‘‹ is a hesitant anti-fuzzy soft positive soft ideal (๐ป, implicative ideal of ๐‘‹.

1 {โ„Ž ฬƒ (๐‘ฅ โˆ— ๐‘ง0 ) 2 ๐ป[๐‘’] 0

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+ {โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— ๐‘ฆ0 ) โˆ— ๐‘ง0 ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ0 โˆ— ๐‘ง0 )}} , it follows that โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ—๐‘ฆ0 )โˆ—๐‘ง0 ) โŠ† ๐œ€0 and โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ0 โˆ—๐‘ง0 ) โŠ† ๐œ€0 , ๐œ€0 ฬƒ which imply that (๐‘ฅ0 โˆ— ๐‘ฆ0 ) โˆ— ๐‘ง0 โˆˆ (๐ป, ๐ด) and ๐‘ฆ0 โˆ— ๐‘ง0 โˆˆ ฬƒ ๐ด)๐œ€0 . But (๐ป, ฬƒ ๐ด)๐œ€0 is a positive implicative ideal of ๐‘‹, and (๐ป, ฬƒ ๐ด)๐œ€0 or โ„Ž ฬƒ (๐‘ฅ0 โˆ— ๐‘ง0 ) โŠ† ๐œ€0 , and this is a thus ๐‘ฅ0 โˆ— ๐‘ง0 โˆˆ (๐ป, ๐ป[๐‘’] contradiction. Therefore โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง) โˆ€๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

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ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive implicative Hence (๐ป, ideal of ๐‘‹.

7. Hesitant Anti-Fuzzy Soft Commutative Ideals Definition 51. A hesitant fuzzy set ๐ป๐‘‹ fl {(๐‘ฅ, โ„Ž๐‘‹ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} in a BCK-algebra ๐‘‹ is called a hesitant anti-fuzzy

Mathematical Problems in Engineering

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commutative ideal (briefly, HAFCI) of ๐‘‹ if it satisfies the following conditions: (HAFCI1) โ„Ž๐‘‹ (0) โŠ† โ„Ž๐‘‹ (๐‘ฅ), for all ๐‘ฅ โˆˆ ๐‘‹. (HAFCI2) โ„Ž๐‘‹ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โŠ† โ„Ž๐‘‹ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐‘‹ (๐‘ง), for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. ฬƒ ๐ด) be a hesitant fuzzy soft set over ๐‘‹, Definition 52. Let (๐ป, ฬƒ ๐ด) is called a where ๐ด is a subset of ๐ธ. Given ๐‘’ โˆˆ ๐ด, (๐ป, hesitant anti-fuzzy soft commutative ideal based on ๐‘’ (briefly, ๐‘’-hesitant anti-fuzzy soft commutative ideal) over ๐‘‹ if the hesitant fuzzy set, ฬƒ [๐‘’] fl {(๐‘ฅ, โ„Ž ฬƒ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} , ๐ป ๐ป[๐‘’]

Proposition 53. Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebra ๐‘‹ is order-preserving. Proof. Let ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด be such that ๐‘ฅ โ‰ค ๐‘ฆ. Since ฬƒ ๐ด) is a hesitant anti-fuzzy soft commutative ideal of ๐‘‹, (๐ป, โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)))

= โ„Ž๐ป[๐‘’] ฬƒ (0 โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) = โ„Ž๐ป[๐‘’] ฬƒ (0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง)

โˆ— 0 ๐‘Ž ๐‘

0 0 ๐‘Ž ๐‘

๐‘Ž 0 0 ๐‘

ฬƒ ๐ด). Table 9: Tabular representation of the hesitant fuzzy soft set (๐ป, ฬƒ ๐ป ๐‘’1 ๐‘’2

0 (0.2, 0.4) (0, 0.2)

๐‘Ž [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‘Ž, ๐‘} be the BCK-algebra with in Table 8 (Cayley). ฬƒ ๐ด) be a Consider a set of parameters ๐ด = {๐‘’1 , ๐‘’2 }. Let (๐ป, hesitant fuzzy soft set over ๐‘‹ which is described in Table 9. ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal (subThen (๐ป, ฬƒ ๐ด) is a algebra) of ๐‘‹. Routine calculations give that (๐ป, ฬƒ 1 ] fl hesitant anti-fuzzy soft ideal (subalgebra) of ๐‘‹, but ๐ป[๐‘’ ฬƒ {(๐‘ฅ, โ„Ž๐ป[๐‘’ (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} and ๐ป[๐‘’ ] fl {(๐‘ฅ, โ„Ž (๐‘ฅ)) | ๐‘ฅ โˆˆ ๐‘‹} ฬƒ 1] ฬƒ 2] 2 ๐ป[๐‘’ on ๐‘‹ is not hesitant anti-fuzzy soft commutative ideal of ๐‘‹, because โ„Ž๐ป[๐‘’ ฬƒ 1 ] (๐‘Ž โˆ— (๐‘ โˆ— (๐‘ โˆ— ๐‘Ž))) = [0.2, 0.7) โŠ†ฬธ (0.2, 0.4) = โ„Ž๐ป[๐‘’ ฬƒ 1 ] ((๐‘Ž โˆ— ๐‘) โˆ— 0) โˆช โ„Ž๐ป[๐‘’ ฬƒ 1 ] (0) .

= โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) .

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Also,

Putting ๐‘ฆ = 0 and replacing ๐‘ง = ๐‘ฆ, we obtain โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ) .

๐‘ 0 0 0

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ฬƒ ๐ด) on ๐‘‹ is a hesitant anti-fuzzy commutative ideal of ๐‘‹. If (๐ป, is an ๐‘’-hesitant anti-fuzzy soft commutative ideal over ๐‘‹, for ฬƒ ๐ด) is a hesitant anti-fuzzy soft all ๐‘’ โˆˆ ๐ด, we say that (๐ป, commutative ideal over ๐‘‹.

โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง)

Table 8

โ„Ž๐ป[๐‘’ ฬƒ 2 ] (๐‘Ž โˆ— (๐‘ โˆ— (๐‘ โˆ— ๐‘Ž))) = (0, 0.5) โŠ†ฬธ (0, 0.2)

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= โ„Ž๐ป[๐‘’ ฬƒ 2 ] ((๐‘Ž โˆ— ๐‘) โˆ— 0) โˆช โ„Ž๐ป[๐‘’ ฬƒ 2 ] (0)

The proof is complete.

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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.

ฬƒ ๐ด) be a Proof. Let ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and let ๐‘’ โˆˆ ๐ด. And let (๐ป, 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‘‹.

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (0 โˆ— (0 โˆ— ๐‘ฅ))) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— 0) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง)

(83)

= โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) โˆช โ„Ž๐‘‹ (๐‘ง) ,

ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal of Proof. Suppose that (๐ป, ฬƒ ๐ด) satisfies a BCK-algebra ๐‘‹. It is sufficient to show that (๐ป, condition (HAFCI2). Let ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹. Then ((๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โˆ— ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง)) โˆ— ๐‘ง

ฬƒ ๐ด) is a hesitant antifor all ๐‘ฅ, ๐‘ง โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. Hence (๐ป, fuzzy soft ideal of ๐‘‹.

= ((๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โˆ— ๐‘ง) โˆ— ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โ‰ค (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โˆ— (๐‘ฅ โˆ— ๐‘ฆ)

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)

= (๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ฆ)) โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) = 0; that is, (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โˆ— ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โ‰ค ๐‘ง.

(87)

12

Mathematical Problems in Engineering

It follows from Proposition 20(c) that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) ,

(88)

ฬƒ ๐ด) is a hesitant antifor all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. Thus (๐ป, fuzzy soft commutative ideal of a BCK-algebra ๐‘‹, and the proof is complete. ฬƒ ๐ด) be a hesitant anti-fuzzy soft ideal of Theorem 59. Let (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft a BCK-algebra ๐‘‹. Then (๐ป, commutative ideal of ๐‘‹ if and only if it satisfies the condition โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— 0) โˆช โ„Ž๐ป[๐‘’] ฬƒ (0) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) .

โˆ€๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

(90)

โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) ,

(91)

(92)

Observing ๐‘ฅ โˆ— ๐‘ฆ โ‰ค ๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) and using Proposition 20(a), we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. Hence, Theorem 59 can be improved as follows. ฬƒ ๐ด) of a BCKTheorem 60. A hesitant anti-fuzzy soft ideal (๐ป, algebra ๐‘‹ is a hesitant anti-fuzzy soft commutative ideal of ๐‘‹ if and only if it satisfies the identity

โˆ€๐‘ฅ, ๐‘ฆ โˆˆ ๐‘‹, ๐‘’ โˆˆ ๐ด.

= โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง))

(94)

โˆ— (๐‘ฅ โˆ— (๐‘ฅ โˆ— ๐‘ง)) , [by Proposition 6 (ii)]

ฬƒ ๐ด) is a hesitant anti-fuzzy soft positive implicaHence (๐ป, tive ideal of ๐‘‹. Also, By Propositions 20 and 6(iii) and Theorem 34(iii), we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) (95)

โˆ— (๐‘ฆ โˆ— (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)))) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) . ฬƒ ๐ด) is hesitant antiIt follows from Theorem 59 that (๐ป, fuzzy soft commutative ideal of ๐‘‹. ฬƒ ๐ด) is both hesitant antiConversely, suppose that (๐ป, fuzzy soft positive implicative ideal of ๐‘‹ and hesitant antifuzzy soft commutative ideal of ๐‘‹. Since (๐‘ฆโˆ—(๐‘ฆโˆ—๐‘ฅ))โˆ—(๐‘ฆโˆ—๐‘ฅ) โ‰ค ๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ), it follows from Proposition 20(a) that โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) (96)

ฬƒ ๐ด) is hesitant antifor all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด. Hence (๐ป, fuzzy soft commutative ideal of ๐‘‹, which completes the proof.

โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) = โ„Ž๐ป[e] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)))

โŠ‡ โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ง) โˆ— ๐‘ง)

= โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))))

Combining (91) and (89), we obtain โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)))

โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— ๐‘ง)

= โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ง) , [by Theorem 34 (iii)] .

ฬƒ ๐ด) satisfies condition (89) as Conversely, assume that (๐ป, ฬƒ ๐ด) is a hesitant anti-fuzzy soft ideal of a BCK-algebra ๐‘‹. (๐ป, Hence โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง)

ฬƒ ๐ด) is a hesitant anti-fuzzy soft Proof. Assume that (๐ป, implicative ideal. By Propositions 6(i) and 20(c), for all ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and ๐‘’ โˆˆ ๐ด, we have

(89)

ฬƒ ๐ด) Proof. Let ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ and let ๐‘’ โˆˆ ๐ด. And suppose that (๐ป, is a hesitant anti-fuzzy soft commutative ideal of ๐‘‹. Taking ๐‘ง = 0, we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)))

ฬƒ ๐ด) of a BCKTheorem 61. A hesitant anti-fuzzy soft ideal (๐ป, algebra ๐‘‹ is a hesitant anti-fuzzy soft implicative ideal if and ฬƒ ๐ด) is both hesitant anti-fuzzy soft commutative ideal only if (๐ป, and hesitant anti-fuzzy soft positive implicative ideal.

Using Theorem 49, we have โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) and so โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) .

(97)

On the other hand, ๐‘ฅโˆ—๐‘ฆ โ‰ค ๐‘ฅโˆ—(๐‘ฆโˆ—๐‘ฅ) implies that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅโˆ— ฬƒ ๐‘ฆ) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)). Since (๐ป, ๐ด) is a hesitant anti-fuzzy soft commutative ideal of ๐‘‹, By Theorem 60, we have โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— ๐‘ฆ) = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) .

(98)

Hence, โ„Ž๐‘‹ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โŠ† โ„Ž๐‘‹ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) .

(99)

Combining (97), we obtain โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โŠ‡ โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)))

(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.

โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))

(100)

โŠ‡ โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) . ฬƒ ๐ด) is a hesitant anti-fuzzy soft implicative ideal of So, (๐ป, ๐‘‹ by Theorem 34. The proof is complete.

Mathematical Problems in Engineering

13

ฬƒ ๐ด) be hesitant anti-fuzzy soft commutaTheorem 62. Let (๐ป, tive ideal of ๐‘‹ if and only if for each ๐œ€ โˆˆ ๐‘ƒ([0, 1]) the hesitant ฬƒ ๐ด)๐œ€ of (๐ป, ฬƒ ๐ด) is a commutative anti-fuzzy soft ๐œ€-level set (๐ป, ideal of ๐‘‹. ฬƒ ๐ด) is a hesitant anti-fuzzy soft comProof. Suppose that (๐ป, mutative ideal of ๐‘‹ and for every ๐œ€ โˆˆ ๐‘ƒ([0, 1]) define the sets ๐œ€

ฬƒ ๐ด) fl {๐‘ฅ โˆˆ ๐‘‹ | โ„Ž ฬƒ (๐‘ฅ) โŠ† ๐œ€} (๐ป, ๐ป[๐‘’]

โˆ€๐‘’ โˆˆ ๐ด.

(101)

ฬƒ ๐ด)๐œ€ =ฬธ โŒ€, let ๐‘ฅ โˆˆ (๐ป, ฬƒ ๐ด)๐œ€ โ‡’ โ„Ž ฬƒ (๐‘ฅ) โŠ† ๐œ€. By Since (๐ป, ๐ป[๐‘’] definition, we have โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ), for all ๐‘’ โˆˆ ๐ด and ฬƒ ๐ด)๐œ€ . Let ๐‘ฅ, ๐‘ฆ, ๐‘ง โˆˆ ๐‘‹ be such ๐‘ฅ โˆˆ ๐‘‹, implying that 0 โˆˆ (๐ป, ฬƒ ๐ด)๐œ€ and ๐‘ง โˆˆ (๐ป, ฬƒ ๐ด)๐œ€ , implying that that (๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง โˆˆ (๐ป, ฬƒ โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โŠ† ๐œ€ and โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) โŠ† ๐œ€. Since (๐ป, ๐ด) is a hesitant anti-fuzzy soft commutative ideal over ๐‘‹, it follows that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง) โŠ† ๐œ€.

(102)

ฬƒ ๐ด)๐œ€ , and (๐ป, ฬƒ ๐ด)๐œ€ is a Namely, ๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ)) โˆˆ (๐ป, commutative ideal of ๐‘‹. ฬƒ ๐ด)๐œ€ is a commutative ideal Conversely, suppose that (๐ป, of ๐‘‹ for all ๐œ€ โˆˆ ๐‘ƒ([0, 1]). Put โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ) = ๐œ€ for any ๐‘ฅ โˆˆ ๐‘‹ and ๐œ€ ฬƒ ๐‘’ โˆˆ ๐ด. Hence 0 โˆˆ (๐ป, ๐ด) โ‡’ โ„Ž๐ป[๐‘’] ฬƒ (0) โŠ† ๐œ€ = โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ), for all ๐‘’ โˆˆ ๐ด and ๐‘ฅ โˆˆ ๐‘‹. Now we prove that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ โˆ— (๐‘ฆ โˆ— (๐‘ฆ โˆ— ๐‘ฅ))) โŠ† โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ โˆ— ๐‘ฆ) โˆ— ๐‘ง) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง). In contrast, there exists ๐‘ฅ0 , ๐‘ฆ0 , ๐‘ง0 โˆˆ ๐‘‹ such that โ„Ž๐ป[๐‘’] ฬƒ (๐‘ฅ0 โˆ— (๐‘ฆ0 โˆ— (๐‘ฆ0 โˆ— ๐‘ฅ0 ))) โŠ‡ โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— ๐‘ฆ0 ) โˆ— ๐‘ง0 ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง0 ) .

(103)

Taking ๐œ€0 =

1 {โ„Ž ฬƒ (๐‘ฅ โˆ— (๐‘ฆ0 โˆ— (๐‘ฆ0 โˆ— ๐‘ฅ0 ))) 2 ๐ป[๐‘’] 0

(104)

+ {โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— ๐‘ฆ0 ) โˆ— ๐‘ง0 ) โˆช โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง0 )}} , it follows that โ„Ž๐ป[๐‘’] ฬƒ ((๐‘ฅ0 โˆ— ๐‘ฆ0 ) โˆ— ๐‘ง0 ) โŠ† ๐œ€0 and โ„Ž๐ป[๐‘’] ฬƒ (๐‘ง0 ) โŠ† ๐œ€0 , ๐œ€0 ฬƒ ฬƒ ๐ด)๐œ€0 . which imply that ((๐‘ฅ0 โˆ—๐‘ฆ0 )โˆ—๐‘ง0 ) โˆˆ (๐ป, ๐ด) and ๐‘ง0 โˆˆ (๐ป, ๐œ€ 0 ฬƒ ๐ด) is a commutative ideal of ๐‘‹, and thus ๐‘ฅ0 โˆ— (๐‘ฆ0 โˆ— But (๐ป, ฬƒ ๐ด)๐œ€0 or โ„Ž ฬƒ (๐‘ฅ0 โˆ— (๐‘ฆ0 โˆ— (๐‘ฆ0 โˆ— ๐‘ฅ0 ))) โŠ† ๐œ€0 , (๐‘ฆ0 โˆ— ๐‘ฅ0 )) โˆˆ (๐ป, ๐ป[๐‘’] ฬƒ ๐ด) is a hesitant antiand this is a contradiction. Therefore (๐ป, 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, โ€œSoft set theory- first results,โ€ Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19โ€“31, 1999. [2] P. K. Maji, R. Biswas, and A. R. Roy, โ€œFuzzy soft sets,โ€ Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589โ€“602, 2001. [3] L. A. Zadeh, โ€œFuzzy sets,โ€ Information and Control, vol. 8, no. 3, pp. 338โ€“353, 1965. [4] V. Torra, โ€œHesitant fuzzy sets,โ€ International Journal of Intelligent Systems, vol. 25, no. 6, pp. 529โ€“539, 2010. [5] K. V. Babitha and S. J. John, โ€œHesitant fuzzy soft sets,โ€ Journal of New Results in Science, vol. 3, pp. 98โ€“107, 2013. [6] Y. B. Jun, S. S. Ahn, and G. Muhiuddin, โ€œHesitant 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, โ€œHesitant fuzzy sets theory applied to in BCK/BCI- algebras,โ€ J. Computational Analysis and Applications, vol. 20, no. 4, pp. 635โ€“646, 2016. [8] K. Isยดeki, โ€œAn algebraic related with a propositional calclus,โ€ Proc. Japan Academy, vol. 42, pp. 26โ€“29, 1996. [9] J. Meng, Y. B. Jun, and H. S. Kim, โ€œFuzzy implicative ideals of BCK-algebras,โ€ Fuzzy Sets and Systems, vol. 89, pp. 243โ€“248, 1997.

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