# Fuzzy Set Theoretic Approach to Generalized Ideals in BCK/BCI-Algebras

Authors: G. Muhiuddin; N. Alam; S. Obeidat; N. M. Khan; H. N. Zaidi; S. A. K. Kirmani; A. Altaleb; J. M. Aqib
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper deals with the study of generalizations of fuzzy subalgebras and fuzzy ideals in BCK/BCI-algebras. In fact, the notions of∈,∈∨κ~∗,qκ~-fuzzy subalgebras, ∈,∈∨κ~∗,qκ~-fuzzy ideals, and ∈∨κ~∗,qκ~,∈∨κ~∗,qκ~-fuzzy ideals in BCK/BCI-algebras are introduced. Some examples are provided to demonstrate the logic of the concepts used in this paper. Moreover, some characterizations of these notions are discussed. In addition, the concept of ∈,∈∨κ~∗,qκ~-fuzzy commutative ideals is introduced, and several significant characteristics are discussed. It is shown that for a BCK-algebra A, every ∈,∈∨κ~∗,qκ~-commutative ideal of a BCK-algebra is an ∈,∈∨κ~∗,qκ~-fuzzy ideal, but the converse does not hold in general; a counter example is constructed.


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## Body

## 1. Introduction

To deal with possible complexity associated with expectations, state imprecision, and desires, a fuzzy set is a valuable method, as proposed by Zadeh [1]. Fuzzy set theory has since become an active research area in a different field. The idea of quasicoincidence of a fuzzy point with a fuzzy set, as stated in [2], was crucial in generating some different types of fuzzy subgroups, known as α,β-fuzzy subgroups, as introduced by Bhakat and Das in [3]. The ∈,∈∨-fuzzy subgroup, in particular, is an important and valuable generalization of Rosenfeld’s fuzzy subgroup. The idea of α,β-fuzzy subalgebras in BCK/BCI-algebras is also interesting and useful generalizations of some well-known ideas of fuzzy subalgebras (see, for example, [4–11]). Several similar ideals based on fuzzy subalgebras and ideals have recently been studied in various algebras (see, for example, [12–18]). Many researchers have also extended fuzzy set theory and related concepts to different fields (see, e.g., [19–24]).Motivated by a lot of work on ideal theory in BCK/BCI-algebras based on the fuzzy set theory, it is reasonable to introduce a generalized version of the fuzzy ideals of BCK/BCI-algebras. To that aim, in Section2, some basic definitions and elementary results are presented. Then, in Section 3, we present the concepts ∈,∈∨κ∗,qκ-fuzzy subalgebras, ∈,∈∨κ∗,qκ-fuzzy ideals, and ∈∨κ∗,qκ,∈∨κ∗,qκ-fuzzy ideals and associated characteristics are investigated. In Section 4, the concept of ∈,∈∨κ∗,qκ-fuzzy commutative ideals is presented and various properties are investigated, as well as their relationship with ∈,∈∨κ∗,qκ-fuzzy ideals.


## 2. Preliminaries

An algebraA=A;∗,0 is called a BCI-algebra if ∀ϖ,Θ,ς∈A,
(1)
ϖ∗Θ∗ϖ∗ς∗ς∗Θ=0(2)
ϖ∗ϖ∗Θ∗Θ=0(3)
ϖ∗ϖ=0(4)
ϖ∗Θ=0 and Θ∗ϖ=0⇒ϖ=ΘIfA satisfies (2), (3), (40), (43), and 0∗ϖ=0, then A is a BCK-algebra.AnyBCK-algebra A satisfies the following:
(5)
ϖ∗0=ϖ(6)
ϖ∗Θ∗ς=ϖ∗ς∗ΘDefine a partially ordered“≤” on A as ϖ≤Θ⇔ϖ∗Θ=0.A subset∅≠S of A is said to be a subalgebra if ∀ϖ,Θ∈A, ϖ∗Θ∈S and is said to be an ideal of A if 0∈S and ∀ϖ,Θ∈A,ϖ∗Θ∈S,Θ∈S implies ϖ∈S.A mappingT^:A⟶0,1 is said to be a fuzzy set (briefly, FS) of A. If T^ϖ∗Θ≥T^ϖ∧T^Θ, ∀ϖ,Θ∈A, then T^ is said to be a fuzzy subalgebra. If T^0≥T^ϖ and T^ϖ≥T^ϖ∗Θ∧T^Θ, ∀ϖ,Θ∈Z, then T^ is said to be a fuzzy ideal.Definition 1.
Letϖ∈A and ε∈0,1. An ordered fuzzy point (briefly, OFP) ϖεofA is defined as
(1)ϖεΘ=ε,ifΘ∈ϖ,0,ifΘ∈ϖ,∀Θ∈A.Clearly,ϖε is a FS of A. For FS T^ of A, we represent ϖε⊆T^ as ϖε∈T^ in the sequel. So, ϖε∈T^⇔T^ϖ≥ε.Lemma 2 (see [4]).
A FST^ in A is an ∈,∈∨q-fuzzy subalgebra of A⇔ it satisfies
(2)∀ϖ,Θ∈AT^ϖ∗Θ≥T^ϖ∧T^Θ∧0.5.Lemma 3 (see [25]).
A FST^ in A is an ∈,∈∨qk-fuzzy subalgebra of A⇔ it satisfies
(3)∀ϖ,Θ∈AT^ϖ∗Θ≥T^ϖ∧μΘ∧1−κ2.


## 3.∈,∈∨κ∗,qκ-Fuzzy Ideals

In what follows,A denotes a BCK/BCI-algebra unless otherwise specified.Definition 4.
Letϖε be an OFP of A and κ∗∈0,1. Then, ϖε is said to be κ∗,q-quasicoincident with a FS T^ of A, represented as ϖεκ∗,qT^, if T^ϖ+ε>κ∗.
Let us take,0≤κ<κ∗≤1. For an OFP ϖε, we define
(1)
ϖεκ∗,qκT^ if T^ϖ+ε+κ>κ∗(2)
ϖε∈∨κ∗,qκT^ if ϖε∈T^ or ϖεκ∗,qκT^(3)
ϖεα¯T^ if ϖεαT^ does not hold for α∈κ∗,qκ, ∈∨κ∗,qκDefinition 5.
A FST^ of A is said to be an ∈,∈∨κ∗,qκ-fuzzy subalgebra (in short ∈,∈∨κ∗,qκ-FSA) of A if ϖε∈T^ and Θδ∈T^ imply ϖ∗Θε∧δ∈∨κ∗,qκT^, ∀ε,δ∈0,1, and ϖ,Θ∈A.Example 6.
Consider aBCI-algebra A=0,1,2,3 with operation ∗, which is described in Table 1. Define T^:A⟶0,1 by the following:
(4)T^ϖ=0.65,ifϖ=0,0.73,ifϖ=1,0.32,ifϖ∈2,3.
Take “κ∗=0.7” and “κ=0.1.” Then, it yields that T^ is an ∈,∈∨κ∗,qκ-FSA of A.Theorem 7.
A FST^ of A is an ∈,∈∨κ∗,qκ-FSA of A⇔(5)T^ϖ∗Θ≥T^ϖ∧T^Θ∧κ∗−κ2,∀ϖ,Θ∈A.Proof.
⇒ On the contrary, assume that T^ϖ∗Θ<T^ϖ∧T^Θ∧κ∗−κ/2, for some ϖ,Θ∈A. Choose ε∈0,1 such that
(6)T^ϖ∗Θ<ε≤T^ϖ∧T^Θ∧κ∗−κ2.
Then,ϖε∈T^ and Θε∈T^, but ϖ∗Θε∈∨κ∗,qκ¯T^, which is not possible. Hence, T^ϖ∗Θ≥T^ϖ∧T^Θ∧κ∗−κ/2.
(⇐) Assume that T^ϖ∗Θ≥T^ϖ∧T^Θ∧κ∗−κ/2, ∀ϖ,Θ∈A. Let ϖε∈T^ and Θδ∈T^, ∀ε,δ∈0,1. Then, T^ϖ≥ε and T^Θ≥δ. So, T^ϖ∗Θ≥T^ϖ∧T^Θ∧κ∗−κ/2≥ε∧δ∧κ∗−κ/2. If ε∧δ≤κ∗−κ/2, then T^ϖ∗Θ≥ε∧δ imply that ϖ∗Θε∧δ∈T^. If ε∧δ>κ∗−κ/2, then T^ϖ∗Θ≥κ∗−κ/2. So, T^ϖ∗Θ+ε∧δ>κ∗−κ/2+κ∗−κ/2=κ∗−κ implies that ϖ∗Θε∧δκ∗,qκT^. Hence, ϖ∗Θε∧δ∈∨κ∗,qκT^. Therefore, T^ is an ∈,∈∨κ∗,qκ-FSA of A.Definition 8.
A FST^ of A is said to be an ∈,∈∨κ∗,qκ-fuzzy ideal (in short ∈,∈∨κ∗,qκ-FI) of A if
(1)
ϖε∈T^ implies 0ε∈∨κ∗,qκT^, and(2)
ϖ∗Θε∈T^ and Θδ∈T^ imply ϖε∧δ∈∨κ∗,qκT^
∀ϖ,Θ∈A and ε,δ∈0,1.Example 9.
Take aBCI-algebra A=0,1,2,3 with operation ∗ which is described in Table 2. Define T^:A⟶0,1 by the following:
(7)T^ϖ=0.9,ifϖ=0,0.3,ifϖ∈1,2,3.
Consider “κ∗=0.2” and “κ=0.1.” It is easy to check that T^ is an ∈,∈∨κ∗,qκ-FI of A.Definition 10 (see [5]).
A FST^ of A is said to be an ∈,∈∨q-fuzzy ideal (briefly ∈,∈∨q-FI) of A if:
(1)
ϖε∈T^ implies 0ε∈∨qT^, and(2)
ϖ∗Θε∈T^ and Θδ∈T^ imply ϖε∧δ∈∨qT^
∀ϖ,Θ∈A and ε,δ∈0,1.Proposition 11.
InA, every ∈,∈∨q-FI is ∈,∈∨κ∗,qκ-FI.Proof.
Suppose thatT^ is any ∈,∈∨q-FI of A. Take ϖε∈T^ for ϖ∈A and ε∈0,1. Then, by hypothesis, 0ε∈∨qT^. It implies that T^0≥ε or T^0+ε≥1, and so T^0≥ε or T^0+κ+ε≥κ∗. Therefore, 0ε∈∨κ∗,qκT^. Next, let ϖ∗εε∈T^ and εδ∈T^. So, ϖε∧δ∈∨qT^ implies T^ϖ≥ε∧δ or T^ϖ+ε∧δ>1. Therefore, T^ϖ≥ε∧δ or T^ϖ+κ+ε∧δ>κ∗. Thus, ϖε∈∨κ∗,qκT^. Hence, T^ is an ∈,∈∨κ∗,qκ-FI of A.Remark 12.
In general, the converse of Proposition 11 is not valid. The following example demonstrates this.Example 13.
Take aBCK-algebra A=0,1,2,3,4 with operation ∗ which is described in Table 3.
DefineT^:A⟶0,1 by
(8)T^ϖ=0.4,ifϖ=0,0.2,ifϖ∈1,2,0.5,ifϖ=3,0.1,ifϖ=4.
Takeκ∗=0.2 and κ=0.1. Then, T^ is an ∈,∈∨κ∗,qκ-FI of A but it is not an ∈,∈∨q-FI of A as 2ε=0.6=4∗2ε=0.6∈T^ and 2δ=0.6∈T^ but 4ε∧δ=0.6∈∨q¯T^.Definition 14.
A FST^ of A is said to be an ∈∨κ∗,qκ,∈∨κ∗,qκ-fuzzy ideal (briefly ∈∨κ∗,qκ,∈∨κ∗,qκ-FI) of A if:
(1)
ϖε∈∨κ∗,qκT^ implies 0ε∈∨κ∗,qκT^, and(2)
ϖ∗Θε∈∨κ∗,qκT^ and Θδ∈∨κ∗,qκT^ imply ϖε∧δ∈∨κ∗,qκT^
∀ϖ,Θ∈A and ε,δ∈0,1.Lemma 15.
InA, every ∈∨κ∗,qκ,∈∨κ∗,qκ-FI is ∈,∈∨κ∗,qκ-FI.Proof.
Suppose thatT^ is an ∈∨κ∗,qκ,∈∨κ∗,qκ-FI of A. Let ϖε∈T^ for ϖ∈A and ε∈0,1. Then, ϖε∈∨κ∗,qκT^. So, by hypothesis, 0ε∈∨κ∗,qκT^. Suppose that ϖ∗εε∈T^ and εδ∈T^. Then, ϖ∗εε∈∨κ∗,qκT^ and εδ∈∨κ∗,qκT^. Therefore, by hypothesis, ϖε∧δ∈∨κ∗,qκT^. Hence, T^ is an ∈,∈∨κ∗,qκ-FI of A.Remark 16.
In general, the converse of Lemma15 is not valid. This is illustrated by the following.Example 17.
From Example13, define T^:A⟶0,1 by
(9)T^ϖ=0.4,ifϖ=0,0.6,ifϖ∈1,3,0.1,ifϖ=2,4.
Take “κ=0” and “κ∗=0.7.” Then, T^ is an ∈,∈∨κ∗,qκ-FI of A but it is not an ∈∨κ∗,qκ,∈∨κ∗,qκ-FI of A as 2ε=0.95=2∗1ε=0.95∈∨κ∗,qκT^ and 1δ=0.5∈∨κ∗,qκT^ but 2ε∧δ=0.5∈∨κ∗,qκ¯T^.Lemma 18.
LetT^ be a FS of A. Then, ϖε∈T^implies0ε∈∨κ∗,qκT^⇔∀ϖ∈A,T^0≥T^ϖ∧κ∗−κ/2.Proof.
⇒ Contrary assume that, for some ϖ∈A, T^0<T^ϖ∧κ∗−κ/2. Take ε∈0,κ∗−κ/2 such that
(10)T^0<ε≤T^ϖ∧κ∗−κ2.
Then,ϖε∈T^, but 0ε∈∨κ∗,qκ¯T^, a contradiction. Hence, T^0≥T^ϖ∧κ∗−κ/2.
(⇐) Let ϖ∈A such that ϖε∈T^. Then, T^ϖ≥ε. So,
(11)T^0≥T^ϖ∧κ∗−κ2≥ε∧κ∗−κ2.
Now, ifε≤κ∗−κ/2, then T^0≥ε. Therefore, 0ε∈T^. Again, if ε>κ∗−κ/2, then T^0≥κ∗−κ/2. So, T^0+ε>κ∗−κ/2+κ∗−κ/2=κ∗−κ. This follows that 0εκ∗,qκT^. Hence, 0ε∈∨κ∗,qκT^.Lemma 19.
LetT^ be FS of A. Then, ϖ∗Θε∈T^ and Θδ∈T^ imply ϖε∧δ∈∨κ∗,qκT^⇔T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ/2.Proof.
⇒ On contrary suppose that T^ϖ<T^ϖ∗Θ∧T^Θ∧κ∗−κ/2 for some ϖ,Θ∈A. Choose ε∈0,κ∗−κ/2 such that T^ϖ<ε≤T^ϖ∗Θ∧T^Θ∧κ∗−κ/2. Then, ϖ∗Θε∈T^, Θε∈T^, but ϖε∈∨κ∗,qκ¯T^, which is not possible. Thus, we have shown that
(12)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ2.
(⇐) Let ϖ∗Θε∈T^ and Θδ∈T^, ∀ε,δ∈0,1. Then, T^ϖ∗Θ≥ε and T^Θ≥δ. Thus,
(13)T^ϖ≥T^ϖ∗Θ∧T^Θ,κ∗−κ2≥ε∧δ∧κ∗−κ2.
Now, ifε∧δ≤κ∗−κ/2, then T^ϖ≥ε∧δ, then ϖε∧δ∈T^; otherwise, i.e., when ε∧δ>κ∗−κ/2, then T^ϖ≥κ∗−κ/2. So, we have
(14)T^ϖ+ε∧δ>κ∗−κ2+κ∗−κ2=κ∗−κ.
This implies thatϖε∧δκ∗,qκT^. Hence, ϖε∧δ∈∨κ∗,qκT^, as required.Table 1∗012300123110322230133210Table 2∗012300003110032220333330Table 3∗01234000000110101222020331303444240We obtain the following theorem by combining Lemmas18 and 19.Theorem 20.
A FST^ of A is an ∈,∈∨κ∗,qκ-FI of A if and only if
(1)
T^0≥T^ϖ∧κ∗−κ/2(2)
T^ϖ≥T^ϖ∗Θ∧T^Θ,κ∗−κ/2,∀ϖ,Θ∈ALemma 21.
LetT^ be an ∈,∈∨κ∗,qκ-FI of A such that ϖ≤Θ. Then, T^ϖ≥T^Θ∧κ∗−κ/2.Proof.
Letϖ≤Θ for ϖ,Θ∈A. Then, ϖ∗Θ=0. By hypothesis, we have
(15)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ2=T^0∧T^Θ∧κ∗−κ2≥T^Θ∧κ∗−κ2.Lemma 22.
LetT^ be an ∈,∈∨κ∗,qκ-FI of A. Then, for any ϖ,Θ,ς∈A,
(16)ϖ∗Θ≤ς⇒T^ϖ≥T^Θ∧T^ς∧κ∗−κ2.Proof.
Letϖ∗Θ≤ς for ϖ,Θ,ς∈A. We have
(17)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ2≥T^ς∧κ∗−κ2∧T^Θ∧κ∗−κ2=T^Θ∧T^ς∧κ∗−κ2.Theorem 23.
Every∈,∈∨κ∗,qκ-FI of BCK-algebra A is an ∈,∈∨κ∗,qκ-FS of A.Proof.
LetT^ be an ∈,∈∨κ∗,qκ-FI of A and ϖ,Θ∈A. As ϖ∗Θ≤Θ in A, so by Lemma 21, we have
(18)T^ϖ∗Θ≥T^Θ∧κ∗−κ2.
SinceT^ is an ∈,∈∨κ∗,qκ-FI of A, we have
(19)T^ϖ∗Θ≥T^Θ∧κ∗−κ2≥T^Θ∗ϖ∧T^ϖ∧κ∗−κ2∧κ∗−κ2≥T^ϖ∧T^Θ∧κ∗−κ2.
Hence,T^ is an ∈,∈∨κ∗,qκ-FS of A.Remark 24.
In general, the converse of Theorem23 is not valid.Example 25.
Take aBCK-algebra A=0,1,2,3 with operation ∗ which is described in Table 4.
Consider the∈,∈∨κ∗,qκ-FS T^ of A, where T^:A⟶0,1 is defined by
(20)T^ϖ=0.3,ifϖ=0,0.1,ifϖ∈1,2,0.2,ifϖ=3.
Take “κ∗=0.8” and “κ=0.1.” Then, T^ is not an “∈,∈∨κ∗,qκ”-FI of A as 0ε=0.3=1∗3ε=0.3∈T^ and 3δ=0.2∈T^ but 1ε∧δ=0.2∈∨κ∗,qκ¯T^.Theorem 26.
LetT^ be an ∈,∈∨κ∗,qκ-FS of A. Then, T^ is an ∈,∈∨κ∗,qκ-FI ⇔∀ϖ,Θ,ς∈A such that ϖ∗Θ≤ς implies T^ϖ≥T^Θ∧T^ς∧κ∗−κ/2.Proof.
⇒ Follows from Lemma 22.
(⇐) Let T^ be an ∈,∈∨κ∗,qκ-FS such that ∀ϖ,Θ,ς∈A with ϖ∗Θ≤ς implies T^ϖ≥T^Θ∧T^ς∧κ∗−κ/2. As ϖ∗ϖ∗Θ≤Θ, so by hypothesis
(21)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ/2.
Hence,T^ is an ∈,∈∨κ∗,qκ-FI of A.Theorem 27.
A FST^ is an ∈,∈∨κ∗,qκ-FI of A⇔ the set T^ε≠∅ is an ideal of A, ∀ε∈0,κ∗−κ/2.Proof.
⇒ Let ε∈0,κ∗−κ/2 such that T^ε≠∅. By Theorem 20, we have
(22)T^0≥T^ϖ∧κ∗−κ2,with ϖ∈T^ε. It follows that T^0≥ε∧κ∗−κ/2=ε. Therefore, 0∈T^ε.
Next, suppose thatϖ∗Θ∈T^ε and Θ∈T^ε. Then, T^ϖ∗Θ≥ε and T^Θ≥ε. Again, by Theorem 20, we have
(23)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ2≥ε,κ∗−κ2=ε.
Therefore,ϖ∈T^ε. Hence, T^ε is an ideal of A.
(⇐) Suppose that T^ε is an ideal of A, ∀ε∈0,κ∗−κ/2. If T^0<T^ϖ∧κ∗−κ/2 for some ϖ∈A. Then, ∃ε∈0,κ∗−κ/2 such that T^0<ε≤T^ϖ∧κ∗−κ/2. It follows that ϖ∈T^ε but 0∈¯T^ε, a contradiction. Therefore, T^0≥T^ϖ∧κ∗−κ/2. Also, if T^ϖ<T^ϖ∗Θ∧T^Θ∧κ∗−κ/2 for some ϖ,Θ∈A. Then, ∃ε∈0,κ∗−κ/2 such that
(24)T^ϖ<ε≤T^ϖ∗Θ∧T^Θ∧κ∗−κ2.
It implies thatϖ∗Θ∈T^ε and Θ∈T^ε but ϖ∈T^ε, a contradiction. Therefore, T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ/2. Hence, T^ is an ∈,∈∨κ∗,qκ-FI of A.Definition 28.
LetT^ be an FS of A. The set
(25)T^ε=ϖ∈Aϖε∈∨κ∗,qκT^,whereε∈0,1,is said to be an ∈∨κ∗,qκ-level subset of T^.Theorem 29.
LetT^ be an FS of A. Then T^ is an ∈,∈∨κ∗,qκ-FI of A⇔ the ∈∨κ∗,qκ-level subset T^ε of T^ is an ideal of A, ∀ε∈0,1.Proof.
⇒ Suppose that T^ is an ∈,∈∨κ∗,qk~-FI of A. Take any ϖ∈T^ε. Then, ϖε∈∨κ∗,qk~T^. So, T^ϖ≥ε or T^ϖ+ε>κ∗−κ. Now, by Theorem 20, we have T^0≥T^ϖ∧κ∗−κ/2. Thus, T^0≥ε∧κ∗−κ/2 when T^ϖ≥ε. If ε>κ∗−κ/2, then T^0≥κ∗−κ/2 implies 0∈T^ε. Also, if ε≤κ∗−κ/2, then T^0≥ε implies 0∈T^ε. Similarly, 0∈T^ε when T^ϖ+ε>κ∗−κ.
Next, take anyϖ∗Θ∈T^ε and Θ∈T^ε. Then, ϖ∗Θε∈∨κ∗,qκT^ and Θε∈∨κ∗,qκT^, i.e, either T^ϖ∗Θ≥ε or T^ϖ∗Θ+ε>κ∗−κ and either T^Θ≥ε or T^Θ+ε>κ∗−κ. By assumption, T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ/2. Thus, we have
Case 1. LetT^ϖ∗Θ≥ε and T^Θ≥ε. If ε>κ∗−κ/2, then
(26)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ2≥ε∧κ∗−κ2=κ∗−κ2,and so, ϖεκ∗,qk~T^. If ε≤κ∗−κ/2, then
(27)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ2≥ε∧κ∗−κ2=ε.
Soϖε∈T^. Hence, ϖε∈∨κ∗,qk~T^.
Case 2. LetT^ϖ∗Θ≥ε and T^Θ+ε≥κ∗−κ. If ε>κ∗−κ/2, then
(28)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ2≥ε∧κ∗−κ−ε∧κ∗−κ2=κ∗−κ−ε,i.e., T^ϖ+ε>κ∗−κ and, thus, ϖεκ∗,qk~T^. If ε≤κ∗−κ/2, then
(29)T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ2≥ε∧κ∗−κ−ε∧κ∗−κ2=ε,and so ϖε∈T^. Hence, ϖε∈∨κ∗,qk~T^.
Similarly, for other cases, i.e., whenT^ϖ∗Θ+ε>κ∗−κ, T^Θ≥ε and T^ϖ∗Θ+ε>κ∗−κ, T^Θ+ε>κ∗−κ, we have ϖε∈∨κ∗,qk~T^. Therefore, for each case, ϖε∈∨κ∗,qk~T^, and hence ϖ∈T^ε.
(⇐) Let T^ε be an ideal of A, ∀ε∈0,1. On contrary, let
(30)T^0<T^ϖ∧κ∗−κ2,with ϖ∈A. Then, ∃ε∈0,1 such that T^0<ε≤T^ϖ∧κ∗−κ/2. It implies that ϖ∈T^ε, but 0∈¯T^ε, which is impossible. Therefore,
(31)T^0≥T^ϖ∧κ∗−κ2.
Also, ifT^ϖ<T^ϖ∗Θ∧T^Θ∧κ∗−κ/2 for some ϖ,Θ∈A. Then, ∃ε∈0,1 such that
(32)T^ϖ<ε≤T^ϖ∗Θ∧T^Θ∧κ∗−κ2.
It follows thatϖ∗Θ∈T^ε and Θ∈T^ε, but ϖ∈¯T^ε, a contradiction. Therefore, T^ϖ≥T^ϖ∗Θ∧T^Θ∧κ∗−κ/2. Hence, T^ is an ∈,∈∨κ∗,qκ-FI of A.Table 4∗012300000110102220033330


## 4.∈,∈∨κ∗,qκ-Fuzzy Commutative Ideals

Throughout this section,A will stand for a BCK-algebra.Definition 30.
LetA be a BCK-algebra. An FS T^ is said to be an ∈,∈∨κ∗,qκ-fuzzy commutative ideal (briefly, ∈,∈∨κ∗,qκ-FCI) if
(1)
ϖε∈T^ implies 0ε∈∨κ∗,qκT^, and(2)
ϖ∗Θ∗ςε∈T^ and ςδ∈T^ imply ϖ∗Θ∗Θ∗ϖε∧δ∈∨κ∗,qκT^
∀ϖ,Θ,ς∈A and ε,δ∈0,1.Example 31.
Take aBCK-algebra A=0,1,2,3 with operation ∗ which is described in Table 5.
DefineT^:A⟶0,1 by
(33)T^ϖ=0.5,ifϖ=0,0.3,ifϖ∈1,2,0.1,ifϖ=3.
Take “κ∗=0.2” and “κ=0.1.” Then, it is easy to get, T^ is an ∈,∈∨κ∗,qκ-FCI of A.Theorem 32.
A FST^ of a BCK-algebra A is an ∈,∈∨κ∗,qκ-FCI of A⇔(1)
T^0≥T^ϖ∧κ∗−κ/2, and(2)
T^ϖ∗Θ∗Θ∗ϖ≥T^ϖ∗Θ∗ς∧T^ς∧κ∗−κ/2,∀ϖ,Θ,ς∈AProof.
⇒ Lemma 18 gives us condition (2). To prove that (3) holds in A, assume that (3) does not hold in A, so we have
(34)T^ϖ∗Θ∗Θ∗ϖ<T^ϖ∗Θ∗ς∧T^ς∧κ∗−κ2,for some ϖ,Θ,p∈A. Choose ε∈0,1 such that T^ϖ∗Θ∗Θ∗ϖ<ε≤T^ϖ∗Θ∗ς∧T^ς∧κ∗−κ/2. Then, ϖ∗Θ∗ςε∈T^ and pε∈T^, but ϖ∗Θ∗Θ∗ϖε∈∨κ∗,qκ¯T^, which is not possible. Thus,
(35)T^ϖ∗Θ∗Θ∗ϖ≥T^ϖ∗Θ∗ς∧T^ς∧κ∗−κ2,∀ϖ,Θ,ς∈A
(⇐) Assume that conditions (2) and (3) hold in A. From condition (2) and Lemma 18, ϖε∈T^ implies 0ε∈∨κ∗,qκT^. Let ϖ∗Θ∗ςε∈T^ and pδ∈T^, ∀ε,δ∈0,1. Then, T^ϖ∗Θ∗ς≥ε and T^ς≥δ. Thus,
(36)T^ϖ∗Θ∗Θ∗ϖ≥T^ϖ∗Θ∗ς∧j∧κ∗−κ2≥ε∧δ∧κ∗−κ2.
Now, ifε∧δ≤κ∗−κ/2, then T^ϖ∗Θ∗Θ∗ϖ≥ε∧δ implies ϖ∗Θ∗Θ∗ϖε∧δ∈T^; otherwise, i.e., when ε∧δ>κ∗−κ/2, then T^ϖ∗Θ∗Θ∗ϖ≥κ∗−κ/2. So, we have
(37)T^ϖ∗Θ∗Θ∗ϖ+ε∧δ>κ∗−κ2+κ∗−κ2=κ∗−κ.
This implies thatϖ∗Θ∗Θ∗ϖε∧δκ∗,qκT^. Therefore, ϖ∗Θ∗Θ∗ϖε∧δ∈∨κ∗,qκT^. Hence, T^ is an ∈,∈∨κ∗,qκ-FCI of A.Theorem 33.
Every∈,∈∨κ∗,qκ-FCI of BCK-algebra A is an ∈,∈∨κ∗,qκ-FI of A.Proof.
LetT^ be an ∈,∈∨κ∗,qκ-FCI of A and ϖ,Θ∈A. So, we have
(38)T^ϖ=T^ϖ∗0∗0∗ϖ≥T^ϖ∗0∗Θ∧T^Θ∧κ∗−κ2=T^ϖ∗Θ∧T^Θ∧κ∗−κ2.
Hence,T^ is an ∈,∈∨κ∗,qκ-FI of A.Remark 34.
The converse of Theorem33 is not valid in general.Example 35.
Take aBCK-algebra A=0,1,2,3,4 with operation ∗ which is described in Table 6.
(39)T^ϖ=0.5,ifϖ=0,0.4,ifϖ=1,0,ifϖ∈2,3,4.
Takeκ∗=0.7 and κ=0.2. Then, T^ is an ∈,∈∨κ∗,qκ-FI of A but it is not an ∈,∈∨κ∗,qκ-FCI of A as 0ε=0.2=2∗3∗3∗2ε=0.2∈T^ and 0δ=0.3∈T^ but 2∗3∗3∗2ε∧δ=0.2=2ε=0.2∈∨κ∗,qκ¯T^.Theorem 36.
LetT^ be an ∈,∈∨κ∗,qκ-FI of BCK-algebra A. Then, T^ is an ∈,∈∨κ∗,qκ-FCI of A⇔∀ϖ,Θ∈A,
(40)T^ϖ∗Θ∗Θ∗ϖ≥T^ϖ∗Θ∧κ∗−κ2.Proof.
⇒ Let T^ be an ∈,∈∨κ∗,qκ-FCI of A. Then, ∀ς,Θ,ϖ∈A, we have
(41)T^ϖ∗Θ∗Θ∗ϖ≥T^ϖ∗Θ∗ς∧T^ς∧κ∗−κ2.
By takingς=0, we have
(42)T^ϖ∗Θ∗Θ∗ϖ≥T^ϖ∗Θ∗0∧T^0∧κ∗−κ2=T^ϖ∗Θ∧κ∗−κ2.
(⇐) Assume that equation (40) holds in A. Let ς,ϖ,Θ∈A. As T^ is an ∈,∈∨κ∗,qκ-FI of A, so we have
(43)T^ϖ∗Θ≥T^ϖ∗Θ∗ς∧T^ς∧κ∗−κ2.
By assumption and (43), we have
(44)T^ϖ∗Θ∗Θ∗ϖ≥T^ϖ∗Θ∧κ∗−κ2≥T^ϖ∗Θ∗ς∧T^ς∧κ∗−κ2.
Hence,T^ is an ∈,∈∨κ∗,qκ-FCI of A.Table 5∗012300000110012210233330Table 6∗01234000000110100222000333300444320


## 5. Conclusion

The main purpose of the present paper is to introduce the concepts of∈,∈∨κ∗,qκ-fuzzy subalgebras and ∈,∈∨κ∗,qκ-fuzzy ideals in BCK/BCI-algebras. We provided some equivalent conditions and different characterizations of the ∈,∈∨κ∗,qκ-fuzzy ideals in terms of level subsets and ∈∨κ∗,qκ-level subsets of BCK/BCI-algebras. It has been shown that in any BCK/BCI-algebras the ∈,∈∨κ∗,qκ-fuzzy ideals are ∈,∈∨κ∗,qκ-fuzzy subalgebras but the converse does not hold and an example provided in this aim. Furthermore, ∈,∈∨κ∗,qκ-fuzzy commutative ideals in BCK-algebras is introduced and some related properties of ∈,∈∨κ∗,qκ-fuzzy ideals and ∈,∈∨κ∗,qκ-fuzzy commutative ideals are considered. We hope that this work will provide a deep impact on the upcoming research in this field and other fuzzy algebraic studies to open up new horizons of interest and innovations. In future study, these notions may be extended to different algebras such as rings, hemirings, LA-semigroups, semihypergroups, semihyperrings, BCK/BCI-algebras, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, and EQ-algebras. Some important issues for future work are (1) to develop strategies for obtaining more valuable results and (2) to apply these notions and results for studying related notions in other algebraic (fuzzy) structures.



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