# Generalizations of Fuzzy q-Ideals ofBCI-Algebras

Authors: G. Muhiuddin; D. Al-Kadi; A. Mahboob; A. Assiry; Abdullah Alsubhi
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In this paper, we introduce the notion of∈,∈∨κ∗,qκ-fuzzy q-ideals of BCI-algebras to propose a more general form of fuzzy q-ideals of BCI-algebras. We prove that ∈,∈∨q-fuzzy q-ideals and ∈∨κ∗,qκ,∈∨κ∗,qκ-fuzzy q-ideals are ∈,∈∨κ∗,qκ-fuzzy q-ideals, but the converse assertion is not valid and examples are given to support this. It is proved that every ∈,∈∨κ∗,qκ-fuzzy q-ideal is an ∈,∈∨κ∗,qκ-fuzzy ideal, but the converse need not be true in general and an example is provided. In addition, correspondence between ∈,∈∨κ∗,qκ-fuzzy q-ideals and q-ideals of BCI-algebras is considered.


---

## Body

## 1. Introduction

A fuzzy set, as defined by Zadeh [1], is a powerful methodology for dealing with possibilistic complexity related to expectations, state imprecision, and preferences. Fuzzy set theory has become an essential study subject in research disciplines such as operation research, statistics, graph theory, social science, management, medical science, computer science, machine learning, multicriteria decision-making, information processing, and optimization. Imai and Iseki proposed the notions of BCK- and BCI-algebras in 1966 [2, 3]. Since then, a large number of studies have been published concerning the theory of BCK/BCI-algebras. Many authors, especially Liu et al. [4], Khalid and Ahmad [5], Jun et al. [6–8], Muhiuddin et al. [9, 10], and Al-Masarwah and Ahmad [11], studied different aspects of BCK/BCI-algebras based on ideal theory.The idea of quasi-coincidence of a fuzzy point with a fuzzy set, as stated in [12], was fundamental in the development of various types of fuzzy subgroups, known as α,β-fuzzy subgroups, as defined by Bhakat and Das in [13]. The concepts of α,β-fuzzy subalgebras and α,β-fuzzy ideals in BCK/BCI-algebras are also important and useful generalizations of fuzzy subalgebras and fuzzy ideals, which were introduced and studied by Jun [14, 15]. Zhang et al. [16] introduced the concepts of ∈,∈∨q-fuzzy p-ideals, ∈,∈∨q-fuzzy q-ideals, and ∈,∈∨q-fuzzy a-ideals in BCI-algebras by using the idea of a quasi-coincidence of a fuzzy point with a fuzzy set in the ideal theory of BCI-algebras. Ma et al. [17] proposed and investigated the concepts of (positive implicative, implicative, and commutative) ∈,∈∨q-interval-valued fuzzy ideals of BCI-algebras. Ma et al. [18] also proposed the concepts of ∈,∈∨q-interval-valued fuzzy (p and q) a-ideals of BCI-algebras. Al-Masarwah et al. [19] proposed a new system of m-polar α,β-fuzzy ideals and m-polar α,β-fuzzy commutative ideals in BCK/BCI-algebras by extending the concept of fuzzy point to m-polar fuzzy sets. Muhiuddin et al. [20] established the concept of m-polar ∈,∈-fuzzy q-ideals in BCI-algebras and explored the characteristics of m-polar α,β-fuzzy q-ideals and m-polar α,β-fuzzy ideals/subalgebras. Many researchers have also extended the fuzzy set theory and related concepts to different algebras and other structures (see, for e.g., [10, 21–27]).It is obvious to provide a generalized version of the existing fuzzy ideals ofBCI-algebras. To do this, we first review some fundamental concepts from the sequel in Section 2. The notions of ∈,∈∨κ∗,qκ-fuzzy q-ideals and ∈∨κ∗,qκ,∈∨κ∗,qκ-fuzzy q-ideals are then introduced and associated properties are investigated in Section 3. Moreover, correspondence between ∈,∈∨κ∗,qκ-fuzzy q-ideals and q-ideals of BCI-algebras is presented.


## 2. Preliminaries

An algebraA˜=A˜;∗,0 of type 2,0 is a BCI-algebra if(1)
ϖ∗Θ∗ϖ∗ϱ∗ϱ∗Θ=0(2)
ϖ∗ϖ∗Θ∗Θ=0(3)
ϖ∗ϖ=0(4)
ϖ∗Θ=0 and Θ∗ϖ=0⇒ϖ=Θ∀ϖ,Θ,ϱ∈A˜AnyBCI-algebra A˜ satisfies the following:(1)
ϖ∗0=ϖ(2)
ϖ∗Θ∗ϱ=ϖ∗ϱ∗ΘDefine an order≤ on A˜ as ϖ≤Θ⇔ϖ∗Θ=0.LetA˜ be a BCI-algebra. A mapping T:A˜⟶0,1 is called a fuzzy subset (briefly, FS) of A˜.Definition 1.
Leta∈A˜ and δ∈0,1. An ordered fuzzy point (briefly, OFP) aδ of A˜ is defined as(1)aδϖ=δ,ifϖ∈a,0,ifϖ∉a,∀ϖ∈A˜.
Consequently,aδ is a FS of A˜. For a FS T of A˜, we write aδ⊆T as aδ∈T in the sequel. So, aδ∈T⇔Ta≥δ.Definition 2.
AFST of A˜ is called an ∈,∈∨κ∗,qκ-fuzzy ideal (briefly, ∈,∈∨κ∗,qκ-FI) of A˜ if ϱδ∈T and ϖϵ∈T imply ϱ∗ϖδ∧ϵ∈∨κ∗,qκT for all δ,ε∈0,1 and ϱ,ϖ∈A˜.Lemma 1.
LetT be a FS of A˜. Then, ϱδ∈Timplies0δ∈∨κ∗,qκT⇔∀ϱ∈A˜,T0≥Tϱ∧κ∗−κ/2.Lemma 2.
LetT be an ∈,∈∨κ∗,qκ-fuzzy ideal of A˜ such that ϱ≤ϖ. Then, Tϱ≥Tϖ∧κ∗−κ/2.Lemma 3.
LetT be an ∈,∈∨κ∗,qκ-fuzzy ideal of A˜. Then, for any ϱ,ϖ,Θ∈A˜, ϱ∗ϖ≤Θ⇒Tϱ≥Tϖ∧TΘ∧κ∗−κ/2.


## 3. (∈, ∈∨(κ∗, qκ))-Fuzzy q-Ideals

Definition 3.
Letaδ be an OFP of A˜ and κ∗∈0,1. Then, aδ is said to be κ∗,q-quasi-coincident with a FS T of A˜, written as aδκ∗,qT, if(2)Ta+δ>κ∗.Let0≤κ<κ∗≤1. For an OFP ϱδ, we write(1)
ϱδκ∗,qκT if Tϱ+δ+κ>κ∗(2)
ϱδ∈∨κ∗,qκT if ϱδ∈T or ϱδκ∗,qκT(3)
ϱδα¯T if ϱδαT does not hold for α∈κ∗,qκ,∈∨κ∗,qκDefinition 4.
AFST of A˜ is called an ∈,∈∨κ∗,qκ-fuzzy q-ideal (briefly, ∈,∈∨κ∗,qκ-FQI) of A˜ if(1)
ϱδ∈T implies 0δ∈∨κ∗,qκT(2)
ϖ∗Θ∗ϱδ∈T and Θϵ∈T imply ϖ∗ϱδ∧ε∈∨κ∗,qκT∀ϖ,Θ,ϱ∈A˜ and δ,ε∈0,1.Example 1.
ConsiderA˜=0,1,j,ϱ,Θ as a BCI-algebra under the operation ∗ which is defined in Table 1.
Define aFST on A˜ as(3)Tϖ=0.6ifϖ=0,0.1ifϖ∈1,Θ,0.3ifϖ=j,0.2ifϖ=ϱ.
Chooseκ∗=0.8 and κ=0.4. It is straightforward to show that T is an ∈,∈∨κ∗,qκ-FQI of A˜.Table 1
Cayley table of the binary operation∗.
∗01jϱΘ0000ϱϱ1101Θϱjjj0ϱϱϱϱϱϱ00ΘΘϱΘ10Definition 5.
AFST of A˜ is called an ∈,∈∨q-fuzzy q-ideal (briefly, ∈,∈∨q-FQI) of A˜ if(1)
ϱδ∈T implies 0δ∈∨qT(2)
ϖ∗Θ∗ϱδ∈T and Θϵ∈T imply ϖ∗ϱδ∧ε∈∨qT,∀ϖ,Θ,ϱ∈A˜ and δ,ε∈0,1.Example 2.
Take aBCI-algebra A˜=0,1,j,ϖ with operation ∗ which is described in Table 2.
Define a FST on A˜ as(4)Tϑ=0.6ifϑ=0,0.1ifϑ∈1,j.
It is straightforward to check thatT is an ∈,∈∨q-FQI of A˜.Table 2
Cayley table of the binary operation∗.
∗01j000j110jjjj0Lemma 4.
InA˜, every ∈,∈∨q-FQI is ∈,∈∨κ∗,qκ-FQI.Proof.
LetT be an ∈,∈∨q-FQI of A˜. Take any ϱδ∈T for ϱ∈A˜ and δ∈0,1. Then, by hypothesis, 0δ∈∨qT. It implies that T0≥δ or T0+δ≥1, and thus, T0≥δ or T0+κ+δ>κ∗. Therefore, 0δ∈∨κ∗,qκT. Next, take any ϖ∗Θ∗ϱδ∈T and Θϵ∈T. So, ϖ∗ϱδ∧ϵ∈∨qT implies Tϖ∗ϱ≥δ∧ε or Tϖ∗ϱ+δ∧ε>1. Therefore, Tϖ∗ϱ≥δ∧ε or Tϖ∗ϱ+κ+δ∧ε>κ∗. Thus, ϖ∗ϱδ∧ϵ∈∨κ∗,qκT. Hence, T is an ∈,∈∨κ∗,qκ-FQI of A˜.In general, the converse of Lemma4 is not valid, as illustrated by the following example.Example 3.
Take aBCI-algebra A˜=0,1,j,ϱ with the operation ∗ described in Table 3.
Define aFST on A˜ as(5)Tϖ=0.7ifϖ=0,0.6ifϖ=1,0.4ifϖ∈j,ϱ.
Chooseκ∗=0.82 and κ=0.02. It is straightforward to show that T is an ∈,∈∨κ∗,qκ-FQI of A˜ but not an ∈,∈∨q-FQI because ϱ∗0∗1δ=0.5∈T and 0ε=0.5∈T but ϱ∗1δ∧ε=0.5∈∨q¯T.Table 3
Cayley table of the binary operation∗.
∗01jϱ00ϱj1110ϱjjj10ϱϱϱj10Definition 6.
AFST of A˜ is called an ∈∨κ∗,qκ,∈∨κ∗,qκ-fuzzy q-ideal (briefly, ∈∨κ∗,qκ,∈∨κ∗,qκ-FQI) of A˜ if(1)
ϱδ∈∨κ∗,qκT implies 0δ∈∨κ∗,qκT(2)
ϖ∗Θ∗ϱδ∈∨κ∗,qκT and Θϵ∈∨κ∗,qκT imply ϖ∗ϱδ∧ε∈∨κ∗,qκT∀ϖ,Θ,ϱ∈A˜ and δ,ε∈0,1Example 4.
Take aBCI-algebra A˜=0,1,j,ϱ with operation ∗ which is described in Table 4.
DefineT:A˜⟶0,1 by(6)Tϖ=0.9,ifϖ=0,0.3,ifϖ∈1,j,ϱ.
Considerκ∗=0.2 and κ=0.1. It is easy to check that T is an ∈,∈∨κ∗,qκ-FI of A˜.Table 4
Cayley table of the binary operation∗.
∗01jϱ0000ϱ1100ϱjjj0ϱϱϱϱϱ0Lemma 5.
InA˜, every ∈∨κ∗,qκ,∈∨κ∗,qκ-FQI is ∈,∈∨κ∗,qκ-FQI.Proof.
LetT be any ∈∨κ∗,qκ,∈∨κ∗,qκ-FQI of A˜. Take any ϱδ∈T for ϱ∈A˜ and δ∈0,1. Then, ϱδ∈∨κ∗,qκT. So, by hypothesis, 0δ∈∨κ∗,qκT. Suppose that ϖ∗Θ∗ϱδ∈T and Θϵ∈T. Then, ϖ∗Θ∗ϱδ∈∨κ∗,qκT and Θϵ∈∨κ∗,qκT. Therefore, by hypothesis, ϖ∗ϱδ∧ε∈∨κ∗,qκT. Hence, T is an ∈,∈∨κ∗,qκ-FQI of A˜.In general, the converse of Lemma5 is not valid, as illustrated in the following example.Example 5.
Take aBCI-algebra A˜=0,1,j,ϱ,Θ with the operation ∗ described in Table 5.
DefineT:A˜⟶0,1 by(7)Tϖ=0.4,ifϖ=0,0.6,ifϖ∈1,ϱ,0.1,ifϖ=j,Θ.
Considerκ=0 and κ∗=0.7. Then, T is an ∈,∈∨κ∗,qκ-FQI of A˜, but it is not an ∈∨κ∗,qκ,∈∨κ∗,qκ-FQI of A˜ as jδ=0.95=j∗0∗1δ=0.95∈∨κ∗,qκT and 0ε=0.5∈∨κ∗,qκT, but j=j∗1δ∧ε=0.5∈∨κ∗,qκ¯T.Table 5
Cayley table of the binary operation∗.
∗01jϱΘ000000110101jjj0j0ϱϱ1ϱ0ϱΘΘΘjΘ0Lemma 6.
LetT be a FS of A˜. Then, ϖ∗Θ∗ϱδ∈T and Θϵ∈T imply ϖ∗ϱδ∧ε∈∨κ∗,qκT⇔Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2.Proof.
⇒ On the contrary, suppose that Tϱ<Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2 for some ϖ,Θ,ϱ∈A˜. Choose δ∈0,κ∗−κ/2 such that Tϱ<δ≤Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2. Then, ϖ∗Θ∗ϱδ∈T and Θδ∈T, but ϱδ∈∨κ∗,qκ¯T, which is not possible. Thus, Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2.
⇐ Let ϖ∗Θ∗ϱδ∈T and Θε∈T, ∀δ,ε∈0,1. Then, Tϖ∗Θ∗ϱ≥δ and TΘ≥ε. Thus,(8)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2≥δ∧ε∧κ∗−κ2.
Now, ifδ∧ε≤κ∗−κ/2, then Tϖ∗ϱ≥δ∧ε; hence, ϖ∗ϱδ∧ε∈T; otherwise, i.e., when δ∧ε>κ∗−κ/2, Tϖ∗ϱ≥κ∗−κ/2. So, we have(9)Tϖ∗ϱ+δ∧ε>κ∗−κ2+κ∗−κ2=κ∗−κ.
This implies thatϖ∗ϱδ∧εκ∗,qκT. Hence, ϖ∗ϱδ∧ε∈∨κ∗,qκT, as required.By combining Lemma1 and Lemma 6, we get the following theorem.Theorem 1.
A FST of A˜ is an ∈,∈∨κ∗,qκ-FQI of A˜⇔(1)
T0≥Tϱ∧κ∗−κ/2(2)
Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2, ∀ϖ,Θ,ϱ∈A˜Theorem 2.
Every∈,∈∨κ∗,qκ-FQI of A˜ is an ∈,∈∨κ∗,qκ-FI.Proof.
LetT be an ∈,∈∨κ∗,qκ-FQI of A˜. Then, ∀ϖ,Θ,ϱ∈A˜, we have(10)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2.
Substituteϱ by 0, to obtain(11)Tϖ∗0≥Tϖ∗Θ∗0∧TΘ∧κ∗−κ2.
Thus,Tϖ≥Tϖ∗Θ∧TΘ∧κ∗−κ/2, as required.In general, the converse of Theorem2 is not valid, as illustrated in the following example.Example 6.
Take aBCI-algebra A˜=0,1,j,ϱ,Θ with the operation ∗ described in Table 6.
Define aFST on A˜ as(12)Tϖ=0.4ifϖ=0,0.2ifϖ=1,0ifϖ∈j,ϱ,Θ.
Chooseκ=0.1 and κ∗=0.9. It is straightforward to check that T is an ∈,∈∨κ∗,qκ-FI of A˜, but it is not an ∈,∈∨κ∗,qκ-FQI because 0=TΘ∗j≱TΘ∗0∗j∧T0∧0.4=T0=0.4.Table 6
Cayley table of the binary operation∗.
∗01jϱΘ000Θϱj110Θϱjjjj0Θϱϱϱϱj0ΘΘΘΘϱj0Theorem 3.
IfT is an ∈,∈∨κ∗,qκ-FI of A˜, then the following statements are equivalent:(1)
T is an ∈,∈∨κ∗,qκ-FQI of A˜(2)
Tϖ∗Θ≥Tϖ∗0∗Θ∧κ∗−κ/2, ∀ϖ,Θ∈A˜(3)
Tϖ∗Θ∗ϱ≥Tϖ∗Θ∗ϱ∧κ∗−κ/2, ∀ϖ,Θ,ϱ∈A˜Proof.
1⇒2. Let T be an ∈,∈∨κ∗,qκ-FQI of A˜. Then, ∀ϖ,Θ,ϱ∈A˜, we have(13)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2.
Replacingϱ by Θ and Θ by ϱ, we get(14)Tϖ∗Θ≥Tϖ∗ϱ∗Θ∧Tϱ∧κ∗−κ2.
Substituteϱ by 0, to obtain(15)Tϖ∗Θ≥Tϖ∗0∗Θ∧T0=Tϖ∗0∗Θ∧κ∗−κ2.
2⇒3. Let ϖ,Θ,ϱ∈A˜. Then,(16)Tϖ∗Θ∗ϱ≥Tϖ∗Θ∗0∗ϱ∧κ∗−κ2.
Now, we have(17)ϖ∗Θ∗0∗ϱ∗ϖ∗Θ∗ϱ=ϖ∗Θ∗ϖ∗Θ∗ϱ∗0∗ϱ≤Θ∗ϱ∗Θ∗0∗ϱ=Θ∗Θ∗ϱ∗0∗ϱ=0∗ϱ∗0∗ϱ=0.
By Lemma3, we have(18)Tϖ∗Θ∗0∗ϱ≥Tϖ∗Θ∗ϱ∧κ∗−κ2.
From (16) and (18), we obtain Tϖ∗Θ∗ϱ≥Tϖ∗Θ∗ϱ∧κ∗−κ/2.
3⇒1. As T is an ∈,∈∨κ∗,qκ-FI of A˜, ∀ϖ,Θ,ϱ∈A˜, we have(19)Tϖ∗ϱ≥Tϖ∗ϱ∗Θ∧TΘ∧κ∗−κ2≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2.Lemma 7.
LetT be an ∈,∈∨κ∗,qκ-FQI of A˜. Then, T0∗ϱ≥Tϱ∧κ∗−κ/2∀ϱ∈A˜.Proof.
Assume thatT is an ∈,∈∨κ∗,qκ-FQI of A˜. Then, ∀ϖ,Θ,ϱ∈A˜, we have(20)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2.
Substituteϖ by 0 and Θ by ϱ, and we have(21)T0∗ϱ≥T0∗ϱ∗ϱ∧Tϱ=Tϱ∧κ∗−κ2.Definition 7.
LetT be a FS of A˜. The set(22)Tδ=ϱ∈A˜|Tϱ≥δ,whereδ∈0,1,is called the level subset of T.Theorem 4.
LetT be a FS of A˜. Then, T is an ∈,∈∨κ∗,qκ-FQI of A˜⇔ the set Tδ≠∅ is a q-ideal of A˜, ∀δ∈0,κ∗−κ/2.Proof.
⇒ Let δ∈0,κ∗−κ/2 with Tδ≠∅. From Theorem 1, we have(23)T0≥Tϱ∧κ∗−κ2,with ϱ∈Tδ. It implies that T0≥δ∧κ∗−κ/2=δ. Therefore, 0∈Tδ.
Next, assume thatϖ∗Θ∗ϱ∈Tδ and Θ∈Tδ. Then, Tϖ∗Θ∗ϱ≥δ and TΘ≥δ. Again, by Theorem 1, we have(24)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2≥δ∧κ∗−κ2=δ.
Therefore,ϖ∗ϱ∈Tδ. Hence, Tδ is a q-ideal of A˜.
⇐ Assume that Tδ is q-ideal of A˜, ∀δ∈0,κ∗−k/2. If T0<Tϱ∧κ∗−κ/2 for some ϱ∈A˜, then ∃δ∈0,κ∗−k/2 such that T0<δ≤Tϱ∧κ∗−κ/2. It follows that ϱ∈Tδ but 0∉Tδ, a contradiction. Therefore, T0≥Tϱ∧κ∗−κ/2. Also, if Tϖ∗ϱ<Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2 for some ϖ,Θ,ϱ∈A˜, then ∃δ∈0,κ∗−k/2 such that(25)Tϖ∗ϱ<δ≤Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2.
It follows thatϖ∗Θ∗ϱ∈Tδ and Θ∈Tδ but ϖ∗ϱ∉Tδ, which is again a contradiction. Therefore, Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2. Hence, T is an ∈,∈∨κ∗,qκ-FQI of A˜.Definition 8.
LetT be a FS of A˜. The set(26)Tδ˜=ϱ∈A˜|ϱδ∈∨κ∗,qκT,whereδ∈0,1,is called an ∈∨κ∗,qκ-level subset of T.Theorem 5.
LetT be a FS of A˜. Then, T is an ∈,∈∨κ∗,qκ-FQI of A˜⇔ the ∈∨κ∗,qκ-level subset Tδ˜ of T is a q-ideal of A˜, ∀δ∈0,1.Proof.
⇒ Suppose T is an ∈,∈∨κ∗,qκ-FQI of A˜. Take any ϱ∈Tδ˜. Then, ϱδ∈∨κ∗,qκT. So, Tϱ≥δ or Tϱ+δ>κ∗−κ. Now, by Theorem 1, we have T0≥Tϱ∧κ∗−κ/2. Thus, T0≥δ∧κ∗−κ/2 when Tϱ≥δ. If δ>κ∗−κ/2, then T0≥κ∗−κ/2 implies 0∈Tδ˜. Also, if δ≤κ∗−κ/2, then T0≥δ implies 0∈Tδ˜. Similarly, 0∈Tδ˜ when Tϱ+δ>κ∗−κ.
Next, take anyϖ∗Θ∗ϱ∈Tδ˜ and Θ∈Tδ˜. Then, ϖ∗Θ∗ϱ∈∨κ∗,qκT and Θ∈∨κ∗,qκT, i.e., either Tϖ∗Θ∗ϱ≥δ or Tϖ∗Θ∗ϱ+δ>κ∗−κ and either TΘ≥δ or TΘ+δ>κ∗−κ. By assumption, Tϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2. Thus, the following cases arise.Case 1.
LetTϖ∗Θ∗ϱ≥δ and TΘ≥δ. If δ>κ∗−κ/2, then(27)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2≥δ∧κ∗−κ2=κ∗−κ2,and so, ϖ∗ϱδ∈κ∗,qκT. If δ≤κ∗−κ/2, then(28)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2≥δ∧κ∗−κ2=δ.
So,ϖ∗ϱδ∈T. Hence, ϱδ∈∨κ∗,qκT.Case 2.
LetTϖ∗Θ∗ϱ≥δ and TΘ+δ≥κ∗−κ. If δ>κ∗−κ/2, then(29)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ,κ∗−κ2≥δ∧κ∗−κ−δ∧κ∗−κ2=κ∗−κ−δ,i.e., Tϖ∗ϱ+δ>κ∗−κ, and thus, ϖ∗ϱδ∈κ∗,qκT. If δ≤κ∗−κ/2, then(30)Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2≥δ∧κ∗−κ−δ∧κ∗−κ2=δ,and so, ϖ∗ϱδ∈T. Hence, ϖ∗ϱδ∈∨κ∗,qκT.
Similarly, for other cases, i.e., whenTϖ∗Θ∗ϱ+δ>κ∗−κ, TΘ≥δ, Tϖ∗Θ∗ϱ+δ>κ∗−κ, and TΘ+δ>κ∗−κ, we have ϖ∗ϱδ∈∨κ∗,qκT. Hence, for each case, ϖ∗ϱδ∈∨κ∗,qκT, and thus, ϖ∗ϱ∈Tδ˜.
⇐ Let Tδ˜ be a q-ideal of A˜, ∀δ∈0,1. On the contrary, let(31)T0<Tϱ∧κ∗−κ2,for some ϱ∈A˜. Then, ∃δ∈0,1 such that T0<δ≤Tϱ∧κ∗−κ/2. It follows that ϱ∈Tδ˜, but 0∉Tδ˜, which is not possible. Therefore,(32)T0≥Tϱ∧κ∗−κ2.
Also, ifTϖ∗ϱ<Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2 for some ϱ,ϖ∈A˜, then ∃δ∈0,1 such that(33)Tϖ∗ϱ<δ≤Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ2.
Thus,ϖ∗Θ∗ϱ∈Tδ˜ and Θ∈Tδ˜, but ϖ∗ϱ∉Tδ˜, which is again a contradiction. Therefore, Tϖ∗ϱ≥Tϖ∗Θ∗ϱ∧TΘ∧κ∗−κ/2. Hence, T is an ∈,∈∨κ∗,qκ-FQI of A˜.


## 4. Conclusion

The main aim of the present paper is to introduce the concept of∈,∈∨κ∗,qκ-FQI in BCI-algebras. We provided some equivalent conditions and different characterizations of the ∈,∈∨κ∗,qκ-FQI in terms of level subsets and ∈∨κ∗,qκ-level subsets of BCI-algebras. It has been shown that in any BCI-algebras, the ∈,∈∨κ∗,qκ-FQI is ∈,∈∨κ∗,qκ-FI but the converse does not hold and an example is provided to support this. Furthermore, relation between ∈,∈∨κ∗,qκ-FQI and q-ideal of BCI-algebras has been considered. In this investigation, we get to the following conclusions:(1)
If we chooseκ∗=1 and k=0, then ∈,∈∨κ∗,qκ-FQI reduces to the notion ∈,∈∨q-FQI of X as established in [16](2)
If we chooseκ∗=1 and k=k, then ∈,∈∨κ∗,qκ-fuzzy subalgebras and ∈,∈∨κ∗,qκ-FQI reduce to the concepts ∈,∈∨qκ-fuzzy subalgebras and ∈,∈∨qκ-FQI of X as introduced in [28]Consequently, the notions introduced in this paper are more general than the existing notions. In future study, these ideas may be extended to other algebraic structures such as rings, hemirings,LA-semigroups, semi-hypergroups, semi-hyper-rings, BL-algebras, MTL-algebras, R0-algebras, EQ-algebras, MV-algebras, and lattice implication algebras.



---

*Source: 2388199-2022-05-06.xml*