Hesitant Anti-Fuzzy Soft Set in BCK-Algebras

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 © 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 Zadeh’s [3] (1965) fuzzy set, allows the membership degree of an element to a set presented by several possible values, and it can express the hesitant information more comprehensively than other extensions of fuzzy set. In [4], Torra introduced the

concept of hesitant fuzzy set and Babitha and John (2013) [5] defined another important soft set, hesitant fuzzy soft set. They introduced basic operations such as intersection, union, and compliment, and De Morgan’s law was proven. In 2014, Jun et al. [6] applied the notion of hesitant fuzzy soft sets to subalgebras and ideals in BCK/BCI-algebras. In this paper, in Section 3, we introduce the concepts of hesitant anti-fuzzy soft set of subalgebra. In Section 4, we define the hesitant 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 (𝑋; ∗, 0) of type (2, 0) is said to be a BCK-algebra if it satisfies the axioms: for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, (BCK-1) ((𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑧)) ∗ (𝑧 ∗ 𝑦) = 0, (BCK-2) (𝑥 ∗ (𝑥 ∗ 𝑦)) ∗ 𝑦 = 0, (BCK-3) 𝑥 ∗ 𝑥 = 0, (BCK-4) 0 ∗ 𝑥 = 0, (BCK-5) 𝑥 ∗ 𝑦 = 0 and 𝑦 ∗ 𝑥 = 0 imply that 𝑥 = 𝑦.

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Mathematical Problems in Engineering

Define a binary relation ≤ on 𝑋 by letting 𝑥 ≤ 𝑦 if and only if 𝑥 ∗ 𝑦 = 0. Then (𝑋; ≤) is a partially ordered set with the least element 0. In any BCK-algebra 𝑋, the following hold: (1) (𝑥 ∗ 𝑦) ∗ 𝑧 = (𝑥 ∗ 𝑧) ∗ 𝑦. (2) 𝑥 ∗ 𝑦 ≤ 𝑥. (3) 𝑥 ∗ 0 = 𝑥.

∗ 0 𝑎 𝑏 𝑐

0 0 𝑎 𝑏 𝑐

𝑎 0 0 𝑏 𝑏

𝑏 0 𝑎 0 𝑎

𝑐 0 0 0 0

Proposition 5 (see [8]). Let 𝐻𝐴 fl {(𝑥, ℎ𝐴 (𝑥)) | 𝑥 ∈ 𝑋} be an 𝐴-hesitant anti-fuzzy ideal of 𝑋. Then the following hold: for all 𝑥, 𝑦, 𝑧 ∈ 𝐴,

(4) (𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧) ≤ 𝑥 ∗ 𝑦. (5) 𝑥 ∗ (𝑥 ∗ (𝑥 ∗ 𝑦)) = 𝑥 ∗ 𝑦. (6) 𝑥 ≤ 𝑦 implies that 𝑥 ∗ 𝑧 ≤ 𝑦 ∗ 𝑧 and 𝑧 ∗ 𝑦 ≤ 𝑧 ∗ 𝑥, for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. A nonempty subset 𝐼 of 𝑋 is called a subalgebra of 𝑋 if, for any 𝑥, 𝑦 ∈ 𝐼, 𝑥∗𝑦 ∈ 𝐼. That is, it is closed under the binary operation ∗ of 𝑋. A nonempty subset 𝐼 of 𝑋 is called an ideal of 𝑋 if (𝐼1 ) 0 ∈ 𝐼; (𝐼2 ) 𝑥 ∗ 𝑦 ∈ 𝐼 and 𝑦 ∈ 𝐼 imply that 𝑥 ∈ 𝐼. A nonempty subset 𝐼 of 𝑋 is called an implicative ideal if it satisfies (𝐼1 ) and (𝐼3 ) 𝑥 ∈ 𝐼 whenever (𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧 ∈ 𝐼 and 𝑧 ∈ 𝐼. It is called a commutative ideal if it satisfies (𝐼1 ) and (𝐼4 ) 𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)) ∈ 𝐼 whenever (𝑥 ∗ 𝑦) ∗ 𝑧 ∈ 𝐼 and 𝑧 ∈ 𝐼; and it is called a positive implicative ideal if it satisfies (𝐼1 ) and (𝐼5 ) 𝑥 ∗ 𝑧 ∈ 𝐼 whenever (𝑥 ∗ 𝑦) ∗ 𝑧 ∈ 𝐼 and 𝑦 ∗ 𝑧 ∈ 𝐼. A BCK-algebra 𝑋 is said to be implicative if it satisfies ∀𝑥, 𝑦 ∈ 𝑋 : 𝑥 = 𝑥 ∗ (𝑦 ∗ 𝑥). A BCK-algebra 𝑋 is said to be positive implicative if it satisfies ∀𝑥, 𝑦, 𝑧 ∈ 𝑋 : (𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧) = (𝑥 ∗ 𝑦) ∗ 𝑧. A BCK-algebra 𝑋 is said to be commutative if it satisfies ∀𝑥, 𝑦 ∈ 𝑋 : 𝑥 ∗ (𝑥 ∗ 𝑦) = 𝑦 ∗ (𝑦 ∗ 𝑥). Definition 1 (see [3]). Let 𝑆 be a set. A fuzzy set in 𝑆 is a function 𝜇 : 𝑆 → [0, 1] Definition 2 (see [2]). Let 𝑈 be an initial universe set and let 𝐸 be a set of parameters. Let F(𝑈) denote the set of all fuzzy ̃ 𝐴) is called a fuzzy soft set over 𝑈, where sets in 𝑈. Then (𝐹, ̃ 𝐴 ⊆ 𝐸 and 𝐹 is a mapping given by 𝐹̃ : 𝐴 → F(𝑈). Definition 3 (see [4, 7]). Let 𝐸 be a reference set. A hesitant fuzzy set on 𝐸 is defined in terms of a function that when applied to 𝐸 returns a subset of [0, 1] which can be viewed as the following mathematical representation: 𝐻𝐸 fl {(𝑒, ℎ𝐸 (𝑒)) : 𝑒 ∈ 𝐸} ,

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Definition 4 (see [7]). Given a nonempty subset 𝐴 of 𝑋, a hesitant fuzzy set 𝐻𝑋 fl {(𝑥; ℎ𝑋 (𝑥) : 𝑥 ∈ 𝑋} on 𝑋 satisfying the condition ∀𝑥 ∉ 𝐴

(a) if 𝑥 ≤ 𝑦, then ℎ𝐴 (𝑥) ⊆ ℎ𝐴 (𝑦), which means that ℎ𝐴 preserves the order, (b) if 𝑥 ∗ 𝑦 ≤ 𝑧, then ℎ𝐴 (𝑥) ⊆ ℎ𝐴 (𝑦) ∪ ℎ𝐴 (𝑧). Proposition 6 (see [9]). In a BCK-algebra 𝑋, the following hold: for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, (i) ((𝑥 ∗ 𝑧) ∗ 𝑧) ∗ (𝑦 ∗ 𝑧) ≤ (𝑥 ∗ 𝑦) ∗ 𝑧, (ii) (𝑥 ∗ 𝑧) ∗ (𝑥 ∗ (𝑥 ∗ 𝑧)) = (𝑥 ∗ 𝑧) ∗ 𝑧, (iii) (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ∗ (𝑦 ∗ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)))) ≤ 𝑥 ∗ 𝑦. Definition 7 (see [5, 6]). Denote by 𝐻F(𝑈) the set of all ̃ 𝐴) is called a hesitant fuzzy hesitant fuzzy sets. A pair (𝐻, ̃ is a mapping given soft set over a reference set 𝑈, where 𝐻 ̃ by 𝐻 : 𝐴 → 𝐻F(𝑈).

3. Hesitant Anti-Fuzzy Soft Subalgebras Definition 8. Given a nonempty subset (subalgebra as much as possible) 𝐴 of 𝑋, let 𝐻𝐴 fl {(𝑥, ℎ𝐴 (𝑥)) | 𝑥 ∈ 𝑋} be an 𝐴hesitant fuzzy set on 𝑋. Then 𝐻𝐴 fl {(𝑥, ℎ𝐴 (𝑥)) | 𝑥 ∈ 𝑋} is called a hesitant anti-fuzzy subalgebra of 𝑋 related to 𝐴 (briefly, 𝐴-hesitant anti-fuzzy subalgebra of 𝑋) if it satisfies the following condition: (ℎ𝐴 (𝑥 ∗ 𝑦) ⊆ ℎ𝐴 (𝑥) ∪ ℎ𝐴 (𝑦))

(∀𝑥, 𝑦 ∈ 𝐴) .

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is called a hesitant fuzzy set related to 𝐴 (briefly, 𝐴-hesitant fuzzy set) on 𝑋 and is represented by 𝐻𝐴 fl {(𝑥, ℎ𝐴 (𝑥)) : 𝑥 ∈ 𝑋}; ℎ𝐴 is a mapping from 𝑋 to 𝑝([0, 1]) with ℎ𝐴 (𝑥) = 𝜙, for all 𝑥 ∉ 𝐴.

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An 𝐴-hesitant anti-fuzzy subalgebra of 𝑋 with 𝐴 = 𝑋 is called a hesitant anti-fuzzy subalgebra of 𝑋. Definition 9. For a subset 𝐴 of 𝐸, a hesitant fuzzy soft set ̃ 𝐴) over 𝑋 is called a hesitant anti-fuzzy soft subalgebra (𝐻, based on 𝑒 ∈ 𝐴 (briefly, 𝑒-hesitant anti-fuzzy soft subalgebra) over 𝑋 if the hesitant fuzzy set, ̃ [𝑒] fl {(𝑥, ℎ ̃ (𝑥)) | 𝑥 ∈ 𝑋} , 𝐻 𝐻[𝑒]

where ℎ𝐸 : 𝐸 → 𝑝([0, 1]).

ℎ𝑋 (𝑥) = 𝜙;

Table 1

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̃ 𝐴) is an on 𝑋 is a hesitant anti-fuzzy subalgebra of 𝑋. If (𝐻, 𝑒-hesitant anti-fuzzy soft subalgebra over 𝑋, for all 𝑒 ∈ 𝐴, we ̃ 𝐴) is a hesitant anti-fuzzy soft subalgebra. say that (𝐻, Example 10. Let 𝑋 = {0, 𝑎, 𝑏, 𝑐} be a BCK-algebra in Table 1 (Cayley). Consider a set of parameters 𝐸 fl {𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 }. Let ̃ 𝐴) be a hesitant fuzzy soft set over 𝑋, where 𝐴 = (𝐻, {𝑒1 , 𝑒2 , 𝑒3 }, which is given in Table 2. It is routine to verify that


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̃ 𝐴). Table 2: Tabular representation of the hesitant fuzzy soft set (𝐻, ̃ 𝐻 𝑒1 𝑒2 𝑒3

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

𝑎 [0.2, 0.8] {0.3} [0.3, 0.7]

𝑏 [0.2, 0.8] [0.1, 0, 2) ∪ (0, 2, 0.8] [0.3, 0.5]

𝑐 [0.2, 0.8] [0.1, 0.9] [0.3, 0.7]

̃ 1 ], 𝐻[𝑒 ̃ 2 ] and 𝐻[𝑒 ̃ 3 ] are hesitant anti-fuzzy subalgebra 𝐻[𝑒 ̃ 𝐴) over 𝑋 based on parameters 𝑒1 , 𝑒2 , and 𝑒3 . Therefore (𝐻, is a hesitant anti-fuzzy soft subalgebra over 𝑋. ̃ 𝐴) is a hesitant anti-fuzzy soft subalgeProposition 11. If (𝐻, bra over 𝑋, then ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥) , ∀𝑥 ∈ 𝑋,

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where 𝑒 is any parameter in 𝐴. Proof. For any 𝑥 ∈ 𝑋 and 𝑒 ∈ 𝐴, we have ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥) ∪ ℎ𝐻[𝑒] ̃ (𝑥) = ℎ𝐻[𝑒] ̃ (𝑥) .

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This completes the proof. ̃ 𝐴) be a hesitant anti-fuzzy soft subalgeTheorem 12. Let (𝐻, ̃ 𝐵 , 𝐵) is a hesitant anti-fuzzy soft bra over 𝑋. If 𝐵 ⊆ 𝐴, then (𝐻| subalgebra over 𝑋. ̃ 𝐴) is a hesitant anti-fuzzy soft subProof. Suppose that (𝐻, ̃𝐴 is a hesitant anti-fuzzy subalgebra algebra over 𝑋. Then 𝐻 of 𝑋, ∀𝑥 ∈ 𝐴. Since 𝐵 ⊆ 𝐴, ℎ𝐵 (𝑏) is a hesitant anti-fuzzy ̃ 𝐵 , 𝐵) is a hesitant antisubalgebra of 𝑋, ∀𝑏 ∈ 𝐵. Hence, (𝐻| fuzzy soft subalgebra over 𝑋. The following example shows that there exists a hesitant ̃ 𝐴) over 𝑋 such that fuzzy soft set (𝐻, ̃ 𝐴) is not a hesitant anti-fuzzy soft subalgebra over (i) (𝐻, 𝑋, ̃ 𝐵 , 𝐵) is a (ii) there exists a subset 𝐵 of 𝐴 such that (𝐻| hesitant anti-fuzzy soft subalgebra over 𝑋. Example 13. Let 𝑋 = {0, 𝑎, 𝑏, 𝑐} be a BCK-algebra in Table 3 (Cayley). ̃ 𝐴) be Consider a set of parameters 𝐴 = {𝑒1 , 𝑒2 , 𝑒3 }. Let (𝐻, a hesitant fuzzy soft set over 𝑋 which is described in Table 4. ̃ 3 ] fl {(𝑥, ℎ ̃ (𝑥)) | 𝑥 ∈ 𝑋} on 𝑋 is not hesitant Then 𝐻[𝑒 𝐻[𝑒3 ] anti-fuzzy subalgebra of 𝑋, because ℎ𝐻[𝑒 ̃ 3 ] (𝑎 ∗ 𝑎) = ℎ𝐻[𝑒 ̃ 3 ] (0) = {0.1} ⊈ (0.1, 0.2) = ℎ𝐻[𝑒 ̃ 3 ] (𝑎) ∪ ℎ𝐻[𝑒 ̃ 3 ] (𝑎) .

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̃ 𝐴) is not hesitant anti-fuzzy soft subalgebra Therefore (𝐻, ̃ 𝐵 , 𝐵) is a hesitant of 𝑋. But if we take 𝐵 = {𝑒1 , 𝑒2 }, then (𝐻| anti-fuzzy soft subalgebra over 𝑋.

Table 3 ∗ 0 𝑎 𝑏 𝑐

0 0 𝑎 𝑏 𝑐

𝑎 0 0 𝑏 𝑐

𝑏 0 𝑎 0 𝑐

𝑐 0 0 0 0

̃ 𝐴). Table 4: Tabular representation of the hesitant fuzzy soft set (𝐻, ̃ 𝐻 𝑒1 𝑒2 𝑒3

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

𝑎 (0.3, 0.8) [0.3, 0.9) (0.1, 0.2)

𝑏 [0.3, 0.8] [0.3, 0.8) [0.1, 0.3]

𝑐 [0.3, 0.5] [0.3, 0.6] [0.1, 0.4]

̃ 𝐴) be a hesitant fuzzy soft set over 𝑋. Definition 14. Let (𝐻, ̃ 𝐴)𝜀 = (𝐻 ̃𝜖 , 𝐴) is called a For each 𝜀 ∈ 𝑃([0, 1]), the set (𝐻, ̃ 𝐴), where 𝐻 ̃𝜖 = {𝑥 ∈ 𝑋 : hesitant anti-𝜀-level soft set of (𝐻, ℎ𝐻[𝑒] ̃ (𝑥) ⊆ 𝜀} for all 𝑒 ∈ 𝐴. ̃ 𝐴) be a hesitant fuzzy soft set over Theorem 15. Let (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft subalgebra of 𝑋 if and 𝑋. (𝐻, ̃ 𝐴)𝜀 is a subalgebra over 𝑋 for each 𝜀 ∈ 𝑃([0, 1]). only if (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft subalProof. Suppose that (𝐻, ̃𝜖 , gebra of 𝑋. For each 𝜀 ∈ 𝑃([0, 1]), 𝑒 ∈ 𝐴 and 𝑥1 , 𝑥2 ∈ 𝐻 ℎ𝐻[𝑒] ̃ (𝑥1 ) ⊆ 𝜀 and ℎ𝐻[𝑒] ̃ (𝑥2 ) ⊆ 𝜀. Thus ℎ𝐻[𝑒] ̃ (𝑥1 ∗ 𝑥2 ) ⊆ ℎ𝐻[𝑒] ̃ (𝑥1 ) ∪ ℎ𝐻[𝑒] ̃ (𝑥) ⊆ 𝜀.

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̃𝜖 . Hence, (𝐻, ̃ 𝐴)𝜀 is a This implies that 𝑥1 ∗ 𝑥2 ∈ 𝐻 subalgebra over 𝑋. ̃ 𝐴)𝜀 is a subalgebra over 𝑋 for Conversely, assume that (𝐻, each 𝜀 ∈ 𝑃([0, 1]). For each 𝑒 ∈ 𝐴 and 𝑥, 𝑦 ∈ 𝑋, let ℎ𝐻[𝑒] ̃ (𝑥) = ̃𝜖 𝜀𝑥 and let ℎ𝐻[𝑒] ̃ (𝑦) = 𝜀𝑦 . Take 𝜀 = 𝜀𝑥 ∪ 𝜀𝑦 . Then 𝑥, 𝑦 ∈ 𝐻 and 𝜖 ̃ . Hence, so 𝑥 ∗ 𝑦 ∈ 𝐻 ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ 𝜀 = 𝜀𝑥 ∪ 𝜀𝑦 = ℎ𝐻[𝑒] ̃ (𝑥) ∪ ℎ𝐻[𝑒] ̃ (𝑦) .

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̃ 𝐴) is a hesitant anti-fuzzy subalgebra over Therefore (𝐻, ̃ 𝐴) is a hesitant 𝑋. Then, by Definition 8, we conclude that (𝐻, anti-fuzzy soft subalgebra of 𝑋. This completes the proof.

4. Hesitant Anti-Fuzzy Soft Ideals Definition 16. Given a nonempty subset (subalgebra as much as possible) 𝐴 of 𝑋, let 𝐻𝐴 fl {(𝑥, ℎ𝐴 (𝑥)) | 𝑥 ∈ 𝑋} be an 𝐴-hesitant fuzzy set on 𝑋. Then 𝐻𝐴 fl {(𝑥, ℎ𝐴 (𝑥)) | 𝑥 ∈ 𝑋} is called a hesitant anti-fuzzy ideal of 𝑋 related to 𝐴 (briefly, 𝐴-hesitant anti-fuzzy ideal of 𝑋) if it satisfies the following conditions: (HAFI1) ℎ𝐴 (0) ⊆ ℎ𝐴 (𝑥). (HAFI2) ℎ𝐴 (𝑥) ⊆ ℎ𝐴 (𝑥 ∗ 𝑦) ∪ ℎ𝐴 (𝑦) for all 𝑥, 𝑦 ∈ 𝐴. An 𝐴-hesitant anti-fuzzy ideal of 𝑋 with 𝐴 = 𝑋 is called a hesitant anti-fuzzy ideal of 𝑋.

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̃ 𝐴). Table 5: Tabular representation of the hesitant fuzzy soft set (𝐻, ̃ 𝐻 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)

̃ 𝐴) be a hesitant fuzzy soft set over 𝑋, Definition 17. Let (𝐻, ̃ 𝐴) is called a where 𝐴 is a subset of 𝐸. Given 𝑒 ∈ 𝐴, (𝐻, hesitant anti-fuzzy soft ideal based on 𝑒 (briefly, 𝑒-hesitant anti-fuzzy soft ideal) over 𝑋, if the hesitant fuzzy set, ̃ [𝑒] fl {(𝑥, ℎ ̃ (𝑥)) | 𝑥 ∈ 𝑋} , 𝐻 𝐻[𝑒]

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̃ 𝐴) is an 𝑒on 𝑋 is a hesitant anti-fuzzy ideal of 𝑋. If (𝐻, hesitant anti-fuzzy soft ideal over 𝑋 for all 𝑒 ∈ 𝐴, we say ̃ 𝐴) is a hesitant anti-fuzzy soft ideal over 𝑋. that (𝐻, Example 18. Let 𝑋 fl {apple, banana, carrot, peach, radish} be a reference set, and consider a soft machine ◊ which produces the following products: {apple if 𝑥 ∈ {carrot, peach, radish} apple ◊𝑥 = { carrot if 𝑥 ∈ {apple, banana} {

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̃𝜖 , and this implies that (𝐻, ̃ 𝐴)𝜀 ̃𝜖 and 𝑥 ∈ 𝐻 Hence, 0 ∈ 𝐻 is an ideal over 𝑋. ̃ 𝐴)𝜀 is an ideal over 𝑋 for Conversely, assume that (𝐻, each 𝜀 ∈ 𝑃([0, 1]). For each 𝑒 ∈ 𝐴 and 𝑥, 𝑦 ∈ 𝑋, let ̃𝜖 ̃ 𝜀 ℎ𝐻[𝑒] ̃ (𝑥) = 𝜀𝑥 , and then 𝑥 ∈ 𝐻 . Since (𝐻, 𝐴) is an ideal ̃𝜖 and so ℎ ̃ (0) ⊆ 𝜀𝑥 = ℎ ̃ (𝑥). over 𝑋, we have 0 ∈ 𝐻 𝐻[𝑒] 𝐻[𝑒] Let ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) = 𝜀𝑥∗𝑦 and let ℎ𝐻[𝑒] ̃ (𝑦) = 𝜀𝑦 . If we take ̃𝜖 and 𝑦 ∈ 𝐻 ̃𝜖 which imply 𝜀 = 𝜀𝑥∗𝑦 ∪ 𝜀𝑦 , then 𝑥 ∗ 𝑦 ∈ 𝐻 ̃𝜖 . Thus, that 𝑥 ∈ 𝐻 ℎ𝐻[𝑒] ̃ (𝑥) ⊆ 𝜀 = 𝜀𝑥∗𝑦 ∪ 𝜀𝑦 = ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) . (14) ̃ 𝐴) is a hesitant anti-fuzzy soft ideal of 𝑋. Therefore (𝐻, ̃ 𝐴) Proposition 20. Every hesitant anti-fuzzy soft ideal (𝐻, over a BCK/BCI-algebra 𝑋 satisfies the following condition for all 𝑒 ∈ 𝐴 and 𝑥, 𝑦, 𝑧 ∈ 𝑋: (b) ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧 ∗ 𝑦). (c) If 𝑥 ∗ 𝑦 ≤ 𝑧, then ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑧). (d) If ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) = ℎ𝐻[𝑒] ̃ (0), then ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑦).

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{peach if 𝑧 ∈ {carrot, radish} peach ◊𝑧 = { carrot if 𝑧 ∈ {apple, banana, peach} {

(e) ℎ𝐻[𝑒] ̃ (0 ∗ (0 ∗ 𝑥)) ⊆ ℎ𝐻[𝑒] ̃ (𝑥). Proof. Let 𝑒 ∈ 𝐴 and 𝑥, 𝑦, 𝑧 ∈ 𝑋. ̃ 𝐴) is a hesitant (a) If 𝑥 ≤ 𝑦, then 𝑥 ∗ 𝑦 = 0. Since (𝐻, anti-fuzzy soft ideal of 𝑋,

{radish if 𝑢 ∈ {carrot, peach} radish ◊𝑢 = { carrot if 𝑢 ∈ {apple, banana, radish} {

ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) = ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑦) = ℎ𝐻[𝑒] ̃ (𝑦) .

carrot ◊V = carrot ∀V ∈ 𝑋. Then 𝑋 is a BCK-algebra under the soft machine ◊. Consider a set of parameters 𝐴 fl {Cat, Cow, Dog, Horse}; let ̃ 𝐴) be a hesitant fuzzy soft set over 𝑋 which is described (𝐻, in Table 5. ̃ 𝐴) is a hesitant anti-fuzzy It is routine to verify that (𝐻, soft ideal over 𝑋 based on parameters “cat,” “cow,” and “dog.” ̃ 𝐴) is not a hesitant anti-fuzzy soft ideal of 𝑋 based on But (𝐻, parameter “horse” because ̃ [horse] (peach) = [0.3, 0.9) ⊈ [0.3, 0.4) 𝐻 ̃ [horse] (peach◊banana) =𝐻

ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) ⊆ 𝜀.

(a) If 𝑥 ≤ 𝑦, then ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑦).

banana ◊𝑦 {banana if 𝑦 ∈ {apple, carrot, peach, radish} ={ carrot if 𝑦 = banana {

̃ 𝐴) is a hesitant anti-fuzzy soft ideal Proof. Suppose that (𝐻, ̃𝜖 such that of 𝑋. For each 𝜀 ∈ 𝑃([0, 1]), 𝑒 ∈ 𝐴 and 𝑥, 𝑦 ∈ 𝐻 𝜖 𝜖 ̃ and 𝑦 ∈ 𝐻 ̃ ; then ℎ ̃ (𝑥∗𝑦) ⊆ 𝜀 and ℎ ̃ (𝑦) ⊆ 𝜀. 𝑥∗𝑦 ∈ 𝐻 𝐻[𝑒] 𝐻[𝑒] Thus,

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̃ [horse] (banana) . ∪𝐻 ̃ 𝐴) be a hesitant fuzzy soft set over Theorem 19. Let (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft ideal of 𝑋 if and only 𝑋. (𝐻, ̃ 𝐴)𝜀 is an ideal over 𝑋 for each 𝜀 ∈ 𝑃([0, 1]). if (𝐻,

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(b) Since (𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑧) ≤ 𝑧 ∗ 𝑦, it follows from (a) that ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑧)) ⊆ ℎ𝐻[𝑒] ̃ (𝑧 ∗ 𝑦). Hence ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑧)) ∪ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧)

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⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧 ∗ 𝑦) , (c) If 𝑥 ∗ 𝑦 ≤ 𝑧, then (𝑥 ∗ 𝑦) ∗ 𝑧 = 0. Since ℎ𝐻[𝑒] ̃ (𝑧) = ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑧) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧)

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⊇ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) , it follows that ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) ⊆ ℎ𝐻[𝑒] ̃ (𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑧).


5

(d) If ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) = ℎ𝐻[𝑒] ̃ (0), then we have ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) = ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑦) = ℎ𝐻[𝑒] ̃ (𝑦) .

Also, (18)

(e) For all 𝑥 ∈ 𝑋,

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= ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑥) = ℎ𝐻[𝑒] ̃ (𝑥) . The proof is complete. ̃ 𝐴) be a hesitant fuzzy soft set over 𝑋 Theorem 21. Let (𝐻, which satisfies condition (5) and (c) from Proposition 20. Then ̃ 𝐴) is a hesitant anti-fuzzy soft ideal over 𝑋. (𝐻, Proof. Let 𝑒 ∈ 𝐴. Since 𝑥 ∗ (𝑥 ∗ 𝑦) ≤ 𝑦, for all 𝑥, 𝑦 ∈ 𝑋, it follows from Proposition 20(c) that ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) .

= ℎ𝐻[𝑒 ̃ 2 ] (𝑎 ∗ 𝑐) ∪ ℎ𝐻[𝑒 ̃ 2 ] (𝑐) .

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We provide a condition for a hesitant anti-fuzzy soft subalgebra over 𝑋 to be a hesitant anti-fuzzy soft ideal over 𝑋.

ℎ𝐻[𝑒] ̃ (0 ∗ (0 ∗ 𝑥)) ⊆ ℎ𝐻[𝑒] ̃ ((0 ∗ (0 ∗ 𝑥)) ∗ 𝑥) ∪ ℎ𝐻[𝑒] ̃ (𝑥)

ℎ𝐻[𝑒 ̃ 2 ] (𝑎) = [0.3, 0.9) ⊈ [0.3, 0.6]

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̃ 𝐴) is a hesitant anti-fuzzy soft ideal over 𝑋. Hence, (𝐻, Theorem 22. Every hesitant anti-fuzzy soft ideal (based on a parameter) over BCK-algebra 𝑋 is a hesitant anti-fuzzy soft subalgebra (based on the same parameter) over 𝑋.

̃ 𝐴) be a hesitant anti-fuzzy soft subalgeTheorem 24. Let (𝐻, ̃ 𝐴) bra over 𝑋. If the inequality 𝑥 ∗ 𝑦 ≤ 𝑧 holds in 𝑋, then (𝐻, is a hesitant anti-fuzzy soft ideal over 𝑋. ̃ 𝐴) is a hesitant anti-fuzzy soft subalProof. Suppose that (𝐻, gebra over 𝑋. Then, from Proposition 11, we have ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥) , ∀𝑥 ∈ 𝑋, 𝑒 ∈ 𝐴.

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Assume that 𝑥 ∗ 𝑦 ≤ 𝑧 holds in 𝑋. Then, by Proposition 20(c), we get ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑧)

∀𝑥, 𝑦, 𝑧 ∈ 𝑋.

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Since 𝑥 ∗ (𝑥 ∗ 𝑦) ≤ 𝑦, ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦)

∀𝑥, 𝑦 ∈ 𝑋.

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̃ 𝐴) is a hesitant anti-fuzzy soft ideal over 𝑋. Hence, (𝐻,

̃ 𝐴) is a hesitant antiProof. For any 𝑒 ∈ 𝐴, assume that (𝐻, fuzzy soft ideal over 𝑋. Then

5. Hesitant Anti-Fuzzy Soft Implicative Ideals

ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑥) ∪ ℎ𝐻[𝑒] ̃ (𝑥) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑥) ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑥) = ℎ𝐻[𝑒] ̃ (0 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑥)

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= ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑥)

(HAFII1) ℎ𝑋 (0) ⊆ ℎ𝑋 (𝑥), for all 𝑥 ∈ 𝑋.

⊆ ℎ𝐻[𝑒] ̃ (𝑥) ∪ ℎ𝐻[𝑒] ̃ (𝑦) , ̃ 𝐴) is a hesitant anti-fuzzy soft for all 𝑥, 𝑦 ∈ 𝑋, and so (𝐻, subalgebra over 𝑋. The following example shows that the converse of Theorem 22 is not true in general. Example 23. Let 𝑋 = {0, 𝑎, 𝑏, 𝑐} in Example 13 and 𝐵 = ̃ 𝐵) is a hesitant anti-fuzzy soft subalge{𝑒1 , 𝑒2 }, and then (𝐻, bra over 𝑋. But it is not a hesitant anti-fuzzy soft ideal over 𝑋 based on parameters “𝑒1 ” and “𝑒2 ” since ℎ𝐻[𝑒 ̃ 1 ] (𝑎) = (0.3, 0.8) ⊈ [0.3, 0.5] = (0.3, 0.5) ∪ [0.3, 0.5] = ℎ𝐻[𝑒 ̃ 1 ] (0) ∪ ℎ𝐻[𝑒 ̃ 1 ] (𝑐) = ℎ𝐻[𝑒 ̃ 1 ] (𝑎 ∗ 𝑐) ∪ ℎ𝐻[𝑒 ̃ 1 ] (𝑐) .

Definition 25. A hesitant fuzzy set 𝐻𝑋 fl {(𝑥, ℎ𝑋 (𝑥)) | 𝑥 ∈ 𝑋} in a BCK-algebra 𝑋 is called a hesitant antifuzzy implicative ideal (briefly, HAFII) of 𝑋 if it satisfies the following conditions:

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(HAFII2) ℎ𝑋 (𝑥) ⊆ ℎ𝑋 ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝑋 (𝑧), for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. ̃ 𝐴) be a hesitant fuzzy soft set over 𝑋, Definition 26. Let (𝐻, ̃ 𝐴) is called a where 𝐴 is a subset of 𝐸. Given 𝑒 ∈ 𝐴, (𝐻, hesitant anti-fuzzy soft implicative ideal based on 𝑒 (briefly, 𝑒-hesitant anti-fuzzy soft implicative ideal) over 𝑋 if the hesitant fuzzy set, ̃ [𝑒] fl {(𝑥, ℎ ̃ (𝑥)) | 𝑥 ∈ 𝑋} , 𝐻 𝐻[𝑒]

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̃ 𝐴) on 𝑋 is a hesitant anti-fuzzy implicative ideal of 𝑋. If (𝐻, is an 𝑒-hesitant anti-fuzzy soft implicative ideal over 𝑋, for ̃ 𝐴) is a hesitant anti-fuzzy soft all 𝑒 ∈ 𝐴, we say that (𝐻, implicative ideal over 𝑋. Proposition 27. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra 𝑋 is order-preserving.

6


̃ 𝐴) be a hesitant anti-fuzzy soft implicative Proof. Let (𝐻, ideal over 𝑋. Let 𝑒 ∈ 𝐴 and 𝑥, 𝑦 ∈ 𝑋 be such that 𝑥 ≤ 𝑦; then ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑧 ∗ 𝑥)) ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ (𝑧 ∗ 𝑥)) ∪ ℎ𝐻[𝑒] ̃ (𝑦) = ℎ𝐻[𝑒] ̃ (0 ∗ (𝑧 ∗ 𝑥)) ∪ ℎ𝐻[𝑒] ̃ (𝑦)

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Table 6 ∗ 0 𝑎 𝑏 𝑐

0 0 𝑎 𝑏 𝑐

Hence, ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑦), and this completes the proof.

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

𝑒1 𝑒2

Proposition 28. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra 𝑋 is a hesitant anti-fuzzy soft ideal of 𝑋. ̃ 𝐴) be a hesitant anti-fuzzy soft implicative Proof. Let (𝐻, ideal over 𝑋. Let 𝑒 ∈ 𝐴 and 𝑥, 𝑦, 𝑧 ∈ 𝑋; then ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) .

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Replace 𝑦 = 𝑥, and using 𝑥 ∗ 𝑥 = 0 we get ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑥 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) = ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) .

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𝑏 0 0 0 𝑐

𝑐 0 𝑎 𝑏 0

̃ 𝐴). Table 7: Tabular representation of the hesitant fuzzy soft set (𝐻, ̃ 𝐻

= ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑦) = ℎ𝐻[𝑒] ̃ (𝑦) .

𝑎 0 0 𝑎 𝑐

𝑎 1 [0, ) 2 1 (0, ) 2

𝑏 1 [0, ) 2 1 [0, ] 2

𝑐 3 [0, ] 4 3 [0, ) 4

Also, 1 ℎ𝐻[𝑒 ̃ 2 ] (𝑎) = (0, ) 2 ⊈ ℎ𝐻[𝑒 ̃ 2 ] ((𝑎 ∗ (𝑏 ∗ 𝑎)) ∗ 0) ∪ ℎ𝐻[𝑒 ̃ 2 ] (0)

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1 = ℎ𝐻[𝑒 ̃ 2 ] (0) ∪ ℎ𝐻[𝑒 ̃ 2 ] (0) = (0, ) . 4 Now we give a condition for a hesitant anti-fuzzy soft ideal over 𝑋 to be a hesitant anti-fuzzy soft implicative ideal over 𝑋.

̃ 𝐴) is a hesitant anti-fuzzy soft ideal. For all 𝑥, 𝑧 ∈ 𝑋, (𝐻,

Theorem 31. If 𝑋 is an implicative BCK-algebra, then every hesitant anti-fuzzy soft ideal over 𝑋 is a hesitant anti-fuzzy soft implicative ideal over 𝑋.

Combining Proposition 28 and Theorem 22 yields the following result.

Proof. Let 𝑋 be an implicative BCK-algebra; it follows that ̃ 𝐴) be hesitant anti𝑥 = 𝑥 ∗ (𝑦 ∗ 𝑥), ∀𝑥, 𝑦 ∈ 𝑋. And let (𝐻, fuzzy soft ideal over 𝑋. Then, for any 𝑒 ∈ 𝐴, we have

Corollary 29. Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebra 𝑋 is a hesitant anti-fuzzy soft subalgebra of 𝑋. The following example shows that the converse of Proposition 28 may not be true in general. Example 30. Let 𝑋 = {0, 𝑎, 𝑏, 𝑐} be a BCK-algebra in Table 6 (Cayley). ̃ 𝐴) be a Consider a set of parameters 𝐴 = {𝑒1 , 𝑒2 }. Let (𝐻, hesitant fuzzy soft set over 𝑋 which is described in Table 7. ̃ 𝐴) is a hesitant anti-fuzzy soft ideal (subalgebra) Then (𝐻, ̃ 1 ] fl {(𝑥, ℎ ̃ (𝑥)) | 𝑥 ∈ 𝑋} and 𝐻[𝑒 ̃ 2 ] fl of 𝑋. But 𝐻[𝑒 𝐻[𝑒1 ] {(𝑥, ℎ𝐻[𝑒 ̃ 2 ] (𝑥)) | 𝑥 ∈ 𝑋} on 𝑋 is not hesitant anti-fuzzy soft implicative ideal of 𝑋, because 1 ℎ𝐻[𝑒 ̃ 1 ] (𝑎) = [0, ) 2 ⊈ ℎ𝐻[𝑒 ̃ 1 ] ((𝑎 ∗ (𝑏 ∗ 𝑎)) ∗ 0) ∪ ℎ𝐻[𝑒 ̃ 1 ] (0) 1 1 = ℎ𝐻[𝑒 ̃ 1 ] (0) ∪ ℎ𝐻[𝑒 ̃ 1 ] (0) = ( , ) . 4 2

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ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧)

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for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Hence, it is a hesitant anti-fuzzy implĩ 𝐴) is hesitant anti-fuzzy soft cative ideal of 𝑋. That is, (𝐻, implicative ideal of 𝑋. ̃ 𝐴) be a hesitant fuzzy soft set in a BCKTheorem 32. Let (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft implicative algebra 𝑋. Then (𝐻, ideal of 𝑋 if and only if for each 𝜀 ∈ 𝑃([0, 1]) the hesitant antĩ 𝐴)𝜀 of (𝐻, ̃ 𝐴) is empty or an implicative fuzzy soft 𝜀-level set (𝐻, ideal of 𝑋. ̃ 𝐴) be a hesitant anti-fuzzy soft implicative Proof. Let (𝐻, ̃𝜀 ≠ ⌀ for every 𝜀 ∈ 𝑃([0, 1]). ideal of 𝑋 and assume that 𝐻 Note that ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥), for all 𝑒 ∈ 𝐴 and 𝑥 ∈ 𝑋. In ̃𝜀 , ℎ ̃ (0) ⊆ ℎ ̃ (𝑥) ⊆ 𝜀 and so particular, for each 𝑥 ∈ 𝐻 𝐻[𝑒] 𝐻[𝑒] ̃𝜀 . Let (𝑥∗(𝑦∗𝑥))∗𝑧 ∈ 𝐻 ̃𝜀 and 𝑧 ∈ 𝐻 ̃𝜀 , and since (𝐻, ̃ 𝐴) 0∈𝐻 is a hesitant anti-fuzzy implicative ideal over 𝑋, it follows that ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) ⊆ 𝜀. (34) ̃𝜀 is an implicative ideal of 𝑋. ̃𝜀 , and 𝐻 Hence 𝑥 ∈ 𝐻


7

̃𝜀 be an implicative ideal of 𝑋; we first Conversely, let 𝐻 show ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥), for all 𝑒 ∈ 𝐴 and 𝑥 ∈ 𝑋. If not, then there exists 𝑥0 ∈ 𝑋 such that ℎ𝐻[𝑒] ̃ (0) ⊇ ℎ𝐻[𝑒] ̃ (𝑥0 ). Let (0) + ℎ (𝑥 )}; then 0 ≤ ℎ 𝜀0 = (1/2){ ℎ𝐻[𝑒] ̃ ̃ ̃ (𝑥0 ) ⊂ 𝜀0 ⊂ 0 𝐻[𝑒] 𝐻[𝑒] 𝜀0 𝜀0 ̃ ̃ ̃𝜀0 is an ℎ𝐻[𝑒] ≠ ⌀. As 𝐻 ̃ (0) ≤ 1. Thus, 𝑥0 ∈ 𝐻 and 𝐻 𝜀 ̃ 0 , which implies that ℎ ̃ (0) ⊆ implicative ideal of 𝑋, 0 ∈ 𝐻 𝐻[𝑒] 𝜀0 . This is a contradiction and so ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥) for all 𝑒 ∈ 𝐴 and 𝑥 ∈ 𝑋. Now assume that ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥))∗𝑧)∪ℎ𝐻[𝑒] ̃ (𝑧) does not hold. Then, ℎ𝐻[𝑒] ̃ (𝑥0 ) ⊇ ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ (𝑦0 ∗ 𝑥0 )) ∗ 𝑧0 ) ∪ ℎ𝐻[𝑒] ̃ (𝑧0 ) for some 𝑥0 , 𝑦0 , 𝑧0 ∈ 𝑋. If we take 𝜀0 =

1 {ℎ ̃ (𝑥 ) 2 𝐻[𝑒] 0

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+ {ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ (𝑦0 ∗ 𝑥0 )) ∗ 𝑧0 ) ∪ ℎ𝐻[𝑒] ̃ (𝑧0 )}} , then 0 ≤ ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ (𝑦0 ∗ 𝑥0 )) ∗ 𝑧0 ) ∪ ℎ𝐻[𝑒] ̃ (𝑧0 ) ⊆ 𝜀0 ⊆ ℎ𝐻[𝑒] ̃ (𝑥0 ) ≤ 1. It follows that ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ (𝑦0 ∗ 𝑥0 )) ∗ 𝑧0 ) ⊆ 𝜀0 ̃𝜀0 and ℎ𝐻[𝑒] ̃ (𝑧0 ) ⊆ 𝜀0 , which imply that (𝑥0 ∗(𝑦0 ∗𝑥0 ))∗𝑧0 ∈ 𝐻 𝜀 𝜀 ̃ 0 . But 𝐻 ̃ 0 is an implicative ideal of 𝑋, so 𝑥0 ∈ and 𝑧0 ∈ 𝐻 𝜀0 ̃ 𝐻 or ℎ𝐻[𝑒] ̃ (𝑥0 ) ⊆ 𝜀0 , and this is a contradiction. Therefore ̃ 𝐴) is a hesitant anti-fuzzy soft implicative ideal of 𝑋. (𝐻, ̃ 𝐴) be a hesitant anti-fuzzy soft ideal of Theorem 33. Let (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft a BCK-algebra 𝑋. Then (𝐻, implicative ideal of 𝑋 if and only if it satisfies the condition ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥))

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for all 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴. ̃ 𝐴) is a hesitant anti-fuzzy soft Proof. Assume that (𝐻, implicative ideal of 𝑋. Take 𝑧 = 0 in ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 0) ∪ ℎ𝐻[𝑒] ̃ (0)

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= ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) . ̃ 𝐴) satisfies the condition. As Conversely, suppose that (𝐻, ̃ (𝐻, 𝐴) is a hesitant anti-fuzzy soft ideal of 𝑋, we have ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) .

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̃ 𝐴) is a hesitant anti-fuzzy soft implicative ideal Then (𝐻, of 𝑋 and the proof is completed. Now we give characterizations of hesitant anti-fuzzy soft implicative ideals. ̃ 𝐴) be a hesitant anti-fuzzy soft ideal of Theorem 34. Let (𝐻, a BCK-algebra 𝑋. Then the following are equivalent: ̃ 𝐴) is a hesitant anti-fuzzy soft implicative ideal of (i) (𝐻, 𝑋. (ii) ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥∗(𝑦∗𝑥)) for all 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴. (iii) ℎ𝐻[𝑒] ̃ (𝑥) = ℎ𝐻[𝑒] ̃ (𝑥∗(𝑦∗𝑥)) for all 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴.

̃ 𝐴) be a hesitant anti-fuzzy soft Proof. (i) ⇒ (ii) Let (𝐻, implicative ideal of 𝑋. Then by Theorem 33 we have ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 0) ∪ ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) ∪ ℎ𝐻[𝑒] ̃ (0)

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= ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) , for all 𝑥, 𝑦 ∈ 𝑋. Hence condition (ii) holds. (ii) ⇒ (iii) Observe that, in BCK-algebra 𝑋, 𝑥∗(𝑦∗𝑥) ≤ 𝑥. Applying Proposition 20(a), we obtain ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) ⊆ ℎ𝐻[𝑒] ̃ (𝑥). Since (ii) holds, ℎ𝐻[𝑒] ̃ (𝑥) = ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) for all 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴. So, condition (iii) holds. ̃ 𝐴) (iii) ⇒ (i) Suppose that condition (iii) holds. Since (𝐻, is a hesitant anti-fuzzy soft ideal of 𝑋, by Definition 16, we have ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) .

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Combining (iii), we obtain ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) .

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̃ 𝐴) is a hesitant anti-fuzzy soft implicative ideal Thus (𝐻, of 𝑋. So, condition (i) holds. ̃ 𝐴) be a hesitant anti-fuzzy soft implicaTheorem 35. Let (𝐻, tive ideal of a BCK-algebra 𝑋; then the set 𝐻ℎ𝐻[𝑒] fl {𝑥 ∈ 𝑋 | ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (𝑥)} ̃

(42)

is an implicative ideal of 𝑋. . Let 𝑥, 𝑦, 𝑧 ∈ 𝐻ℎ𝐻[𝑒] be such that Proof. Clearly, 0 ∈ 𝐻ℎ𝐻[𝑒] ̃ ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧 ∈ 𝐻ℎ𝐻[𝑒] and 𝑧 ∈ 𝐻ℎ𝐻[𝑒] , for all 𝑒 ∈ 𝐴; then ̃ ̃ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (𝑧) .

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̃ 𝐴) is a hesitant anti-fuzzy soft implicative ideal Since (𝐻, of 𝑋, it follows that ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) = ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (0) .

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Combining Definition 25(HAFII1), we get ℎ𝐻[𝑒] ̃ (𝑥) = . Therefore 𝐻ℎ𝐻[𝑒] is an ℎ𝐻[𝑒] ̃ (0) and hence 𝑥 ∈ 𝐻ℎ𝐻[𝑒] ̃ ̃ implicative ideal of 𝑋. Theorem 36. A hesitant anti-fuzzy soft subalgebra ℎ𝐻[𝑒] of 𝑋 ̃ is a hesitant anti-fuzzy soft implicative ideal if and only if it satisfies the condition (𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧 ≤ 𝑢 implying that ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑢) for all 𝑥, 𝑦, 𝑧, 𝑢 ∈ 𝑋 and ∀𝑒 ∈ 𝐴. is a hesitant anti-fuzzy soft implicaProof. Assume that ℎ𝐻[𝑒] ̃ tive ideal of 𝑋 and let 𝑥, 𝑦, 𝑧, 𝑢 ∈ 𝑋 be such that (𝑥 ∗ (𝑦 ∗

8


𝑥)) ∗ 𝑧 ≤ 𝑢. Since ℎ𝐻[𝑒] is also hesitant anti-fuzzy soft ideal of ̃ 𝑋, by Proposition 28, it follows from Proposition 20(c) that ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) ⊆ ℎ𝐻[𝑒] ̃ (𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑢) .

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Making use of Theorem 34, we obtain ℎ𝐻[𝑒] ⊆ ̃ (𝑥) ℎ𝐻[𝑒] ̃ (𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑢). satisfies the condition in Conversely, suppose that ℎ𝐻[𝑒] ̃ satisfies (5) from Proposition 11. theorem. Obviously, ℎ𝐻[𝑒] ̃ Since (𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ≤ 𝑧, it follows from Proposition 20(c) and Theorem 34 that ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑥)) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) ,

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which shows that ℎ𝐻[𝑒] satisfies conditions of a hesitant antĩ is a hesitant antifuzzy implicative ideal of 𝑋 and so ℎ𝐻[𝑒] ̃ fuzzy soft implicative ideal of 𝑋. The proof is complete.

Definition 37. A hesitant fuzzy set 𝐻𝑋 fl {(𝑥, ℎ𝑋 (𝑥)) | 𝑥 ∈ 𝑋} in a BCK-algebra 𝑋 is called a hesitant anti-fuzzy positive implicative ideal (briefly, HAFPII) of 𝑋 if it satisfies the following conditions: (HAFPII1) ℎ𝑋 (0) ⊆ ℎ𝑋 (𝑥), for all 𝑥 ∈ 𝑋. (HAFPII2) ℎ𝑋 (𝑥 ∗ 𝑧) ⊆ ℎ𝑋 ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝑋 (𝑦 ∗ 𝑧), for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. ̃ 𝐴) be a hesitant fuzzy soft set over 𝑋, Definition 38. Let (𝐻, ̃ 𝐴) is called a where 𝐴 is a subset of 𝐸. Given 𝑒 ∈ 𝐴, (𝐻, hesitant anti-fuzzy soft positive implicative ideal based on 𝑒 (briefly, 𝑒-hesitant anti-fuzzy soft positive implicative ideal) over 𝑋 if the hesitant fuzzy set, (47)

on 𝑋 is a hesitant anti-fuzzy positive implicative ideal of 𝑋. ̃ 𝐴) is an 𝑒-hesitant anti-fuzzy soft positive implicative If (𝐻, ̃ 𝐴) is a hesitant ideal over 𝑋, for all 𝑒 ∈ 𝐴, we say that (𝐻, anti-fuzzy soft positive implicative ideal over 𝑋. Proposition 39. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra 𝑋 is order-preserving. Proof. Let 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴 be such that 𝑥 ≤ 𝑦. Since ̃ 𝐴) is a hesitant anti-fuzzy soft positive implicative ideal (𝐻, of 𝑋, ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (0 ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧)

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(50)

Putting 𝑧 = 0, ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) .

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̃ 𝐴) is a hesitant anti-fuzzy soft ideal of 𝑋. Therefore (𝐻,

Proposition 41. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra 𝑋 is a hesitant anti-fuzzy soft subalgebra of 𝑋. Remark 42. A hesitant anti-fuzzy soft ideal (subalgebra) of a BCK-algebra 𝑋 may not be a hesitant anti-fuzzy soft positive implicative ideal of 𝑋 as shown in the following example. Example 43. Let 𝑋 be the BCK-algebra in Example 30. ̃ 𝐴) is a hesitant anti-fuzzy Routine calculations give that (𝐻, soft ideal (subalgebra) of 𝑋, but it is not a hesitant anti-fuzzy soft positive implicative ideal of 𝑋, because 1 ℎ𝐻[𝑒 ̃ 1 ] (𝑏 ∗ 𝑎) = ℎ𝐻[𝑒 ̃ 1 ] (𝑎) = [0, ) 2 ⊈ ℎ𝐻[𝑒 ̃ 1 ] ((𝑏 ∗ 𝑎) ∗ 𝑎) ∪ ℎ𝐻[𝑒 ̃ 1 ] (𝑎 ∗ 𝑎)

The converse of Proposition 40 is valid if the following condition holds. Proposition 44. If 𝑋 is a positive implicative BCK-algebra, then every hesitant anti-fuzzy soft ideal of 𝑋 is a hesitant antifuzzy soft positive implicative ideal of 𝑋. ̃ 𝐴) is a hesitant anti-fuzzy soft ideal Proof. Assume that (𝐻, of a positive implicative BCK-algebra 𝑋, for all 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴, and then (53)

By replacing 𝑥 by 𝑥 ∗ 𝑧 and 𝑦 by 𝑦 ∗ 𝑧, we get

∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) . (49)

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1 1 = ℎ𝐻[𝑒 ̃ 1 ] (0) ∪ ℎ𝐻[𝑒 ̃ 1 ] (0) = ( , ) . 4 2

ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧))

Putting 𝑧 = 0,

The proof is complete.

ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) .

ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎH[𝑒] ̃ (𝑥 ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦) .

= ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) . ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑦) .

̃ 𝐴) be a hesitant anti-fuzzy soft positive Proof. Let (𝐻, implicative ideal of a BCK-algebra 𝑋, so for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑒 ∈ 𝐴 we have

Combining Theorem 21 and Proposition 39 yields the following result.

6. Hesitant Anti-Fuzzy Soft Positive Implicative Ideals

̃ [𝑒] fl {(𝑥, ℎ ̃ (𝑥)) | 𝑥 ∈ 𝑋} , 𝐻 𝐻[𝑒]

Proposition 40. Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra 𝑋 is a hesitant anti-fuzzy soft ideal of 𝑋.

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Since 𝑋 is a positive implicative BCK-algebra, (𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧) = (𝑥 ∗ 𝑦) ∗ 𝑧 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Hence, ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) .

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9

̃ 𝐴) is a hesitant anti-fuzzy soft positive This shows that (𝐻, implicative ideal of 𝑋. ̃ 𝐴) be a hesitant anti-fuzzy soft ideal over Theorem 45. Let (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft positive implicative 𝑋; then (𝐻, ideal of 𝑋 if and only if ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ,

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̃ 𝐴) is a hesitant anti-fuzzy soft ideal Proof. Suppose that (𝐻, over 𝑋 and (57)

for all 𝑒 ∈ 𝐴 and 𝑥, 𝑦, 𝑧 ∈ 𝑋.

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̃ 𝐴) is a hesitant anti-fuzzy soft positive implicaThus (𝐻, tive ideal of 𝑋. ̃ 𝐴) is a hesitant anti-fuzzy Conversely, assume that (𝐻, ̃ 𝐴) is soft positive implicative ideal of 𝑋 implying that (𝐻, a hesitant anti-fuzzy soft ideal of 𝑋 by Proposition 40. Let 𝑎 = 𝑥 ∗ (𝑦 ∗ 𝑧) and let 𝑏 = 𝑥 ∗ 𝑦; since (59)

fl {𝑥 ∈ 𝑋 | ℎ𝐻[𝑒] 𝐻ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (𝑥)} ̃

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. Let 𝑥, 𝑦, 𝑧 ∈ 𝐻ℎ𝐻[𝑒] be Proof. Let 𝑒 ∈ 𝐴. Clearly, 0 ∈ 𝐻ℎ𝐻[𝑒] ̃ ̃ such that (𝑥 ∗ 𝑦) ∗ 𝑧 ∈ 𝐻ℎ𝐻[𝑒] and 𝑦 ∗ 𝑧 ∈ 𝐻 , and then ℎ ̃ ̃ 𝐻[𝑒] (65)

It follows that ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (0) ,

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̃ 𝐴) is a hesitant anti-fuzzy positive implicative ideal since (𝐻, of 𝑋, and then ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (0) and hence 𝑥 ∗ 𝑧 ∈ .Therefore 𝐻 is a positive implicative ideal of 𝑋. 𝐻ℎ𝐻[𝑒] ℎ𝐻[𝑒] ̃ ̃ ̃ 𝐴) be a hesitant anti-fuzzy soft ideal over Theorem 48. Let (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft positive implicative 𝑋; then (𝐻, ideal of 𝑋 if and only if it satisfies the inequalities

using Proposition 20(a), we have ℎ𝐻[𝑒] ̃ ((𝑎 ∗ 𝑏) ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ (𝑥 ∗ 𝑦)) ∗ 𝑧)

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̃ 𝐴) be a hesitant anti-fuzzy soft positive Theorem 47. Let (𝐻, implicative ideal of a BCK-algebra 𝑋, for all 𝑒 ∈ 𝐴. Then the set

ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) .

⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) .

((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ (𝑥 ∗ 𝑦)) ≤ 𝑦 ∗ (𝑦 ∗ 𝑧) ,

⊆ ℎ𝐻[𝑒] ̃ (𝑎) ∪ ℎ𝐻[𝑒] ̃ (𝑏) .

is a positive implicative ideal of 𝑋.

ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧)

ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧)

This completes the proof.


ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ,

Proof. Let 𝑥, 𝑦, 𝑧, 𝑎, 𝑏 ∈ 𝑋 be such that ((𝑥 ∗ 𝑦) ∗ 𝑧) ∗ 𝑎 ≤ 𝑏. ̃ 𝐴) is a hesitant anti-fuzzy soft positive implicative Since (𝐻, ideal of 𝑋, it follows from Theorem 45 and Proposition 20(c) that

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ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦)

⊆ ℎ𝐻[𝑒] ̃ ((𝑦 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (0) .

∀𝑥, 𝑦 ∈ 𝑋, 𝑒 ∈ 𝐴,

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And so ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (𝑎 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑎 ∗ 𝑏) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑏 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑏 ∗ 𝑧) = ℎ𝐻[𝑒] ̃ (𝑏 ∗ 𝑧)

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̃ 𝐴) Proof. Suppose that the hesitant anti-fuzzy soft ideal (𝐻, of a BCK-algebra 𝑋 is a hesitant anti-fuzzy soft filter positive implicative ideal of 𝑋. So ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧)

= ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) .

∀𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝑒 ∈ 𝐴.

Therefore ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧), for all 𝑒 ∈ 𝐴 and 𝑥, 𝑦, 𝑧 ∈ 𝑋. ̃ 𝐴) be a hesitant anti-fuzzy soft positive Theorem 46. Let (𝐻, implicative ideal of 𝑋. Then, for all 𝑥, 𝑦, 𝑧, 𝑎, 𝑏 ∈ 𝑋 and 𝑒 ∈ 𝐴, ((𝑥 ∗ 𝑦) ∗ 𝑧) ∗ 𝑎 ≤ 𝑏 󳨐⇒ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ⊆ ℎ𝐻[𝑒] ̃ (𝑎) ∪ ℎ𝐻[𝑒] ̃ (𝑏) .

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Substituting 𝑧 = 𝑦, we have ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑦) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦) ∪ ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦) , for all 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴.

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10


̃ 𝐴) is a hesitant anti-fuzzy Conversely, suppose that (𝐻, soft ideal over 𝑋 and satisfies inequality (67). ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦) ∀𝑥, 𝑦 ∈ 𝑋, 𝑒 ∈ 𝐴,

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since ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥). Now we can prove that ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑒 ∈ 𝐴. In contrast, there exist 𝑥,̀ 𝑦̀ ∈ 𝑋 such that ̀ ⊇ ℎ𝐻[𝑒] ̀ ∗ 𝑦)̀ ∪ ℎ𝐻[𝑒] ̀ ℎ𝐻[𝑒] ̃ (𝑥̀ ∗ 𝑦) ̃ ((𝑥̀ ∗ 𝑦) ̃ (𝑦̀ ∗ 𝑦) ̀ ∗ 𝑦)̀ ∪ ℎ𝐻[𝑒] = ℎ𝐻[𝑒] ̃ ((𝑥̀ ∗ 𝑦) ̃ (0)

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̀ ∗ 𝑦)̀ , = ℎ𝐻[𝑒] ̃ ((𝑥̀ ∗ 𝑦) which is a contradiction. Therefore ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑒 ∈ 𝐴. Thus ̃ 𝐴) is a hesitant anti-fuzzy soft positive implicative ideal (𝐻, of 𝑋. Now we give the condition that makes equality in Theorem 48 hold. Theorem 49. If 𝑋 is positive implicative, then a hesitant antifuzzy soft ideal of 𝑋 is a hesitant anti-fuzzy soft positive implicative ideal of 𝑋 if and only if it satisfies ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦) ∀𝑥, 𝑦 ∈ 𝑋, 𝑒 ∈ 𝐴.

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̃ 𝐴) Proof. Suppose that the hesitant anti-fuzzy soft ideal (𝐻, of a BCK-algebra 𝑋 is a hesitant anti-fuzzy soft positive implicative ideal of 𝑋. So by Theorem 48 we have ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦) ∀𝑥, 𝑦 ∈ 𝑋, 𝑒 ∈ 𝐴.

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∀𝑥, 𝑦 ∈ 𝑋, 𝑒 ∈ 𝐴.

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Conversely, assume that ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦) for all 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴. We want to prove that a hesitant ̃ 𝐴) of a BCK-algebra 𝑋 is a hesitant anti-fuzzy soft ideal (𝐻, anti-fuzzy soft positive implicative ideal of 𝑋. It is clear that ̃ ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥) for all 𝑥 ∈ 𝑋 and 𝑒 ∈ 𝐴. Since (𝐻, 𝐴) is a hesitant anti-fuzzy soft ideal and 𝑋 is positive implicative, ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) ,

̃ 𝐴) be a hesitant anti-fuzzy soft positive Theorem 50. Let (𝐻, implicative ideal of 𝑋 if and only if for each 𝜀 ∈ 𝑃([0, 1]) the ̃ 𝐴)𝜀 of (𝐻, ̃ 𝐴) is a positive hesitant anti-fuzzy soft 𝜀-level set (𝐻, implicative ideal of 𝑋. ̃ 𝐴) is a hesitant anti-fuzzy soft posiProof. Suppose that (𝐻, tive implicative ideal of 𝑋 and for every 𝜀 ∈ 𝑃([0, 1]) define the sets 𝜀

̃ 𝐴) fl {𝑥 ∈ 𝑋 | ℎ ̃ (𝑥) ⊆ 𝜀} (𝐻, 𝐻[𝑒]

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∀𝑒 ∈ 𝐴.

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̃ 𝐴)𝜀 ⇒ ℎ ̃ (𝑥) ⊆ 𝜀. By ̃ 𝐴)𝜀 ≠ ⌀, let 𝑥 ∈ (𝐻, Since (𝐻, 𝐻[𝑒] Definition 37, we have ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥), for all 𝑒 ∈ 𝐴 and ̃ 𝐴)𝜀 . Let 𝑥, 𝑦, 𝑧 ∈ 𝑋 be such 𝑥 ∈ 𝑋, implying that 0 ∈ (𝐻, 𝜀 ̃ 𝐴) and 𝑦 ∗ 𝑧 ∈ (𝐻, ̃ 𝐴)𝜀 , implying that that (𝑥 ∗ 𝑦) ∗ 𝑧 ∈ (𝐻, ̃ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ⊆ 𝜀 and ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) ⊆ 𝜀. Since (𝐻, 𝐴) is a hesitant anti-fuzzy soft positive implicative ideal over 𝑋, it follows that ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) ⊆ 𝜀.

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̃ 𝐴)𝜀 , and (𝐻, ̃ 𝐴)𝜀 is a positive Hence 𝑥 ∗ 𝑧 ∈ (𝐻, implicative ideal of 𝑋. ̃ 𝐴)𝜀 is a positive implicative Conversely, suppose that (𝐻, ideal of 𝑋 for all 𝜀 ∈ 𝑃([0, 1]). Put ℎ𝐻[𝑒] ̃ (𝑥) = 𝜀 for any 𝑥 ∈ 𝑋. ̃ 𝐴)𝜀 ⇒ ℎ ̃ (0) ⊆ 𝜀 = ℎ ̃ (𝑥), for all 𝑒 ∈ 𝐴 Hence 0 ∈ (𝐻, 𝐻[𝑒] 𝐻[𝑒] and 𝑥 ∈ 𝑋. Now we prove that ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧). In contrast, there exists 𝑥0 , 𝑦0 , 𝑧0 ∈ 𝑋 such that ℎ𝐻[𝑒] ̃ (𝑥0 ∗ 𝑧0 ) ⊇ ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ 𝑦0 ) ∗ 𝑧0 ) ∪ ℎ𝐻[𝑒] ̃ (𝑦0 ∗ 𝑧0 ). Taking 𝜀0 =

On the other hand, since (𝑥 ∗ 𝑦) ∗ 𝑦 ≤ 𝑥 ∗ 𝑦, it follows from Proposition 20(a) that ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦). Thus we have ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) = ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑦)

for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑒 ∈ 𝐴. Therefore a hesitant anti-fuzzy ̃ 𝐴) of 𝑋 is a hesitant anti-fuzzy soft positive soft ideal (𝐻, implicative ideal of 𝑋.

1 {ℎ ̃ (𝑥 ∗ 𝑧0 ) 2 𝐻[𝑒] 0

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+ {ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ 𝑦0 ) ∗ 𝑧0 ) ∪ ℎ𝐻[𝑒] ̃ (𝑦0 ∗ 𝑧0 )}} , it follows that ℎ𝐻[𝑒] ̃ ((𝑥0 ∗𝑦0 )∗𝑧0 ) ⊆ 𝜀0 and ℎ𝐻[𝑒] ̃ (𝑦0 ∗𝑧0 ) ⊆ 𝜀0 , 𝜀0 ̃ which imply that (𝑥0 ∗ 𝑦0 ) ∗ 𝑧0 ∈ (𝐻, 𝐴) and 𝑦0 ∗ 𝑧0 ∈ ̃ 𝐴)𝜀0 . But (𝐻, ̃ 𝐴)𝜀0 is a positive implicative ideal of 𝑋, and (𝐻, ̃ 𝐴)𝜀0 or ℎ ̃ (𝑥0 ∗ 𝑧0 ) ⊆ 𝜀0 , and this is a thus 𝑥0 ∗ 𝑧0 ∈ (𝐻, 𝐻[𝑒] contradiction. Therefore ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧) ∀𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝑒 ∈ 𝐴.

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̃ 𝐴) is a hesitant anti-fuzzy soft positive implicative Hence (𝐻, ideal of 𝑋.

7. Hesitant Anti-Fuzzy Soft Commutative Ideals Definition 51. A hesitant fuzzy set 𝐻𝑋 fl {(𝑥, ℎ𝑋 (𝑥)) | 𝑥 ∈ 𝑋} in a BCK-algebra 𝑋 is called a hesitant anti-fuzzy


11

commutative ideal (briefly, HAFCI) of 𝑋 if it satisfies the following conditions: (HAFCI1) ℎ𝑋 (0) ⊆ ℎ𝑋 (𝑥), for all 𝑥 ∈ 𝑋. (HAFCI2) ℎ𝑋 (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ⊆ ℎ𝑋 ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝑋 (𝑧), for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. ̃ 𝐴) be a hesitant fuzzy soft set over 𝑋, Definition 52. Let (𝐻, ̃ 𝐴) is called a where 𝐴 is a subset of 𝐸. Given 𝑒 ∈ 𝐴, (𝐻, hesitant anti-fuzzy soft commutative ideal based on 𝑒 (briefly, 𝑒-hesitant anti-fuzzy soft commutative ideal) over 𝑋 if the hesitant fuzzy set, ̃ [𝑒] fl {(𝑥, ℎ ̃ (𝑥)) | 𝑥 ∈ 𝑋} , 𝐻 𝐻[𝑒]

Proposition 53. Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebra 𝑋 is order-preserving. Proof. Let 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴 be such that 𝑥 ≤ 𝑦. Since ̃ 𝐴) is a hesitant anti-fuzzy soft commutative ideal of 𝑋, (𝐻, ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)))

= ℎ𝐻[𝑒] ̃ (0 ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) = ℎ𝐻[𝑒] ̃ (0) ∪ ℎ𝐻[𝑒] ̃ (𝑧)

∗ 0 𝑎 𝑏

0 0 𝑎 𝑏

𝑎 0 0 𝑏

̃ 𝐴). Table 9: Tabular representation of the hesitant fuzzy soft set (𝐻, ̃ 𝐻 𝑒1 𝑒2

0 (0.2, 0.4) (0, 0.2)

𝑎 [0.2, 0.7) (0, 0.5)

𝑏 [0, 1) [0, 0.7)

(81)

Remark 56. A hesitant anti-fuzzy soft ideal (subalgebra) of a BCK-algebra 𝑋 may not be a hesitant anti-fuzzy soft commutative ideal of 𝑋 as shown in the following example. Example 57. Let 𝑋 = {0, 𝑎, 𝑏} be the BCK-algebra with in Table 8 (Cayley). ̃ 𝐴) be a Consider a set of parameters 𝐴 = {𝑒1 , 𝑒2 }. Let (𝐻, hesitant fuzzy soft set over 𝑋 which is described in Table 9. ̃ 𝐴) is a hesitant anti-fuzzy soft ideal (subThen (𝐻, ̃ 𝐴) is a algebra) of 𝑋. Routine calculations give that (𝐻, ̃ 1 ] fl hesitant anti-fuzzy soft ideal (subalgebra) of 𝑋, but 𝐻[𝑒 ̃ {(𝑥, ℎ𝐻[𝑒 (𝑥)) | 𝑥 ∈ 𝑋} and 𝐻[𝑒 ] fl {(𝑥, ℎ (𝑥)) | 𝑥 ∈ 𝑋} ̃ 1] ̃ 2] 2 𝐻[𝑒 on 𝑋 is not hesitant anti-fuzzy soft commutative ideal of 𝑋, because ℎ𝐻[𝑒 ̃ 1 ] (𝑎 ∗ (𝑏 ∗ (𝑏 ∗ 𝑎))) = [0.2, 0.7) ⊈ (0.2, 0.4) = ℎ𝐻[𝑒 ̃ 1 ] ((𝑎 ∗ 𝑏) ∗ 0) ∪ ℎ𝐻[𝑒 ̃ 1 ] (0) .

= ℎ𝐻[𝑒] ̃ (𝑧) .

(84)

Also,

Putting 𝑦 = 0 and replacing 𝑧 = 𝑦, we obtain ℎ𝐻[𝑒] ̃ (𝑥) ⊆ ℎ𝐻[𝑒] ̃ (𝑦) .

𝑏 0 0 0

(80)

̃ 𝐴) on 𝑋 is a hesitant anti-fuzzy commutative ideal of 𝑋. If (𝐻, is an 𝑒-hesitant anti-fuzzy soft commutative ideal over 𝑋, for ̃ 𝐴) is a hesitant anti-fuzzy soft all 𝑒 ∈ 𝐴, we say that (𝐻, commutative ideal over 𝑋.

⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧)

Table 8

ℎ𝐻[𝑒 ̃ 2 ] (𝑎 ∗ (𝑏 ∗ (𝑏 ∗ 𝑎))) = (0, 0.5) ⊈ (0, 0.2)

(82)

= ℎ𝐻[𝑒 ̃ 2 ] ((𝑎 ∗ 𝑏) ∗ 0) ∪ ℎ𝐻[𝑒 ̃ 2 ] (0)

The proof is complete.

(85)

Theorem 54. Any hesitant anti-fuzzy soft commutative ideal of BCK-algebra 𝑋 is hesitant anti-fuzzy soft ideal of 𝑋.

The condition in the following theorem makes converse Theorem 54 valid.

̃ 𝐴) be a Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝑋 and let 𝑒 ∈ 𝐴. And let (𝐻, hesitant anti-fuzzy soft commutative ideal of a BCK-algebra 𝑋, so we have

Theorem 58. In a commutative BCK-algebra 𝑋. Every hesitant anti-fuzzy soft ideal of 𝑋 is a hesitant anti-fuzzy soft commutative ideal of 𝑋.

ℎ𝐻[𝑒] ̃ (𝑥) = ℎ𝐻[𝑒] ̃ (𝑥 ∗ (0 ∗ (0 ∗ 𝑥))) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 0) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧)

(83)

= ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) ∪ ℎ𝑋 (𝑧) ,

̃ 𝐴) is a hesitant anti-fuzzy soft ideal of Proof. Suppose that (𝐻, ̃ 𝐴) satisfies a BCK-algebra 𝑋. It is sufficient to show that (𝐻, condition (HAFCI2). Let 𝑥, 𝑦, 𝑧 ∈ 𝑋. Then ((𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ∗ ((𝑥 ∗ 𝑦) ∗ 𝑧)) ∗ 𝑧

̃ 𝐴) is a hesitant antifor all 𝑥, 𝑧 ∈ 𝑋 and 𝑒 ∈ 𝐴. Hence (𝐻, fuzzy soft ideal of 𝑋.

= ((𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ∗ 𝑧) ∗ ((𝑥 ∗ 𝑦) ∗ 𝑧) ≤ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ∗ (𝑥 ∗ 𝑦)

Combining Theorems 22 and 54 yields the following result. Corollary 55. Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebra 𝑋 is a hesitant anti-fuzzy soft subalgebra of 𝑋.

(86)

= (𝑥 ∗ (𝑥 ∗ 𝑦)) ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)) = 0; that is, (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ∗ ((𝑥 ∗ 𝑦) ∗ 𝑧) ≤ 𝑧.

(87)

12


It follows from Proposition 20(c) that ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) ,

(88)

̃ 𝐴) is a hesitant antifor all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑒 ∈ 𝐴. Thus (𝐻, fuzzy soft commutative ideal of a BCK-algebra 𝑋, and the proof is complete. ̃ 𝐴) be a hesitant anti-fuzzy soft ideal of Theorem 59. Let (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft a BCK-algebra 𝑋. Then (𝐻, commutative ideal of 𝑋 if and only if it satisfies the condition ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ∀𝑥, 𝑦 ∈ 𝑋, 𝑒 ∈ 𝐴.

⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 0) ∪ ℎ𝐻[𝑒] ̃ (0) = ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) .

∀𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝑒 ∈ 𝐴.

(90)

⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) ,

(91)

(92)

Observing 𝑥 ∗ 𝑦 ≤ 𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)) and using Proposition 20(a), we have ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) for all 𝑥, 𝑦 ∈ 𝑋 and 𝑒 ∈ 𝐴. Hence, Theorem 59 can be improved as follows. ̃ 𝐴) of a BCKTheorem 60. A hesitant anti-fuzzy soft ideal (𝐻, algebra 𝑋 is a hesitant anti-fuzzy soft commutative ideal of 𝑋 if and only if it satisfies the identity

∀𝑥, 𝑦 ∈ 𝑋, 𝑒 ∈ 𝐴.

= ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧))

(94)

∗ (𝑥 ∗ (𝑥 ∗ 𝑧)) , [by Proposition 6 (ii)]

̃ 𝐴) is a hesitant anti-fuzzy soft positive implicaHence (𝐻, tive ideal of 𝑋. Also, By Propositions 20 and 6(iii) and Theorem 34(iii), we have ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) (95)

∗ (𝑦 ∗ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)))) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) . ̃ 𝐴) is hesitant antiIt follows from Theorem 59 that (𝐻, fuzzy soft commutative ideal of 𝑋. ̃ 𝐴) is both hesitant antiConversely, suppose that (𝐻, fuzzy soft positive implicative ideal of 𝑋 and hesitant antifuzzy soft commutative ideal of 𝑋. Since (𝑦∗(𝑦∗𝑥))∗(𝑦∗𝑥) ≤ 𝑥 ∗ (𝑦 ∗ 𝑥), it follows from Proposition 20(a) that ℎ𝐻[𝑒] ̃ ((𝑦 ∗ (𝑦 ∗ 𝑥)) ∗ (𝑦 ∗ 𝑥)) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) (96)

̃ 𝐴) is hesitant antifor all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑒 ∈ 𝐴. Hence (𝐻, fuzzy soft commutative ideal of 𝑋, which completes the proof.

ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) = ℎ𝐻[e] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)))

⊇ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑧) ∗ 𝑧)

= ℎ𝐻[𝑒] ̃ ((𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))))

Combining (91) and (89), we obtain ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)))

ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ 𝑧)

= ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑧) , [by Theorem 34 (iii)] .

̃ 𝐴) satisfies condition (89) as Conversely, assume that (𝐻, ̃ 𝐴) is a hesitant anti-fuzzy soft ideal of a BCK-algebra 𝑋. (𝐻, Hence ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧)

̃ 𝐴) is a hesitant anti-fuzzy soft Proof. Assume that (𝐻, implicative ideal. By Propositions 6(i) and 20(c), for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and 𝑒 ∈ 𝐴, we have

(89)

̃ 𝐴) Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝑋 and let 𝑒 ∈ 𝐴. And suppose that (𝐻, is a hesitant anti-fuzzy soft commutative ideal of 𝑋. Taking 𝑧 = 0, we have ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)))

̃ 𝐴) of a BCKTheorem 61. A hesitant anti-fuzzy soft ideal (𝐻, algebra 𝑋 is a hesitant anti-fuzzy soft implicative ideal if and ̃ 𝐴) is both hesitant anti-fuzzy soft commutative ideal only if (𝐻, and hesitant anti-fuzzy soft positive implicative ideal.

Using Theorem 49, we have ℎ𝐻[𝑒] ̃ ((𝑦 ∗ (𝑦 ∗ 𝑥)) ∗ (𝑦 ∗ 𝑥)) = ℎ𝐻[𝑒] ̃ (𝑦 ∗ (𝑦 ∗ 𝑥)) and so ℎ𝐻[𝑒] ̃ (𝑦 ∗ (𝑦 ∗ 𝑥)) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) .

(97)

On the other hand, 𝑥∗𝑦 ≤ 𝑥∗(𝑦∗𝑥) implies that ℎ𝐻[𝑒] ̃ (𝑥∗ ̃ 𝑦) ⊆ ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)). Since (𝐻, 𝐴) is a hesitant anti-fuzzy soft commutative ideal of 𝑋, By Theorem 60, we have ℎ𝐻[𝑒] ̃ (𝑥 ∗ 𝑦) = ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) .

(98)

Hence, ℎ𝑋 (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ⊆ ℎ𝑋 (𝑥 ∗ (𝑦 ∗ 𝑥)) .

(99)

Combining (97), we obtain ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ 𝑥)) ⊇ ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)))

(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.

∪ ℎ𝐻[𝑒] ̃ (𝑦 ∗ (𝑦 ∗ 𝑥))

(100)

⊇ ℎ𝐻[𝑒] ̃ (𝑥) . ̃ 𝐴) is a hesitant anti-fuzzy soft implicative ideal of So, (𝐻, 𝑋 by Theorem 34. The proof is complete.


13

̃ 𝐴) be hesitant anti-fuzzy soft commutaTheorem 62. Let (𝐻, tive ideal of 𝑋 if and only if for each 𝜀 ∈ 𝑃([0, 1]) the hesitant ̃ 𝐴)𝜀 of (𝐻, ̃ 𝐴) is a commutative anti-fuzzy soft 𝜀-level set (𝐻, ideal of 𝑋. ̃ 𝐴) is a hesitant anti-fuzzy soft comProof. Suppose that (𝐻, mutative ideal of 𝑋 and for every 𝜀 ∈ 𝑃([0, 1]) define the sets 𝜀

̃ 𝐴) fl {𝑥 ∈ 𝑋 | ℎ ̃ (𝑥) ⊆ 𝜀} (𝐻, 𝐻[𝑒]

∀𝑒 ∈ 𝐴.

(101)

̃ 𝐴)𝜀 ≠ ⌀, let 𝑥 ∈ (𝐻, ̃ 𝐴)𝜀 ⇒ ℎ ̃ (𝑥) ⊆ 𝜀. By Since (𝐻, 𝐻[𝑒] definition, we have ℎ𝐻[𝑒] ̃ (0) ⊆ ℎ𝐻[𝑒] ̃ (𝑥), for all 𝑒 ∈ 𝐴 and ̃ 𝐴)𝜀 . Let 𝑥, 𝑦, 𝑧 ∈ 𝑋 be such 𝑥 ∈ 𝑋, implying that 0 ∈ (𝐻, ̃ 𝐴)𝜀 and 𝑧 ∈ (𝐻, ̃ 𝐴)𝜀 , implying that that (𝑥 ∗ 𝑦) ∗ 𝑧 ∈ (𝐻, ̃ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ⊆ 𝜀 and ℎ𝐻[𝑒] ̃ (𝑧) ⊆ 𝜀. Since (𝐻, 𝐴) is a hesitant anti-fuzzy soft commutative ideal over 𝑋, it follows that ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧) ⊆ 𝜀.

(102)

̃ 𝐴)𝜀 , and (𝐻, ̃ 𝐴)𝜀 is a Namely, 𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥)) ∈ (𝐻, commutative ideal of 𝑋. ̃ 𝐴)𝜀 is a commutative ideal Conversely, suppose that (𝐻, of 𝑋 for all 𝜀 ∈ 𝑃([0, 1]). Put ℎ𝐻[𝑒] ̃ (𝑥) = 𝜀 for any 𝑥 ∈ 𝑋 and 𝜀 ̃ 𝑒 ∈ 𝐴. Hence 0 ∈ (𝐻, 𝐴) ⇒ ℎ𝐻[𝑒] ̃ (0) ⊆ 𝜀 = ℎ𝐻[𝑒] ̃ (𝑥), for all 𝑒 ∈ 𝐴 and 𝑥 ∈ 𝑋. Now we prove that ℎ𝐻[𝑒] ̃ (𝑥 ∗ (𝑦 ∗ (𝑦 ∗ 𝑥))) ⊆ ℎ𝐻[𝑒] ̃ ((𝑥 ∗ 𝑦) ∗ 𝑧) ∪ ℎ𝐻[𝑒] ̃ (𝑧). In contrast, there exists 𝑥0 , 𝑦0 , 𝑧0 ∈ 𝑋 such that ℎ𝐻[𝑒] ̃ (𝑥0 ∗ (𝑦0 ∗ (𝑦0 ∗ 𝑥0 ))) ⊇ ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ 𝑦0 ) ∗ 𝑧0 ) ∪ ℎ𝐻[𝑒] ̃ (𝑧0 ) .

(103)

Taking 𝜀0 =

1 {ℎ ̃ (𝑥 ∗ (𝑦0 ∗ (𝑦0 ∗ 𝑥0 ))) 2 𝐻[𝑒] 0

(104)

+ {ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ 𝑦0 ) ∗ 𝑧0 ) ∪ ℎ𝐻[𝑒] ̃ (𝑧0 )}} , it follows that ℎ𝐻[𝑒] ̃ ((𝑥0 ∗ 𝑦0 ) ∗ 𝑧0 ) ⊆ 𝜀0 and ℎ𝐻[𝑒] ̃ (𝑧0 ) ⊆ 𝜀0 , 𝜀0 ̃ ̃ 𝐴)𝜀0 . which imply that ((𝑥0 ∗𝑦0 )∗𝑧0 ) ∈ (𝐻, 𝐴) and 𝑧0 ∈ (𝐻, 𝜀 0 ̃ 𝐴) is a commutative ideal of 𝑋, and thus 𝑥0 ∗ (𝑦0 ∗ But (𝐻, ̃ 𝐴)𝜀0 or ℎ ̃ (𝑥0 ∗ (𝑦0 ∗ (𝑦0 ∗ 𝑥0 ))) ⊆ 𝜀0 , (𝑦0 ∗ 𝑥0 )) ∈ (𝐻, 𝐻[𝑒] ̃ 𝐴) is a hesitant antiand this is a contradiction. Therefore (𝐻, fuzzy soft commutative ideal of 𝑋.

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, “Soft set theory- first results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999. [2] P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589–602, 2001. [3] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965. [4] V. Torra, “Hesitant fuzzy sets,” International Journal of Intelligent Systems, vol. 25, no. 6, pp. 529–539, 2010. [5] K. V. Babitha and S. J. John, “Hesitant fuzzy soft sets,” Journal of New Results in Science, vol. 3, pp. 98–107, 2013. [6] Y. B. Jun, S. S. Ahn, and G. Muhiuddin, “Hesitant fuzzy soft subalgebras and ideals in BCK/BCI -algebras,” Scientific World Journal, vol. 2014, Article ID 763929, 2014. [7] Y. B. Jun and S. S. Ahn, “Hesitant fuzzy sets theory applied to in BCK/BCI- algebras,” J. Computational Analysis and Applications, vol. 20, no. 4, pp. 635–646, 2016. [8] K. Is´eki, “An algebraic related with a propositional calclus,” Proc. Japan Academy, vol. 42, pp. 26–29, 1996. [9] J. Meng, Y. B. Jun, and H. S. Kim, “Fuzzy implicative ideals of BCK-algebras,” Fuzzy Sets and Systems, vol. 89, pp. 243–248, 1997.

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