# (∈,∈∨q˘)-Bipolar Fuzzy b-Ideals of BCK/BCI-Algebras

Authors: M. Mursaleen; M. Balamurugan; K. Loganathan; Kottakkaran Sooppy Nisar
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, the idea of∈,∈∨q˘-bipolar fuzzy b-ideals and an ∈,∈∨q˘-bipolar fuzzy ideals of BCK/BCI-algebras is delivered, and their related properties are investigated with the aid of some examples. We also provide the connection between ∈,∈∨q˘-bipolar fuzzy ideals and bipolar fuzzy ideals and ∈,∈∨q˘-bipolar fuzzy b-ideals and bipolar fuzzy b-ideals by way of counterexamples.


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## Body

## 1. Introduction

In the real world, there are several difficult problems in engineering, medical science, economics, environment, social science, and other various fields involving incalculable data. Problems of this nature which are frequently encountered in our daily lives cannot be solved by classical mathematical methods. In 1965, the belief of fuzzy sets turned into was first delivered with the aid of Zadeh [1], which is an effective hand set for modelling indecision and elusiveness in numerous issues springing up inside the field of technology. For the ultimate four decades, the fuzzy idea has ended up as a very lively area of study, and plenty of developments have been made within the concept of fuzzy sets to find the fuzzy analogues of the classical set theory. The study of BCK-algebras was initiated by Imai and Iséki [2, 3] in 1966 as showed in the concept of set-theoretic difference and propositional calculi. After that, many researchers investigated the fuzzification of ideals and subalgebras in BCK/BCI-algebras.Bipolar fuzzy sets are an extension of fuzzy sets whose membership degree range is [1, 1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [1, 0) of an element indicates that the element somewhat satisfies the implicit counterproperty. The idea which lies behind such description is connected with the existence of “bipolar information” (e.g., positive information and negative information) about the given set. Positive information represents what is granted to be possible, while negative information represents what is considered to be impossible. Actually, a wide variety of human decision-making is based on double-sided or bipolar judgmental thinking on a positive side and a negative side. For instance, cooperation and competition, friendship and hostility, common interests and conflict of interests, effect and side effect, likelihood and unlikelihood, and feedforward and feedback are often the two sides in decision and coordination. In traditional Chinese medicine, “yin” and “yang” are the two sides. Yin is the feminine or negative side of a system, and yang is the masculine or positive side of a system. The coexistence, equilibrium, and harmony of the two sides are considered a key for the mental and physical health of a person as well as for the stability and prosperity of a social system. Thus, bipolar fuzzy sets indeed have potential impacts on many fields, including artificial intelligence, computer science, information science, cognitive science, decision science, management science, economics, neural science, quantum computing, medical science, and social science. In 1998, the notion of bipolar fuzzy sets was proposed by Zhang [5, 6] as a generalization of fuzzy sets [1].Bipolar-valued fuzzy units, which can be added by means of Lee [7], are an augmentation of fuzzy sets whose participation degree extend is broadened from the stretch [0, 1] to −1,1. Lee [8] involved the research with interval-valued fuzzy units, intuitionistic fuzzy units, and bipolar-valued fuzzy sets. Bipolar fuzzy sets have various applications in fuzzy algebras. For example, bipolar fuzzy ideals [4] in LA-semigroups, bipolar fuzzy subalgebras and ideals [7] of BCK/BCI-algebras, bipolar fuzzy a-ideals in BCK/BCI-algebras, and bipolar valued fuzzy BCK/BCI-algebras [9] are some of them. Bhakat and Das [10, 11] utilized the idea of α˘,β˘-fuzzy subgroups by the usage of ∈ and q˘ among a fuzzy factor and a fuzzy subset. It is visible that ∈,∈∨q˘-fuzzy subgroups are a critical generalization of Rosenfeld’s [12] fuzzy subgroup. Similar types of ∈,∈∨q-fuzzy ideals of BCI-algebras are introduced by Zhan et al. [5]. Zhan et al. [13, 14] introduced three-way multiattribute decision-making based on outranking relations and multicriteria decision-making method based on a fuzzy rough set model with fuzzy α-neighborhoods. Senapati et al. [15] introduced cubic intuitionistic implicative ideals of BCK/BCI-algebras.Jana et al. [16, 17] introduced generalizations of ∈,∈∨q-intuitionistic fuzzy subalgebras and ideals of BCK/BCI-algebras with thresholds. Jana et al. [18] introduced the concept of ∈,∈∨q˘-bipolar fuzzy BCK/BCI-subalgebras and ∈,∈∨q˘-bipolar fuzzy ideals of BCK/BCI-algebras. These works are enough to motivate us, and, to the best of our knowledge, no other works are available on ∈,∈∨q˘-bipolar fuzzy ideals and b-ideals in BCK/BCI-algebras and other fuzzy algebraic structures. For this reason, we have developed the theoretical study of ∈,∈∨q˘-bipolar fuzzy ideals of BCK/BCI-algebras and ∈,∈∨q˘-bipolar fuzzy b-ideals of BCK/BCI-algebras. In this paper, the concepts of ∈,∈∨q˘-bipolar fuzzy ideals are presented, and properties are established. Moreover, ∈,∈∨q˘-bipolar fuzzy b-ideals of BCI/BCK-algebras are proposed, and their properties are examined in detail.


## 2. Preliminaries

Definition 1.
An algebraX˘;⋄,0 of kind 2,0 could be a BCK-algebra if it fulfills for all x˘,y˘,z˘∈X˘,
K1x˘⋄y˘⋄x˘⋄z˘⋄z˘⋄y˘=0,
K2x˘⋄x˘⋄y˘⋄y˘=0,
K3x˘⋄x˘=0,
K40⋄x˘=0,
K5x˘⋄y˘=0 and y˘⋄x˘=0⇒x˘=y˘.
We remind the reader thatX˘ signifies a BCK/BCI-algebra unless otherwise specified.
A nonempty subsetA˘ of X˘ is called an ideal of X˘ if it satisfies
(I1) 0∈A˘,
(I2) ∀x˘,y˘∈X˘,x˘⋄y˘∈A,y˘∈A⇒x˘∈A˘.
A nonempty subsetA˘ of X˘ is called a b-ideal of X˘ if it satisfies I1 and
(I3) ∀x˘,y˘,z˘∈X˘,x˘∗z˘⋄y˘∈A˘,y˘∈A˘⇒x˘∈A˘.
A bipolar fuzzy set (BFS) in X˘ is denoted by ξ=X˘;ξn˘,ξp˘, where ξn˘,ξp˘ are the maps from X˘ to −1,0 and from X˘ to 0,1, respectively.


## 3.∈,∈∨q˘-Bipolar Fuzzy Ideals

In this section, we investigate an∈,∈∨q˘-bipolar fuzzy ideals of BCK/BCI-algebras.Definition 2 (see [18]).
ABFSξ=X˘;ξn˘,ξp˘ is a BFI of X˘ if it satisfies the subsequent assertions:
(i)
∀x˘∈X˘ξn˘0≤ξn˘x˘,ξp˘0≥ξp˘x˘(ii)
∀x˘,y˘∈X˘ξn˘x˘≤ξn˘x˘⋄y˘∨ξn˘y˘(iii)
∀x˘,y˘∈X˘ξp˘x˘≥ξp˘x˘⋄y˘∧ξp˘y˘Definition 3.
Letξ=X˘;ξn˘,ξp˘ be a BFS in X˘ of the form
(1)ξn˘y˘=m˘∈−1,0,ify˘=x˘,0,ify˘≠x˘,ξp˘y˘=s˘∈0,1,ify˘=x˘,0,ify˘≠x˘.
A bipolar fuzzy point with helpx˘ and values m˘ and s˘ is signified by x˘,m˘,s˘. In a bipolar fuzzy point x˘,m˘,s˘ and a BFSξ=X˘;ξn˘,ξp˘ in a set X˘, we offer significance to the symbol x˘,m˘Φξn˘,x˘,s˘Φξp˘, where Φ∈∈,q˘,∈∨q˘,∈∧q˘.
To say thatx˘,m˘∈ξn˘respectively,x˘,m˘q˘ξn˘ and x˘,s˘∈ξp˘respectively,x˘,s˘q˘ξp˘ means that ξn˘x˘≤m˘respectively,ξn˘x˘+m˘<−1 and ξp˘x˘≥s˘respectively,ξp˘x˘+s˘>1, and in this case, we say that x˘,m˘ and x˘,s˘ are said to belong to (respectively, be quasicoincident with) a BFSξ=X˘;ξn˘,ξp˘.
To say thatx˘,m˘∈∧q˘respectively,x˘,m˘∈∧q˘ and x˘,s˘∈∧q˘respectively,x˘,s˘∈∧q˘ and imply x˘,m˘∈ξn˘ or x˘,m˘q˘ξn˘respectively,x˘,m˘∈ξn˘ and x˘,s˘∈ξp˘ or x˘,s˘q˘ξp˘respectively,x˘,s˘∈ξn˘).
To say thatx˘,m˘Φ¯ξn˘,x˘,s˘Φ¯ξp˘ imply x˘,m˘Φξn˘ does not hold and x˘,s˘Φξp˘ does not hold, where Φ∈∈,q˘,∈∨q˘,∈∧q˘.Definition 4.
ABFSξ=X˘;ξn˘,ξp˘ is an ∈,∈∨q˘−BFI of X˘ if it satisfies the subsequent assertions:
(i)
x˘⋄y˘,m˘∈ξn˘,y˘,n˘∈ξn˘⇒x˘,m˘∨n˘∈∨q˘ξn˘, for all x˘,y˘∈X˘ and m˘,n˘∈−1,0(ii)
x˘⋄y˘,t˘∈ξn˘,y˘,s˘∈ξn˘⇒x˘,t˘∧s˘∈∨q˘ξp˘, for all x˘,y˘∈X˘ and t˘,s˘∈0,1Example 5.
Consider aBCI/BCK-algebra X˘=0,w˘,x˘,y˘,z˘ with the subsequent Cayley table:
(2)
Define aBFSξ of X˘ as follows:
(3)
Hence,ξ=X˘;ξn˘,ξp˘ is an ∈,∈∨q˘−BFI of X˘.Theorem 6.
ABFSξ=X˘;ξn˘,ξp˘ is an ∈,∈∨q˘−BFI of X˘ if and only if it satisfiesif and only if it satisfies
(4)ξn˘x˘≤ξn˘x˘⋄y˘∨ξn˘y˘∨−12,ξp˘x˘≥ξp˘x˘⋄y˘∧ξp˘y˘∧12for all x˘,y˘∈X˘.Proof.
Letξ=X˘;ξn˘,ξp˘ be an ∈,∈∨q˘−BFI of X˘ and x˘,y˘∈X˘. If ξn˘x˘⋄y˘∨ξn˘y˘>−1/2 and ξp˘x˘⋄y˘∧ξp˘y˘<1/2, then ξn˘x˘≤ξn˘x˘⋄y˘∨ξn˘y˘ and ξp˘x˘≥ξp˘x˘⋄y˘∧ξp˘y˘.
Assume thatξn˘x˘>ξn˘x˘⋄y˘∨ξn˘y˘ and ξp˘x˘<ξp˘x˘⋄y˘∧ξp˘y˘. Let us take m˘∈¬ξ and t∈ξ such that ξn˘x˘>m˘>ξn˘x˘⋄y˘∨ξn˘y˘ and ξp˘x˘<t˘<ξp˘x˘⋄y˘∧ξp˘y˘. Then, x˘⋄y˘,m˘∈ξn˘,y˘,m˘∈ξn˘ and x˘⋄y˘,t˘∈ξp˘,y˘,t˘∈ξp˘ but x˘,m˘∨m˘=x˘,m˘∈,∈∨q˘ξn˘¯ and x˘,t˘∧t˘=x˘,t˘∈,∈∨q˘ξp˘¯, a contradiction.
Hence,ξn˘x˘≤ξn˘x˘⋄y˘∨ξn˘y˘ whenever ξn˘x˘⋄y˘∨ξn˘y˘>−1/2 and ξp˘x˘≥ξp˘x˘⋄y˘∧ξp˘y˘ whenever ξp˘x˘⋄y˘∧ξp˘y˘<1/2.
Suppose thatξn˘x˘⋄y˘∨ξn˘y˘≤−1/2 and ξp˘x˘⋄y˘∧ξp˘y˘≥1/2. Then, x˘⋄y˘,−1/2∈ξn˘, y˘,−1/2∈ξn˘ and x˘⋄y˘,1/2∈ξp˘, y˘,1/2∈ξp˘, which imply that
(5)x˘,−12∧−12=x˘,−12∈∨q˘ξn˘,x˘,12∧12=x˘,12∈∨q˘ξp˘.
Thus,ξn˘x˘≤−1/2 and ξp˘x˘≥1/2. Otherwise, ξn˘x˘−1/2>−1/2−1/2=−1 and ξp˘x˘+1/2<1/2+1/2=1, a contradiction. And so,
(6)ξn˘x˘≤ξn˘x˘⋄y˘∨ξn˘y˘∨−12,ξp˘x˘≥ξp˘x˘⋄y˘∧ξp˘y˘∧12for all x˘,y˘∈X˘.
On the contrary, assume that∈,∈∨q˘−BFI of X˘ is valid. Let x˘,y˘∈X˘ and s˘,t˘∈0,1 and m˘,n˘∈¬ξ such that x˘⋄y˘,m˘∈ξn˘,y˘,n˘∈ξn˘ and x˘⋄y˘,s˘∈ξp˘,y˘,t˘∈ξp˘. Then, ξn˘x˘⋄y˘≤m˘,ξn˘x˘⋄y˘≤n˘ and ξp˘x˘⋄y˘≥s˘,ξp˘y˘≤t˘.
Ifξn˘x˘>m˘∨n and ξp˘x˘<s˘∨t˘, then ξn˘x˘⋄y˘∨ξn˘y˘≤−1/2 and ξp˘x˘⋄y˘∧ξp˘y˘≥1/2.
Otherwise, we get(7)ξn˘x˘≤ξn˘x˘⋄y˘∨ξn˘y˘∨−12≤ξn˘x˘⋄y˘∨ξn˘y˘≤m˘∨n˘,ξp˘x˘≥ξp˘x˘⋄y˘∧ξp˘y˘∧12≥ξp˘x˘⋄y˘∧ξp˘y˘≥s˘∧t˘,a contradiction. In that case,
(8)ξn˘x˘+m˘∨n˘<2ξn˘x˘≤2ξn˘x˘⋄y˘∨ξn˘y˘∨−12=−1,ξn˘x˘+s˘∧t˘>2ξp˘x˘≥2ξp˘x˘⋄y˘∧ξp˘y˘∧12=1.
Hence,x˘,m˘∨n˘∈∨q˘ξn˘ and x˘,s˘∧t˘∈∨q˘ξp˘.
Therefore,ξ is an ∈,∈∨q˘−BFI of X˘.Lemma 7.
EveryBFI is an ∈,∈∨q˘−BFI of X˘.
The opposite of Lemma7 is not correct in general, justified in the subsequent Example 8.Example 8.
Consider aBCI-algebra X˘=0,w˘,x˘,y˘ with Cayley table:
(9)
Define aBFSξ of X˘ as follows:
(10)
Hence,∈,∈∨q˘−BFI of X˘ but is not BFI of X˘ because ξp˘y˘=0.55⪈0.65=ξn˘y˘⋄w˘∧ξw˘.


## 4.∈,∈∨q˘-Bipolar Fuzzy b-Ideals

In this section, we investigate an∈,∈∨q˘-bipolar fuzzy b-ideals of BCK/BCI-algebras.Definition 9.
ABFSξ=X˘;ξn˘,ξp˘ is a BFBI of X˘ if it satisfies Definition 2 (i) and the subsequent assertions:
(i)
∀x˘,y˘∈X˘ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘(ii)
∀x˘,y˘∈X˘ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘Definition 10.
ABFSξ=X˘;ξn˘,ξp˘ is called an ∈,∈∨q˘-BFBI of X˘ if it satisfies the subsequent assertions:
(i)
x˘⋄z˘⋄y˘,m˘∈ξn˘,y˘,n˘∈ξn˘⇒x˘,m˘∨n˘∈∨q˘ξn˘, for all x˘,y˘,z˘∈X˘ and m˘,n˘∈−1,0(ii)
x˘⋄z˘⋄y˘,s˘∈ξn˘,y˘,t˘∈ξn˘⇒x˘,s˘∧t˘∈∨q˘ξp˘, for all x˘,y˘,z˘∈X˘ and s˘,t˘∈0,1Example 11.
LetX˘=0,w˘,x˘,y˘,z˘ be a BCK-algebra in Example 5 and a BFSξ of X˘ defined by
(11)
Hence,ξ=X˘;ξn˘,ξp˘ is an ∈,∈∨q˘-BFBI of X˘ as well as BFBI of X˘.Theorem 12.
ABFSξ=X˘;ξn˘,ξp˘ is a BFBI of X˘ if and only if the following assertions are valid:
(i)
x˘,m˘∈ξn˘⇒0,m˘∈ξn˘ and x˘,s˘∈ξp˘⇒0,s˘∈ξp˘, for all x˘∈X˘,m˘∈−1,0,s˘∈0,1(ii)
x˘⋄z˘⋄y˘,m˘∈ξn˘,y˘,n˘∈ξn˘⇒x˘,m˘∨n˘∈ξn˘, for all x˘,y˘∈X˘ and m˘,n˘∈−1,0(iii)
x˘⋄z˘⋄y˘,s˘∈ξp˘,y˘,t˘∈ξp˘⇒x˘,s˘∧t˘∈ξp˘, for all x˘,y˘∈X˘ and s˘,t˘∈0,1Proof.
Assume that Definition2(i) is valid and x˘∈X˘, s˘∈0,1, m˘∈−1,0 such that x˘,m˘∈ξn˘ and x˘,s˘∈ξp˘. Then, ξn˘0≤ξn˘x˘≤m˘ and ξp˘0≥ξp˘x˘≥s˘, and so 0,m˘∈ξn˘ and 0,s˘∈ξp˘.
Sincex˘,ξx˘∈ξn˘ and x˘,ξx˘∈ξp˘ for all x˘∈X˘, it follows from (i) that 0,ξx˘∈ξn˘ and 0,ξx˘∈ξp˘ so that ξn˘0≤ξn˘x˘ and ξp˘0≥ξp˘x˘ for all x˘∈X˘. Assume that Definition 9 holds.
Letx˘,y˘,z˘∈X˘, and m˘,n˘∈−1,0, s˘,t˘∈0,1 be such that x˘⋄z˘⋄y˘,m˘∈ξn˘, y˘,n˘∈ξn˘, and x˘⋄z˘⋄y˘,s˘∈ξp˘,y˘,t˘∈ξp˘. Then, ξn˘x˘⋄z˘⋄y˘≤m˘,ξn˘y˘≤n˘ and ξp˘x˘⋄z˘⋄y˘≥s˘,ξp˘y˘≥t˘. It follows from Definition 9:
(12)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘≤m˘∨n˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘≥s˘∧t˘.
So thatx˘,m˘∨n˘∈ξn˘ and x˘,s˘∧t˘∈ξp˘.
Again, suppose that (ii) and (iii) are valid. Also, for everyx˘,y˘,z˘∈X˘x˘⋄z˘⋄y˘,ξn˘x˘⋄z˘⋄y˘∈ξn˘,y˘,ξn˘y˘∈ξn˘ and x˘⋄z˘⋄y˘,ξp˘x˘⋄z˘⋄y˘∈ξp˘,y˘,ξp˘y˘. Hence, x˘,ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∈ξn˘ and x˘,ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∈ξp˘ by (ii) and (iii), respectively, and thus,
(13)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘.Theorem 13.
ABFSξ=X˘;ξn˘,ξp˘ is an ∈,∈∨q˘-BFBI of X˘ if and only if it satisfies the subsequent assertions:
(i)
ξn˘0≤ξn˘x˘∨−1/2 and ξp˘0≥ξp˘x˘∧1/2(ii)
ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−1/2(iii)
ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧1/2for all x˘,y˘,z˘∈X˘.Proof.
Supposeξ=X˘;ξn˘,ξp˘ be an ∈,∈∨q˘-BFBI of X˘. Let x˘∈X˘ be such that ξn˘x˘>−1/2 and ξp˘x˘<1/2. If ξn˘0>ξn˘x˘ and ξp˘0<ξp˘x˘, ξn˘0>m˘>ξn˘x˘ and ξp˘0<s˘<ξp˘x˘ for every m˘∈−1/2,0 and s˘∈0,1/2, so we get x˘,m˘∈ξn˘, 0,m˘∈¯ξn˘ and x˘,s˘∈ξp˘, 0,s˘∈¯ξp˘.
Sinceξn˘0+m˘>−1 and ξp˘0+s˘<1, so we have 0,m˘q˘¯ξn˘ and 0,s˘q˘¯ξp˘. It follows that 0,m˘∈∨q˘¯ξn˘ and 0,s˘∈∨q˘¯ξp˘, a contradiction. Hence, ξn˘0≤ξn˘x˘ and ξp˘0≥ξp˘x˘. Now, if ξn˘x˘≤−1/2 and ξp˘x˘≥1/2, then x˘,−1/2∈ξn˘ and x˘,1/2∈ξp˘. Thus, 0,−1/2∈∨q˘ξn˘ and 0,1/2∈∨q˘ξp˘. Thus, ξn˘0≤−1/2 and ξp˘0≥1/2.
Otherwise,ξn˘x˘−1/2>−1/2−1/2=−1 and ξp˘x˘+1/2<1/2+1/2=1, a contradiction. Consequently, ξn˘0≤ξn˘x˘,−1/2 and ξp˘0≥ξp˘x˘,1/2 for all x˘∈X˘.
Letx˘,y˘,z˘∈X˘. Suppose that ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘>−1/2 and ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘<1/2. Then, ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘ and ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘.
If not, thenξn˘x˘>m˘>ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘ and ξp˘x˘<s˘<ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘, for some m˘∈−1/2,0,s˘∈0,1/2.
It follows thatx˘⋄z˘⋄y˘,m˘∈ξn˘ and y˘,m˘∈ξn˘ but x˘,m˘∨m˘=x˘,m˘∈∨q˘ξn˘ and x˘⋄z˘⋄y˘,s˘∈ξp˘ and y˘,s˘∈ξp˘ but x˘,s˘∨s˘=x˘,s˘∈∨q˘ξp˘ which is a contradiction.
Hence,ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘ whenever ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘>−1/2 and ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘ whenever ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘<1/2.
Ifξn˘x˘⋄y˘∨ξn˘y˘≤−1/2, then x˘⋄z˘⋄y˘,−1/2∈ξn˘ and y˘,−1/2∈ξn˘, which imply that x˘,−1/2=x˘,−1/2∨−1/2∈∨q˘ξn˘ and if ξp˘x˘⋄y˘∧ξp˘y˘≥1/2, then x˘⋄z˘⋄y˘,1/2∈ξp˘ and y˘,1/2∈ξp˘, which imply that x˘,1/2=x˘,1/2∧1/2∈∨q˘ξp˘.
Therefore,ξn˘x˘≤−1/2 and ξp˘x˘≥1/2, because if ξn˘x˘>−1/2 and ξp˘x˘<1/2, then ξn˘x˘−1/2>−1/2−1/2=−1 and ξp˘x˘+1/2<1/2+1/2=1, which is a contradiction. Hence,
(14)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−1/2,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧1/2for all x˘,y˘,z˘∈X˘.
Conversely, assume thatξ satisfies the conditions of (i), (ii), and (iii). Let x˘∈X˘ and s˘∈0,1 and m˘∈−1,0 be such that x˘,m˘∈ξn˘ and x˘,s˘∈ξp˘. Then, ξn˘x˘≤m˘ and ξp˘x˘≥s˘.
Suppose thatξn˘0≥m˘ and ξp˘0≤s˘. If ξn˘x˘>−1/2 and ξp˘x˘<1/2, then ξn˘0≤ξn˘x˘∨−1/2=ξn˘x˘≤m˘ and ξp˘0≥ξp˘x˘∧1/2=ξp˘x˘≥s˘, a contradiction. Hence, we know that ξn˘x˘≤−1/2 and ξp˘x˘≥1/2, and so, we get
(15)ξn˘0+m˘<2ξn˘0≤ξn˘x˘∨−12=−1,ξp˘0+t˘>2ξn˘0≥ξp˘x˘∧12=1.
Thus,0,m˘∈∨ξn˘ and 0,s˘∈∨ξp˘.
Letx˘,y˘∈X˘, s˘,t˘∈0,1 and m˘,n˘∈1,0 be such that x˘⋄z˘⋄y˘,m˘∈ξn˘, y˘,n˘∈ξn˘ and x˘⋄z˘⋄y˘,s˘∈ξp˘, y˘,t˘∈ξp˘. Then, ξn˘x˘⋄z˘⋄y˘≤m˘, ξn˘y˘≤n and ξp˘x˘⋄z˘⋄y˘≥s˘, ξp˘y˘≥t˘.
Suppose thatξn˘x˘>m˘∨n and ξp˘x˘<s˘∧t˘. If ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘≥−1/2 and ξp˘x˘⋄z˘⋄y˘∧ξy˘≤1/2. Then,
(16)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12≤ξn˘x˘⋄y˘∨ξn˘y˘≤m˘∨n,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12≥ξp˘x˘⋄y˘∧ξp˘y˘≥s˘∧t˘,a contradiction. Thus, ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘≤−1/2 and ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘≥1/2. In that case,
(17)ξn˘x˘+m˘∨n˘<2ξn˘x˘≤2ξn˘x˘⋄y˘∨ξn˘y˘∨−12=−1,ξn˘x˘+s˘∧t˘>2ξp˘x˘≥2ξp˘x˘⋄y˘∧ξp˘y˘∧12=1.
Hence,x˘,m˘∨n˘∈∨q˘ξn˘ and x˘,s˘∧t˘∈∨q˘ξp˘. So, ξ is an ∈,∈∨q˘-BFBI of X˘.Definition 14.
Letξ=X˘;ξn˘,ξp˘ be a BFS of X˘ and m˘,s˘∈−1,0×0,1, we define Uξn˘,ξp˘;m˘,s˘=x∈X˘∣ξn˘x≤m˘andξp˘x≥s˘ is called a m˘-level cut of ξn˘ and s˘-level cut of ξp˘ of the BFSξ=X˘;ξn˘,ξp˘.Theorem 15.
ABFSξ=X˘;ξn˘,ξp˘ is an ∈,∈∨q˘−BFI of X˘ if and only if the level subset
(18)Uξn˘,ξp˘;m˘,s˘=x˘∈X˘∣ξn˘x˘≤m˘andξp˘x˘≥s˘is a BFI of X˘ for all m˘∈−1/2,0 and for all s˘∈0,1/2.Proof.
Assume that aBFSξ=X˘;ξn˘,ξp˘ is an ∈,∈∨q˘−BFI of X˘. Let x˘⋄z˘⋄y˘,y˘∈Uξn˘,ξp˘;m˘,s˘ with m˘∈−1/2,0 and s˘∈0,1/2. Then ξn˘x˘⋄z˘⋄y˘≤m˘,ξn˘y˘≤m˘ and ξp˘x˘⋄z˘⋄y˘≥s˘,ξp˘y˘≥s˘. Therefore, from Theorem 6 that
(19)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘x˘∨−12≤m˘∨−12=m˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘x˘∧12≥s˘∧12=s˘,so that x˘∈Uξn˘,ξp˘;m˘,s˘. Therefore, Uξn˘,ξp˘;m˘,s˘ is a b-ideal of X˘.
Conversely, letξ be a BFS of X˘ such that the set Uξn˘,ξp˘;m˘,s˘=x˘∈X˘∣ξn˘≤m˘andξp˘≥s˘ is a b-ideal of X˘ for all m˘∈−1/2,0 and s˘∈0,1/2. If there exist x˘⋄z˘⋄y˘,y˘∈X˘ such that ξn˘>ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−1/2 and ξp˘x˘<ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧1/2, then we take m˘∈−1,0 and s˘∈0,1 such that ξn˘x>m˘>ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−1/2 and ξp˘x˘<s˘<ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧1/2.
Thus,x˘⋄z˘⋄y˘,y˘∈Uξn˘,ξp˘;m˘,s˘ with s˘<1/2 and m˘>1/2, and so x˘∈Uξn˘,ξp˘;m˘,s˘, i.e., ξn˘x˘≤m˘andξp˘x˘≥s˘ which is a contradiction. Therefore,
(20)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12,for all x˘,y˘∈X˘. Using Theorem 6, we conclude that ξ is an ∈,∈∨q˘-BFBI of X˘.Theorem 16.
Letξ=X˘;ξn˘,ξp˘ be an ∈,∈∨q−BFBI of X˘, where ξn˘x˘>−1/2 and ξp˘x˘<1/2 for all x˘∈X˘. Then, ξ=X˘;ξn˘,ξp˘ is an ∈,∈-BFBI of X˘.Proof.
The proof is straightforward using Theorem6.Theorem 17.
Let∧ be an index set and ξin˘,ξip˘∣i∈∧ be a family of ∈,∈∨q˘−BFBI of X˘. Then, ξ=⋂i∈∧ξin˘,ξip˘ is an ∈,∈∨q˘−BFBI of X˘.Proof.
Let us takex˘⋄z˘⋄y˘,y˘∈X˘ and m˘1,m˘2∈−1,0, and s˘1,s˘2∈0,1 be such that ξn˘x˘⋄z˘⋄y˘≤m˘1 and ξn˘x˘⋄z˘⋄y˘≤m˘2, ξp˘x˘⋄z˘⋄y˘≥s˘1 and ξp˘x˘⋄z˘⋄y˘≥s˘2.
Assume thatx˘m˘1∨m˘2∈∨qξn˘¯ and x˘s˘1∧s˘2∈∨qξp˘¯. Then, ξn˘x˘>m˘1∨m˘2 and ξn˘x˘+m˘1∨m˘2≥−1, and ξp˘x˘<s˘1∧s˘2 and ξp˘x˘+s˘1∧s˘2≤1, which implies
(21)ξn˘x˘>−12,ξp˘x˘<12.
Now, we define(22)Δ1=i∈∧∣x˘m˘1∨m˘2∈ξin˘andx˘s˘1∧s˘2∈ξip˘,Δ2=i∈∧∣x˘m˘1∨m˘2qξin˘∩j∈∧∣x˘m˘1∨m˘2∈ξin˘¯andi∈∧∣x˘s˘1∧s˘2qξip˘∩j∈∧∣x˘s˘1∧s˘2∈ξip˘¯.
Then,∧=Δ1∪Δ2 and Δ1∩Δ2=∅.
IfΔ2=∅, then x˘m˘1∨m˘2∈ξin˘andx˘s˘1∧s˘2∈ξip˘ for all i∈∧, i.e., ξin˘x˘≤m˘1∨m˘2 and ξip˘x˘≥s˘1∧s˘2 for all i∈∧, which indicate
(23)ξin˘x˘≤m˘1∨m˘2andξip˘x˘≥s˘1∧s˘2.
This is a contradiction. Hence,Δ2≠∅, and so for every i∈Δ2, we have ξin˘x˘>m˘1∨m˘2 and ξin˘x˘+m˘1∨m˘2<−1, and ξip˘x˘<s˘1∧s˘2 and ξip˘x˘+s˘1∧s˘2>1.
It follows thatm˘1∨m˘2<−1/2 and s˘1∧s˘2>1/2.
Now,x˘m˘1∈ξn˘ and x˘s˘1∈ξp˘ implies that ξn˘x˘⋄z˘⋄y˘≤m˘1 and ξp˘x˘⋄z˘⋄z˘≥s˘1, and thus,
(24)ξin˘x˘⋄z˘⋄y˘≤ξn˘x˘⋄z˘⋄y˘≤m˘1≤m˘1∨m˘2<−12,ξip˘x˘⋄z˘⋄y˘≥ξp˘x˘⋄z˘⋄y˘≥s˘1≥s˘1∧s˘2>12for all i∈∧.
Similarly, we getξin˘<−1/2 and ξip˘y˘>1/2 for all i∈∧.
We suppose thatm˘=ξin˘>−1/2 and s˘=ξip˘<1/2.
Taking thatm˘>n>−1/2 and s˘<t<1/2, we get
(25)x˘⋄y˘⋄z˘n∈ξin˘andy˘n∈ξin˘,butx˘n∨n=x˘n∈∨q¯ξin.˘,(26)x˘⋄y˘⋄z˘t∈ξip˘andy˘t∈ξip˘,butx˘t∧t=x˘t∈∨q¯ξip˘.
This contradicts thatξ=ξn˘,ξp˘ is an ∈,∈∨q˘−BFBI of X˘.
Hence,ξin˘≤−1/2 and ξip˘≥1/2 for all i∈∧, so ξn˘≤−1/2 and ξp˘≥1/2 which contradicts (21).
Therefore,x˘m˘1∨m˘2∈∨qξn˘ and x˘s˘1∧s˘2∈∨qξp˘, and consequently, ξ=ξn˘,ξp˘ is an ∈,∈∨q−BFBI of X˘.
For anyBFSξ in X˘, where m∈1,0 and s∈0,1, we denote
(27)ξm˘n˘=x˘∈X˘∣x˘m˘∈q˘ξn˘,ξs˘p˘=x˘∈X˘∣x˘s˘∈q˘ξp˘,ξm˘,s˘=x˘∈X˘∣x˘m˘∈q˘ξn˘andx˘s˘∈q˘ξp˘.
Then, it is obvious thatξm˘,s˘=Uξn˘,ξp˘;m˘,s˘∪ξm˘n˘∪ξs˘p˘. Here, ξm˘,s˘ is an ∈∨q-level b-ideal of ξ.Theorem 18.
Letξ=ξn˘,ξp˘ be a BFS in ξ. Then, ξ=ξn˘,ξp˘ is an ∈,∈∨q˘−BFBI of X˘ if and only if ξm˘,s˘ is a b-ideal of ξ for all m˘∈−1,0 and s˘∈0,1.Proof.
Suppose thatξ=ξn˘,ξp˘ is an ∈,∈∨q˘−BFBI of X˘ and let x˘,y˘∈ξm˘,s˘ for m˘∈−1,0 and s˘∈0,1. Then x˘m˘∈q˘ξn˘,y˘m˘∈q˘ξn˘ and x˘s˘∈q˘ξp˘,y˘s˘∈q˘ξp˘. That is, ξn˘x˘≤m˘ or ξn˘x˘+m˘<−1, ξn˘y˘≤m˘ or ξn˘y˘+m˘<−1 and ξp˘x˘≥s˘ or ξp˘x˘+s˘>1, ξp˘y˘≥s˘ or ξp˘y˘+s˘>1. Using Theorem 6, we get
(28)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12.Case 19.
ξn˘x˘≤m˘, ξn˘y˘≤m˘ and ξp˘x˘≥s˘, ξp˘y˘≥s˘. If m˘<−1/2 and t>1/2, then
(29)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12=−12,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12=12.
Hence,(30)ξn˘x˘+m˘<−12−12=−1,ξp˘x˘+s˘>12+12=1,and so, x˘∈q˘ξn˘ and x˘∈q˘ξp˘. If m˘≥−1/2 and s˘≤1/2, then
(31)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12≤m˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12≥s˘.
Thus,x˘m˘∈∨q˘ξn˘ and x˘s˘∈∨q˘ξp˘. Therefore, ξm˘,s˘.Case 20.
ξn˘x˘≤m˘, ξn˘y˘+m˘<−1 and ξp˘x˘≥s˘, ξp˘y˘+s˘>1. If m˘<−1/2 and t>1/2, then
(32)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12=ξn˘y˘∨−12=−1−m˘∨−12=−1−m˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12=ξp˘y˘∧12=1−t∧12=1−t.
Hence,(33)ξn˘x˘+m˘<−12−12=−1,ξp˘x˘+s˘>12+12=1,and so, x˘∈q˘ξn˘ and x˘∈q˘ξp˘. If m˘≥−1/2 and s˘≤1/2, then
(34)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12≤m˘∨−1−m˘∨−12=m˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12≥s˘∧1−s˘∧12=s˘.
Thus,x˘m˘∈∨q˘ξn˘ and x˘s˘∈∨q˘ξp˘. Therefore, ξm˘,s˘.Case 21.
ξn˘x˘+m˘<−1, ξn˘y˘≤m˘ and ξp˘x˘+s˘>1, ξp˘y˘≥s˘. If m˘<−1/2 and t>1/2, then
(35)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12=ξn˘x˘∨−12=−1−m˘∨−12=−1−m˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12=ξp˘x˘∧12=1−t∧12=1−t.
Hence,(36)ξn˘x˘+m˘<−12−12=−1,ξp˘x˘+s˘>12+12=1,and so, x˘∈q˘ξn˘ and x˘∈q˘ξp˘. If m˘≥−1/2 and s˘≤1/2, then
(37)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12≤−1−m˘∨m˘∨−12=m˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12≥1−s˘∧s˘∧12=s˘.
Thus,x˘m˘∈∨q˘ξn˘ and x˘s˘∈∨q˘ξp˘. Therefore, ξm˘,s˘.Case 22.
ξn˘x˘+m˘<−1, ξn˘y˘+m˘<−1 and ξp˘x˘+s˘>1, ξp˘y˘+s˘>1. If m˘<−1/2 and t>1/2, then
(38)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12=ξn˘x˘∨−12=−1−m˘∨−12=−1−m˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12=ξp˘x˘∧12=1−t∧12=1−t.
Hence,(39)ξn˘x˘+m˘<−12−12=−1,ξp˘x˘+s˘>12+12=1,
and so,x˘∈q˘ξn˘ and x˘∈q˘ξp˘. If m˘≥−1/2 and s˘≤1/2, then
(40)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12≤−1−m˘∨−12=−12≤m˘,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12≥1−s˘∧12=12≥s˘.
Therefore,x˘m˘∈∨q˘ξn˘ and x˘s˘∈∨q˘ξp˘. Hence, ξm˘,s˘. Therefore, ξm˘,s˘ is a b-ideal of ξ.
Conversely, letξ=ξn˘,ξp˘ be a BFS in ξ and m˘∈−1,0,s˘∈0,1. Then, ξ=ξn˘,ξp˘ is an ∈,∈∨q˘−BFBI of X˘ such that ξm˘,s˘ is a b-ideal of ξ. If possible, let
(41)ξn˘x˘>m˘≥ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12,ξp˘x˘<s˘≤ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12,for some m˘∈−1,0,s˘∈0,s˘. Then, x˘⋄z˘⋄y˘,y˘∈Uξn˘,ξp˘;m˘,s˘⊆ξm˘,s˘, which indicate x˘∈ξm˘,s˘. Thus, ξn˘x≤m˘ or ξn˘x˘+m˘<−1, ξn˘y˘≤m˘ or ξn˘y˘+m˘<−1 and ξp˘x˘≥s˘ or ξp˘x˘+s˘>1, ξp˘y˘≥s˘ or ξp˘y˘+s˘>1, and these are a contradiction. Hence,
(42)ξn˘x˘≤ξn˘x˘⋄z˘⋄y˘∨ξn˘y˘∨−12,ξp˘x˘≥ξp˘x˘⋄z˘⋄y˘∧ξp˘y˘∧12for all x˘,y˘∈X˘. Now, by using Theorem 6, we conclude that ξ=ξn˘,ξp˘ is an ∈,∈∨q˘−BFBI of X˘.


## 5. Conclusion

In this paper, the thought of∈,∈∨q˘-bipolar fuzzy ideals, and ∈,∈∨q˘-bipolar fuzzy b-ideals are presented and described their valuable properties. We investigated the connection between ∈,∈∨q˘-bipolar fuzzy b-ideals and bipolar fuzzy b-ideals and also the relation of their corresponding ideals. In our future study of a bipolar fuzzy structure of BCK/BCI-algebra, we may consider the following topics: (i) bipolar fuzzy soft subalgebra of BCK/BCI-algebra, (ii) bipolar ∈,∈∨q˘-fuzzy soft subalgebra of BCK/BCI-algebra, and (iii) ∈,∈∨q˘-bipolar fuzzy soft (p and a)-ideals of BCK/BCI-algebra and their relations.



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