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Mathematical Problems in Engineering

Volume 2017, Article ID 3634258, 13 pages

https://doi.org/10.1155/2017/3634258

## Hesitant Anti-Fuzzy Soft Set in BCK-Algebras

^{1}Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia^{2}Department of Natural and Engineering Science, Faculty of Applied Studies and Community Service, King Saud University, Riyadh, Saudi Arabia

Correspondence should be addressed to Halimah Alshehri; as.ude.usk@irhehslaah

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.

#### Abstract

We introduce the notions of hesitant anti-fuzzy soft set (subalgebras and ideals) and provide relation between them. However, we study new types of hesitant anti-fuzzy soft ideals (implicative, positive implicative, and commutative). Also, we stated and proved some theorems which determine the relationship between these notions.

#### 1. Introduction

In the real world, there are many complicated problems in economic science, engineering, environment, social science, and management science. They are characterized by uncertainty, imprecision, and vagueness. We cannot successfully utilize the classical methods to deal with these problems because there are various types of uncertainties involved in these problems. Moreover, although there are many theories, such as theory of probability, theory of fuzzy sets, theory of interval mathematics, and theory of rough sets, to be considered as mathematical tools to deal with uncertainties, Molodtsov [1] pointed out that all these theories had their own limitations. Also, in order to overcome these difficulties, Molodtsov [1] firstly proposed a new mathematical tool named soft set theory to deal with uncertainty and imprecision. This theory has been demonstrated to be a useful tool in many applications such as decision-making, measurement theory, and game theory.

The soft set model can be combined with other mathematical models. Maji et al. [2] firstly presented the concept of fuzzy soft set by combining the theories of fuzzy set and soft set together. The hesitant fuzzy set, as one of the extensions of Zadeh’s [3] (1965) fuzzy set, allows the membership degree of an element to a set presented by several possible values, and it can express the hesitant information more comprehensively than other extensions of fuzzy set. In [4], Torra introduced the concept of hesitant fuzzy set and Babitha and John (2013) [5] defined another important soft set, hesitant fuzzy soft set. They introduced basic operations such as intersection, union, and compliment, and De Morgan’s law was proven. In 2014, Jun et al. [6] applied the notion of hesitant fuzzy soft sets to subalgebras and ideals in BCK/BCI-algebras. In this paper, in Section 3, we introduce the concepts of hesitant anti-fuzzy soft set of subalgebra. In Section 4, we define the hesitant anti-fuzzy soft ideal in BCK-algebras and give some basic relations. In Section 5, we discuss notion of hesitant anti-fuzzy soft implicative ideals and provide some properties. In Section 6, we investigate concept of hesitant anti-fuzzy soft positive implicative ideals and give some relations. In Section 7, we introduce the notion of hesitant anti-fuzzy soft commutative ideals in BCK-algebras and related properties are investigated. Finally, conclusions are presented in the last section.

#### 2. Preliminaries

An algebra of type (2, 0) is said to be a BCK-algebra if it satisfies the axioms: for all , (BCK-1) , (BCK-2) , (BCK-3) , (BCK-4) , (BCK-5) and imply that .

Define a binary relation on by letting if and only if

Then is a partially ordered set with the least element . In any BCK-algebra , the following hold:(1).(2).(3).(4).(5).(6) implies that and , for all .

A nonempty subset of is called a subalgebra of if, for any , That is, it is closed under the binary operation of

A nonempty subset of is called an ideal of if () and imply that .

A nonempty subset of is called an implicative ideal if it satisfies and whenever and .

It is called a commutative ideal if it satisfies and whenever and ; and it is called a positive implicative ideal if it satisfies and whenever and .

A BCK-algebra is said to be implicative if it satisfies

A BCK-algebra is said to be positive implicative if it satisfies

A BCK-algebra is said to be commutative if it satisfies

*Definition 1 (see [3]). *Let be a set. A fuzzy set in is a function

*Definition 2 (see [2]). *Let be an initial universe set and let be a set of parameters. Let denote the set of all fuzzy sets in . Then is called a fuzzy soft set over , where and is a mapping given by

*Definition 3 (see [4, 7]). *Let be a reference set. A hesitant fuzzy set on is defined in terms of a function that when applied to returns a subset of which can be viewed as the following mathematical representation:where

*Definition 4 (see [7]). *Given a nonempty subset of , a hesitant fuzzy set on satisfying the conditionis called a hesitant fuzzy set related to (briefly, -hesitant fuzzy set) on and is represented by ; is a mapping from to with , for all

Proposition 5 (see [8]). *Let be an -hesitant anti-fuzzy ideal of . Then the following hold: for all ,*(a)*if , then , which means that preserves the order,*(b)*if , then *

*Proposition 6 (see [9]). In a BCK-algebra , the following hold: for all ,(i),(ii),(iii)*

*Definition 7 (see [5, 6]). *Denote by the set of all hesitant fuzzy sets. A pair is called a hesitant fuzzy soft set over a reference set , where is a mapping given by

*3. Hesitant Anti-Fuzzy Soft Subalgebras*

*3. Hesitant Anti-Fuzzy Soft Subalgebras*

*Definition 8. *Given a nonempty subset (subalgebra as much as possible) of , let be an -hesitant fuzzy set on . Then is called a hesitant anti-fuzzy subalgebra of related to (briefly, -hesitant anti-fuzzy subalgebra of ) if it satisfies the following condition:An -hesitant anti-fuzzy subalgebra of with is called a hesitant anti-fuzzy subalgebra of

*Definition 9. *For a subset of , a hesitant fuzzy soft set over is called a hesitant anti-fuzzy soft subalgebra based on (briefly, -hesitant anti-fuzzy soft subalgebra) over if the hesitant fuzzy set,on is a hesitant anti-fuzzy subalgebra of . If is an -hesitant anti-fuzzy soft subalgebra over , for all , we say that is a hesitant anti-fuzzy soft subalgebra.

*Example 10. *Let be a BCK-algebra in Table 1 (Cayley).

Consider a set of parameters Let be a hesitant fuzzy soft set over , where , which is given in Table 2. It is routine to verify that and are hesitant anti-fuzzy subalgebra over based on parameters , and Therefore is a hesitant anti-fuzzy soft subalgebra over .