# Bipolar Fuzzy Implicative Ideals of BCK-Algebras

Authors: G. Muhiuddin; D. Al-Kadi
Journal: Journal of Mathematics
 (2021)

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## Abstract

The notion of bipolar fuzzy implicative ideals of a BCK-algebra is introduced, and several properties are investigated. The relation between a bipolar fuzzy ideal and a bipolar fuzzy implicative ideal is studied. Characterizations of a bipolar fuzzy implicative ideal are given. Conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal are provided. Extension property for a bipolar fuzzy implicative ideal is stated.


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## Body

## 1. Introduction

Fuzzy sets are characterized by a membership function which associates elements with real numbers in the interval [0, 1] that represents its membership degree to the fuzzy set. Several kinds of fuzzy set extensions have been introduced such as interval-valued, intuitionistic, and bipolar-valued fuzzy sets. The bipolar-valued fuzzy set notion [1] was introduced to treat imprecision as in traditional fuzzy sets, where the degree of membership belongs to the interval [0, 1], and we cannot tell apart unrelated elements from the opposite elements. The extension here enlarges the range of the membership degree from the interval [0, 1] to the interval [−1, 1] to solve such a problem (we refer the reader to [2–4]). The membership degrees which lie in the interval [−1, 1] represent the satisfaction degree to the corresponding property in a fuzzy set and its counter property as follows: having a membership degree in the interval [−1, 0) means that the elements are satisfying implicit counter property, having (0, 1] means that the elements are satisfying the property, and having 0 means that the elements are unrelated to the corresponding property.The bipolar-valued fuzzification has been used to study different notions in BCK/BCI-algebras such as subalgebras and ideals of BCK/BCI-algebras [5], a-ideals of BCI-algebras [6], and more, see the references [7–10]. Other researches also added their contribution to the study in this field on different branches of algebra in various aspects (see, e.g., [11–26]). Also, some more general concepts on bipolar fuzzy have been studied in [27–31].Recently, the bipolar fuzzy BCI-implicative ideals of BCI-algebras were studied in [32]. Moreover, new types of bipolar fuzzy ideals of BCK-algebras have been investigated in [33], typically bipolar fuzzy (closed, positive implicative, and implicative) ideals. Moreover, some related concepts on fuzzy sets and their useful generalizations were applied in various algebraic structures (see, e.g., [33–53]).In this paper, we apply the notion of a bipolar-valued fuzzy set to implicative ideals of BCK-algebras and obtain further results in this manner. Furthermore, we consider the relation of a bipolar fuzzy ideal with a bipolar fuzzy implicative ideal. We provide characterizations of a bipolar fuzzy implicative ideal. Moreover, we display conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal. Finally, we discuss extension property for a bipolar fuzzy implicative ideal.


## 2. Preliminaries

The basic results on BCK-algebras are given in this section.By aBCK-algebra, we mean an algebra Ł;∗,0 of type (2, 0) satisfying the axioms:(a1)
∀ϰ,ℓ,υ∈Łϰ∗ℓ∗ϰ∗υ∗υ∗ℓ=0(a2)
∀ϰ,ℓ∈Łϰ∗ϰ∗ℓ∗ℓ=0(a3)
∀ϰ∈Łϰ∗ϰ=0,0∗ϰ=0(a4)
∀ϰ,ℓ∈Łϰ∗ℓ=0,ℓ∗ϰ=0⟹ϰ=ℓWe can define a partial ordering≤ by ϰ≤ℓ if and only if ϰ∗ℓ=0.In any BCK-algebraŁ, the following hold:(b1)
∀ϰ∈Łϰ∗0=ϰ(b2)
∀ϰ,ℓ,υ∈Łϰ∗ℓ∗υ=ϰ∗υ∗ℓ(b3)
∀ϰ,ℓ,υ∈Łϰ∗υ∗ℓ∗υ≤ϰ∗ℓ(b4)
∀ϰ,ℓ,υ∈Łϰ≤ℓ⇒ϰ∗υ≤ℓ∗υ,υ∗ℓ≤υ∗ϰLet us consider a subset∅≠I of a BCK-algebra Ł. We say I is an ideal if(c1)0∈I, (c2)∀ϰ∈Ł∀ℓ∈Iϰ∗ℓ∈I⇒ϰ∈IA nonempty subsetI of a BCK-algebra Ł is called an implicative ideal of Ł if it satisfies (c1) and(c3)∀ϰ,ℓ,υ∈Łϰ∗ℓ∗υ∈I,ℓ∗υ∈I⇒ϰ∗υ∈I


## 3. Bipolar Fuzzy Ideals

In the following sections,Ł denotes a BCK-algebra.For any familyδi|i∈Δ of real numbers, we define(1)∨δi|i∈Δ≔maxδi|i∈Δ,ifΔis finite,supδi|i∈Δ,otherwise,∧δi|i∈Δ≔minδi|i∈Δ,ifΔis finite,infδi|i∈Δotherwise.Moreover, ifΔ=1,2,…,n, then ∨δi|i∈Δ and ∧δi|i∈Δ are denoted by δ1∨δ2∨⋯∨δn and δ1∧δ2∧⋯∧δn, respectively.For a bipolar fuzzy setq=Ł;qn,qp, we define negativeα-cut of q=Ł;qn,qp and the positiveβ-cut of q=Ł;qn,qp, respectively, as follows:(2)Nq;α≔ϰ∈Ł|qnϰ≤α,Pq;β≔ϰ∈Ł|qpϰ≥β,where α,β∈−1,0×0,1. The set(3)Cq;α,β≔Nq;α∩Pq;βis called the α,β-cut of q=Ł;qn,qp. For every k∈0,1, if α,β=−k,k, then the set(4)Cq;k≔Nq;−k∩Pq;kis called the k-cut of q=Ł;qn,qp.Definition 1.
(see [5]). A bipolar fuzzy set q=Ł;qn,qp in a BCK-algebra Ł is called a bipolar fuzzy ideal of Ł if it satisfies the following assertions:(i)
∀ϰ∈Łqn0≤qnϰ,qp0≥qpϰ(ii)
∀ϰ,ℓ∈Łqnϰ≤qnϰ∗ℓ∨qnℓ,qpϰ≥qpϰ∗ℓ∧qpℓ.
For anyw∈Ł and any bipolar fuzzy set q=Ł;qn,qp in Ł, we let(5)Iw=ϰ∈Ł|qnϰ≤qnw,qpϰ≥qpw.
Obviously,w∈Iw. If q=Ł;qn,qp is a bipolar fuzzy ideal of Ł, then 0∈Iw. The following is our question: For a bipolar fuzzy setq=Ł;qn,qpinŁsatisfying Definition 1 (i), isIwan ideal ofŁ? The following example provides a negative answer; that is, there exists an element w∈Ł such that Iw is not an ideal of Ł.Example 1.
LetŁ=θ,ℓ,υ,ω,δ be a set with a Cayley table which is given in Table 1.
Then,Ł;∗,θ is a BCK-algebra. Let q=Ł;qn,qp be a bipolar fuzzy set in Ł defined by
Then,q=Ł;qn,qp satisfies Definition 1 (i), and it is not a bipolar fuzzy ideal of Ł because(6)qnυ=−0.3>−0.4=qnυ∗δ∨qnδ,and/or(7)qpυ=0.4<0.5=qpυ∗δ∧qpδ.
Then,Iδ=θ,ℓ,δ is not an ideal of Ł since υ∗δ=θ∈Iδ and δ∈Iδ, while υ∉Iδ. Note that Iυ=θ,ℓ,υ,δ is an ideal of Ł.Table 1
Cayley table.
∗θℓυωδθθθθθθℓℓθℓθθυυυθυθωωωωθωδδδδδθWe give conditions for the setIw to be an ideal.Theorem 1.
Letw∈Ł. If q=Ł;qn,qp is a bipolar fuzzy ideal of Ł, then Iw is an ideal of Ł.Proof.
We recall that0∈Iw. Let ϰ,ℓ∈Ł such that ϰ∗ℓ∈Iw and ℓ∈Iw. Then, qnw≥qnϰ∗ℓ, qpw≤qpϰ∗ℓ, qnw≥qnℓ and qpw≤qpℓ. Since q=Ł;qn,qp is a bipolar fuzzy ideal of Ł, we have from Definition 1 (ii) that(8)qnϰ≤qnϰ∗ℓ∨qnℓ≤qnw,qpϰ≥qpϰ∗ℓ∧qpℓ≥qpw,and so, ϰ∈Iw. Therefore, Iw is an ideal of Ł.Theorem 2.
Letq=Ł;qn,qp be a bipolar fuzzy set in Ł and w∈Ł.(1)
IfIw is an ideal of Ł, then q=Ł;qn,qp satisfies the following implications for all ϰ,ℓ,υ∈Ł:(9)qnϰ≥qnℓ∗υ∨qnυ⇒qnϰ≥qnℓ,qpϰ≤qpℓ∗υ∧qpυ⇒qpϰ≤qpℓ.(2)
Ifq=Ł;qn,qp satisfies Definition 1(i) and (9), then Iw is an ideal of Ł.Proof.
(1)
We assume thatIw is an ideal of Ł for each w∈Ł. We suppose that qnϰ≥qnℓ∗υ∨qnυ and qpϰ≤qpℓ∗υ∧qpυ for all ϰ,ℓ,υ∈Ł. Then, ℓ∗υ∈Iϰ and υ∈Iϰ. Since Iϰ is an ideal of Ł, it follows that ℓ∈Iϰ, that is, qnϰ≥qnℓ and qpϰ≤qpℓ.(2)
We suppose thatq=Ł;qn,qp satisfies Definition 1 (i) and (9). For each w∈Ł, let ϰ,ℓ∈Ł such that ϰ∗ℓ∈Iw and ℓ∈Iw. Then, qnϰ∗ℓ≤qnw, qpϰ∗ℓ≥qpw, qnℓ≤qnw, and qpℓ≥qpw, which imply that qnw≥qnϰ∗ℓ∨qnℓ and qpw≤qpϰ∗ℓ∧qpℓ. Using (9), we have qnw≥qnϰ and qpw≤qpϰ, and so, ϰ∈Iw. Since q=Ł;qn,qp satisfies Definition 1 (i), it follows that 0∈Iw. Therefore, Iw is an ideal of Ł.Lemma 1.
(see [5]). Every bipolar fuzzy ideal q=Ł;qn,qp of Ł satisfies the following implication:(10)∀ϰ,ℓ∈Łϰ≤ℓ⇒qnϰ≤qnℓ,qpϰ≥qpℓ.Proposition 1.
For any bipolar fuzzy idealq=Ł;qn,qp of Ł, the following are equivalent:(1)
∀ϰ,ℓ∈Łqnϰ∗ℓ≤qnϰ∗ℓ∗ℓ,qpϰ∗ℓ≥qpϰ∗ℓ∗ℓ.(2)
∀ϰ,ℓ,υ∈Łqnϰ∗υ∗ℓ∗υ≤qnϰ∗ℓ∗υ,qpϰ∗υ∗ℓ∗υ≥qpϰ∗ℓ∗υ.Proof.
We assume that condition (2) is valid. Note that(11)ϰ∗ℓ∗υ∗υ∗υ=ϰ∗υ∗ℓ∗υ∗υ≤ϰ∗ℓ∗υ,for all ϰ,ℓ,υ∈Ł by using (b2), (b3), and (b4). It follows from Lemma 1 that(12)qnϰ∗ℓ∗υ≥qnϰ∗ℓ∗υ∗υ∗υ,qpϰ∗ℓ∗υ≤qpϰ∗ℓ∗υ∗υ∗υ.
So, from (b2) and (2), it follows that(13)qnϰ∗υ∗ℓ∗υ=qnϰ∗ℓ∗υ∗υ≤qnϰ∗ℓ∗υ∗υ∗υ≤qnϰ∗ℓ∗υ,qpϰ∗υ∗ℓ∗υ=qpϰ∗ℓ∗υ∗υ≥qpϰ∗ℓ∗υ∗υ∗υ≥qpϰ∗ℓ∗υ.
Thus, (9) holds. Now, we suppose that (9) is valid. Using (b1), (a3), and (9) with replacing υ by ℓ, we have(14)qnϰ∗ℓ=qnϰ∗ℓ∗0=qnϰ∗ℓ∗ℓ∗ℓ≤qnϰ∗ℓ∗ℓ,qpϰ∗ℓ=qpϰ∗ℓ∗0=qpϰ∗ℓ∗ℓ∗ℓ≥qpϰ∗ℓ∗ℓ,which proves (2).Proposition 2.
(see [5]). A bipolar fuzzy set q=Ł;qn,qp in Ł is a bipolar fuzzy ideal of Ł if and only if for all ϰ,ℓ,υ∈Ł, ϰ∗ℓ∗υ=0 implies qnϰ≤qnℓ∨qnυ and qpϰ≥qpℓ∧qpυ.As a generalization of Proposition2, we have the following results.Theorem 3.
If a bipolar fuzzy setq=Ł;qn,qp in Ł is a bipolar fuzzy ideal of Ł, then for all ϰ,w1,w2,…,wn∈Ł,(15)∏i=1nϰ∗wi=0⇒qnϰ≤qnw1∨qnw2∨⋯∨qnwn,qpϰ≥qpw1∧qpw2∧⋯∧qpwn,where ∏i=1nϰ∗wi=⋯ϰ∗w1∗w2∗⋯∗wn.Proof.
The proof is by induction onn. Let q=Ł;qn,qp be a bipolar fuzzy ideal of Ł. Lemma 1 and Proposition 2 show that condition (15) is valid for n=1,2. We assume that q=Ł;qn,qp satisfies condition (15) for n=k, that is, for all ϰ,w1,w2,…,wk∈Ł, ∏i=1kϰ∗wi=0 implies(16)qnϰ≤qnw1∨qnw2∨⋯∨qnwk,qpϰ≥qpw1∧qpw2∧⋯∧qpwk.
Letϰ,w1,w2,…,wk,wk+1∈Ł such that ∏i=1k+1ϰ∗wi=0. Then,(17)qnϰ∗w1≤qnw2∨qnw3∨⋯∨qnwk+1,qpϰ∗w1≥qpw2∧qpw3∧⋯∧qpwk+1.
Sinceq=Ł;qn,qp is a bipolar fuzzy ideal of Ł, it follows from Definition 1 (ii) that(18)qnϰ≤qnϰ∗w1∨qnw1≤qnw1∨qnw2∨⋯∨qnwk+1,qpϰ≥qpϰ∗w1∧qpw1≥qpw1∧qpw2∧⋯∧qpwk+1.
This completes the proof.Now, we consider the converse of Theorem3.Theorem 4.
Letq=Ł;qn,qp be a bipolar fuzzy set in Ł satisfying condition (15). Then, q=Ł;qn,qp is a bipolar fuzzy ideal of Ł.Proof.
Note that⋯0∗ϰ∗ϰ∗⋯∗ϰ︸ntimes=0 for all ϰ∈Ł. It follows from (15) that qn0≤qnϰ and qp0≥qpϰ for all ϰ∈Ł. Let ϰ,ℓ,υ∈Ł such that ϰ∗ℓ≤υ. Then,(19)0=ϰ∗ℓ∗υ=⋯ϰ∗ℓ∗υ∗0∗⋯∗0︸n−2times,and so,(20)qnϰ≤qnℓ∨qnυ∨qn0=qnℓ∨qnυ,qpϰ≥qpℓ∧qpυ∧qp0=qpℓ∧qpυ.
Hence, by Proposition2, we conclude that q=Ł;qn,qp is a bipolar fuzzy ideal of Ł.


## 4. Bipolar Fuzzy Implicative Ideals

Definition 2.
A bipolar fuzzy setq=Ł;qn,qp in Ł is called a bipolar fuzzy implicative ideal of Ł if both the nonempty negative α-cut and the nonempty positive β-cut of q=Ł;qn,qp are implicative ideals of Ł for all α,β∈−1,0×0,1.Example 2.
LetŁ=θ,ℓ,υ be a set in which the operation ∗ is defined by Table 2.
Then,Ł;∗,θ is a BCK-algebra. Let t0,s0,t1,s1∈−1,0×0,1 satisfy t0,s0>t1,s1, that is, t0<t1 and s0>s1. Let q=Ł;qn,qp be a bipolar fuzzy set in Ł given by
By routine calculations, we know thatq=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł.Table 2
Cayley table.
∗θℓυθθθθℓℓθθυυυθTheorem 5.
A bipolar fuzzy setq=Ł;qn,qpinŁis a bipolar fuzzy implicative ideal ofŁif and only if it satisfies Definition 1 (i) and the following assertions:(21)∀ϰ,ℓ,υ∈Łqnϰ∗υ≤qnϰ∗ℓ∗υ∨qnℓ∗υ,qpϰ∗υ≥qpϰ∗ℓ∗υ∧qpℓ∗υ.Proof.
We suppose thatq=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł. If qn0>qnb or qp0<qpd for some b,d∈Ł, then 0∉Nq;qnb or 0∉Pq;qpd, which contradicts the fact. Hence, qn0≤qnϰ and qp0≥qpϰ∀ϰ∈Ł. For some b,d,c∈Ł, we assume that we have the following relation:(22)qnb∗c>qnb∗d∗c∨qnd∗c=s.
Then,b∗d∗c∈Nq;s and d∗c∈Nq;s, but b∗c∉Nq;s. This is not possible; therefore, we have(23)qnϰ∗υ≤qnϰ∗ℓ∗υ∨qnℓ∗υ,for all ϰ,ℓ,υ∈Ł. If qpb∗c<qpb∗d∗c∧qpd∗c=t for some b,d,c∈Ł, then b∗d∗c∈Pq;t and d∗c∈Pq;t, but b∗c∉Pq;t. We reach a contradiction because Pq;t is an implicative ideal of Ł. Henceforth,(24)qpϰ∗υ≥qpϰ∗ℓ∗υ∧qpℓ∗υ,for all ϰ,ℓ,υ∈Ł. Consequently, a bipolar fuzzy implicative ideal q=Ł;qn,qp satisfies Definition 1 (i) and (21).
Conversely, we suppose thatq=Ł;qn,qp satisfies Definition 1 (i) and (21) and let α,β∈−1,0×0,1 s.th. Nq;α≠∅ and Pq;β≠∅. It is clear that 0∈Nq;α∩Pq;β. Let ϰ,ℓ,υ∈Ł be such that ϰ∗ℓ∗υ∈Nq;α and ℓ∗υ∈Nq;α. Then, qnϰ∗ℓ∗υ≤α and qnℓ∗υ≤α. It follows from (21) that(25)qnϰ∗υ≤qnϰ∗ℓ∗υ∨qnℓ∗υ≤α,and so, ϰ∗υ∈Nq;α. Hence, Nq;α is an implicative ideal of Ł. Similarly, we can show that(26)qpϰ∗υ≥qpϰ∗ℓ∗υ∧qpℓ∗υ≥β,for all ϰ,ℓ,υ∈Ł, and so, ϰ∗υ∈Pq;β. Therefore, Pq;β is an implicative ideal of Ł. Consequently, q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł.Next, we have the following theorems.Theorem 6.
A bipolar fuzzy idealq=Ł;qn,qp of Ł is a bipolar fuzzy implicative ideal of Ł if and only if it satisfies Proposition 1 (1).Proof.
Letq=Ł;qn,qp be a bipolar fuzzy implicative ideal of Ł. If υ is replaced by ℓ in (21), then(27)qnϰ∗ℓ≤qnϰ∗ℓ∗ℓ∨qnℓ∗ℓ,=qnϰ∗ℓ∗ℓ∨qn0,=qnϰ∗ℓ∗ℓ,qpϰ∗ℓ≥qpϰ∗ℓ∗ℓ∧qpℓ∗ℓ,=qpϰ∗ℓ∗ℓ∧qp0,=qpϰ∗ℓ∗ℓ,which is Proposition 1 (1). Conversely, let q=Ł;qn,qp be a bipolar fuzzy ideal of Ł satisfying Proposition 1 (1). Note that(28)ϰ∗υ∗υ∗ℓ∗υ≤ϰ∗υ∗ℓ=ϰ∗ℓ∗υ,for all ϰ,ℓ,υ∈Ł. Using Lemma 1, we have(29)qnϰ∗ℓ∗υ≥qnϰ∗υ∗υ∗ℓ∗υ,qpϰ∗ℓ∗υ≤qpϰ∗υ∗υ∗ℓ∗υ.It follows from Definition 1 (ii) and Proposition 1 (1) that(30)qnϰ∗υ≤qnϰ∗υ∗υ≤qnϰ∗υ∗υ∗ℓ∗υ∨qnℓ∗υ≤qnϰ∗ℓ∗υ∨qnℓ∗υ,qpϰ∗υ≥qpϰ∗υ∗υ≥qpϰ∗υ∗υ∗ℓ∗υ∧qpℓ∗υ≥qpϰ∗ℓ∗υ∧qpℓ∗υ.
Thus,q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł.Combining Proposition1 and Theorem 6, we have the following characterization of a bipolar fuzzy implicative ideal.Theorem 7.
Letq=Ł;qn,qp be a bipolar fuzzy ideal of Ł. Then, it is a bipolar fuzzy implicative ideal of Ł if and only if it satisfies Proposition 1 (2).Theorem 8.
(see [33]). Every bipolar fuzzy implicative ideal is a bipolar fuzzy ideal.Theorem 9.
Letq=Ł;qn,qp be a bipolar fuzzy set in Ł. Then, q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł if and only if it satisfies Definition 1 (i) and(31)∀ϰ,ℓ,υ∈Łqnϰ∗ℓ≤qnϰ∗ℓ∗ℓ∗υ∨qnυ,qpϰ∗ℓ≥qpϰ∗ℓ∗ℓ∗υ∧qpυ.Proof.
We suppose thatq=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł. Then, q=Ł;qn,qp is a bipolar fuzzy ideal of Ł by Theorem 8, and so, Definition 1 (i) is true. From Theorem 7, it follows that q=Ł;qn,qp satisfies Proposition 1 (2). Thus,(32)qnϰ∗ℓ≤qnϰ∗ℓ∗υ∨qnυ,=qnϰ∗υ∗ℓ∗ℓ∗ℓ∨qnυ≤qnϰ∗υ∗ℓ∗ℓ∨qnυ,=qnϰ∗ℓ∗ℓ∗υ∨qnυqpϰ∗ℓ≥qpϰ∗ℓ∗υ∧qpυ,=qpϰ∗υ∗ℓ∗ℓ∗ℓ∧qpυ≥qpϰ∗υ∗ℓ∗ℓ∧qpυ,=qpϰ∗ℓ∗ℓ∗υ∧qpυ,which proves (31). Conversely, let q=Ł;qn,qp be a bipolar fuzzy set in Ł satisfying Definition 1 (i) and (31). Then,(33)qnϰ=qnϰ∗0≤qnϰ∗0∗0∗υ∨qnυ=qnϰ∗υ∨qnυ,qpϰ=qpϰ∗0≥qpϰ∗0∗0∗υ∧qpυ=qpϰ∗υ∧qpυ.
Thus,q=Ł;qn,qp is a bipolar fuzzy ideal of Ł. Now, we take υ=0 in (31) and use (b1) and Definition 1 (i) to get(34)qnϰ∗ℓ≤qnϰ∗ℓ∗ℓ∗0∨qn0,=qnϰ∗ℓ∗ℓ∨qn0,=qnϰ∗ℓ∗ℓ,qpϰ∗ℓ≥qpϰ∗ℓ∗ℓ∗0∧qp0,=qpϰ∗ℓ∗ℓ∧qp0,=qpϰ∗ℓ∗ℓ.
It follows from Theorem6 that q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł.Summarizing the abovementioned results, we have a characterization of a bipolar fuzzy implicative ideal ofŁ.Theorem 10.
Letq=Ł;qn,qp be a bipolar fuzzy set in Ł. Then, the following assertions are equivalent:(1)
q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł(2)
q=Ł;qn,qp satisfies Definition 1 (i) and (21)(3)
q=Ł;qn,qp is a bipolar fuzzy ideal of Ł satisfying Proposition 1 (1)(4)
q=Ł;qn,qp is a bipolar fuzzy ideal of Ł satisfying Proposition 1 (2)(5)
q=Ł;qn,qp satisfies Definition 1 (i) and (31)Theorem 11.
Letw∈Ł. If q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł, then Iw is an implicative ideal of Ł.Proof.
We recall that0∈Iw. Let ϰ,ℓ,υ∈Ł such that ϰ∗ℓ∗υ∈Iw and ℓ∗υ∈Iw. Then, qnw≥qnϰ∗ℓ∗υ, qpw≤qpϰ∗ℓ∗υ, qnw≥qnℓ∗υ, and qpw≤qpℓ∗υ. Since q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł, it follows from (21) that(35)qnϰ∗υ≤qnϰ∗ℓ∗υ∨qnℓ∗υ≤qnw,qpϰ∗υ≥qpϰ∗ℓ∗υ∧qpℓ∗υ≥qpw,so that ϰ∗υ∈Iw. Therefore, Iw is an implicative ideal of Ł.Theorem 12.
Ifq=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł, then for all ϰ,ℓ,υ,a,b∈Ł,(1)
ϰ∗ℓ∗ℓ∗a≤b⇒qnϰ∗ℓ≤qna∨qnb,qpϰ∗ℓ≥qpa∧qpb.(2)
ϰ∗ℓ∗υ∗a≤b⇒qnϰ∗υ∗ℓ∗υ≤qna∨qnb,qpϰ∗υ∗ℓ∗υ≥qpa∧qpb.Proof.
Letϰ,ℓ,a,b∈Ł such that ϰ∗ℓ∗ℓ∗a≤b. Using Proposition 2, we have qnϰ∗ℓ∗ℓ≤qna∨qnb and qpϰ∗ℓ∗ℓ≥qpa∧qpb. It follows that(36)qnϰ∗ℓ≤qnϰ∗ℓ∗ℓ∨qnℓ∗ℓ,=qnϰ∗ℓ∗ℓ∨qn0,=qnϰ∗ℓ∗ℓ≤qna∨qnb,qpϰ∗ℓ≥qpϰ∗ℓ∗ℓ∧qpℓ∗ℓ,=qpϰ∗ℓ∗ℓ∧qp0,=qpϰ∗ℓ∗ℓ≥qpa∧qpb.
Now, letϰ,ℓ,υ,a,b∈Ł such that ϰ∗ℓ∗υ∗a≤b, that is,(37)ϰ∗ℓ∗υ∗a∗b=0.
Sinceq=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł, it follows from Theorem 7 and Proposition 2 that(38)qnϰ∗υ∗ℓ∗υ≤qnϰ∗ℓ∗υ≤qna∨qnb,qpϰ∗υ∗ℓ∗υ≥qpϰ∗ℓ∗υ≥qpa∧qpb.
This completes the proof.Theorem 13.
Letq=Ł;qn,qp be a bipolar fuzzy set in Ł satisfying Theorem 12 (1). Then, q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł.Proof.
We first prove thatq=Ł;qn,qp is a bipolar fuzzy ideal of Ł. Let ϰ,ℓ,υ∈Ł such that ϰ∗ℓ≤υ. Then,(39)ϰ∗0∗0∗ℓ∗υ=ϰ∗ℓ∗υ=0,that is,ϰ∗0∗0∗ℓ≤υ,which implies from (b1) and Theorem 12 (1) that qnϰ=qnϰ∗0≤qnℓ∨qnυ and qpϰ=qpϰ∗0≥qpℓ∧qpυ. Therefore, by Proposition 2, we know that q=Ł;qn,qp is a bipolar fuzzy ideal of Ł. Note that ϰ∗ℓ∗ℓ∗ϰ∗ℓ∗ℓ∗0=0 for all ϰ,ℓ∈Ł. Using Theorem 12 (1) and Definition 1 (i), we have(40)qnϰ∗ℓ≤qnϰ∗ℓ∗ℓ∨qn0=qnϰ∗ℓ∗ℓ,qpϰ∗ℓ≥qpϰ∗ℓ∗ℓ∧qp0=qpϰ∗ℓ∗ℓ,and so, q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł by Theorem 6.Theorem 14.
Letq=Ł;qn,qp be a bipolar fuzzy set in Ł satisfying Theorem 12 (2). Then, q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł.Proof.
Letϰ,ℓ,a,b∈Ł such that ϰ∗ℓ∗ℓ∗a≤b, that is,(41)ϰ∗ℓ∗ℓ∗a∗b=0.Then,(42)qnϰ∗ℓ=qnϰ∗ℓ∗0=qnϰ∗ℓ∗ℓ∗ℓ≤qna∨qnb,qpϰ∗ℓ=qpϰ∗ℓ∗0=qpϰ∗ℓ∗ℓ∗ℓ≥qpa∧qpb,and so, q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł by Theorem 13.Corollary 1.
Ifq=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł, then(43)qnϰ∗υ∗ℓ∗υ≤∨qnwi|i=1,2,⋯,n,qpϰ∗υ∗ℓ∗υ≥∧qpwi|i=1,2,⋯,n,whenever ∏i=1nϰ∗ℓ∗υ∗wi=0 for all ϰ,ℓ,υ,w1,…,wn∈Ł.Proof.
Letϰ,ℓ,υ,w1,…,wn∈Ł such that ∏i=1nϰ∗ℓ∗υ∗wi=0. Then,(44)qnϰ∗υ∗ℓ∗υ≤qnϰ∗ℓ∗υ≤∨qnwi|i=1,2,⋯,n,qpϰ∗υ∗ℓ∗υ≥qpϰ∗ℓ∗υ≥∧qpwi|i=1,2,⋯,n.
This completes the proof.Theorem 15.
(Extension Property). Letq=Ł;qn,qp and g=Ł;gn,gp be bipolar fuzzy ideals of Ł such that qn0=gn0 and qp0=gp0 and qnϰ≥gnϰ and qpϰ≤gpϰ for all ϰ∈Ł. If q=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł, then so is g=Ł;gn,gp.Proof.
We assume thatq=Ł;qn,qp is a bipolar fuzzy implicative ideal of Ł. For any ϰ,ℓ,υ∈Ł, we have(45)gnϰ∗υ∗ℓ∗υ∗ϰ∗ℓ∗υ,=gnϰ∗υ∗ϰ∗ℓ∗υ∗ℓ∗υ,=gnϰ∗ϰ∗ℓ∗υ∗υ∗ℓ∗υ≤qnϰ∗ϰ∗ℓ∗υ∗υ∗ℓ∗υ≤qnϰ∗ϰ∗ℓ∗υ∗ℓ∗υ,=qnϰ∗ℓ∗ϰ∗ℓ∗υ∗υ,=qnϰ∗ℓ∗υ∗ϰ∗ℓ∗υ,=qn0=gn0,gpϰ∗υ∗ℓ∗υ∗ϰ∗ℓ∗υ,=gpϰ∗υ∗ϰ∗ℓ∗υ∗ℓ∗υ,=gpϰ∗ϰ∗ℓ∗υ∗υ∗ℓ∗υ≥qpϰ∗ϰ∗ℓ∗υ∗υ∗ℓ∗υ≥qpϰ∗ϰ∗ℓ∗υ∗ℓ∗υ,=qpϰ∗ℓ∗ϰ∗ℓ∗υ∗υ,=qpϰ∗ℓ∗υ∗ϰ∗ℓ∗υ,=qp0=gp0.
It follows from Definition1 (i) and (ii) that(46)gnϰ∗υ∗ℓ∗υ≤gnϰ∗υ∗ℓ∗υ∗ϰ∗ℓ∗υ∨gnϰ∗ℓ∗υ≤gn0∨gnϰ∗ℓ∗υ,=gnϰ∗ℓ∗υ,gpϰ∗υ∗ℓ∗υ≥gpϰ∗υ∗ℓ∗υ∗ϰ∗ℓ∗υ∧gpϰ∗ℓ∗υ≥gp0∧gpϰ∗ℓ∗υ,=gpϰ∗ℓ∗υ,for all ϰ,ℓ,υ∈Ł. Hence, by Theorem 7, g=Ł;gn,gp is a bipolar fuzzy implicative ideal of Ł.


## 5. Conclusions

In the present paper, we apply the notion of a bipolar-valued fuzzy set to implicative ideals of BCK-algebras and obtain more related results. We considered the relation of a bipolar fuzzy ideal with a bipolar fuzzy implicative ideal and provided characterizations of a bipolar fuzzy implicative ideal. Also, we studied conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal. Furthermore, an extension property for a bipolar fuzzy implicative ideal is discussed.We hope that this work will give a deep impact on the upcoming research in this field and other fuzzy algebraic study to open up new horizons of interest and innovations. One may apply this concept to study some application fields such as decision making, knowledge base system, and data analysis. In our opinion, these definitions and main results can be similarly extended to some other algebraic systems such as subtraction algebras, B-algebras, MV-algebras, d-algebras, and Q-algebras.



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