# More General Form of Interval-Valued Fuzzy Ideals of BCK/BCI-Algebras

Authors: G. Muhiuddin; D. Al-Kadi; A. Mahboob
Journal: Security and Communication Networks
 (2021)

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## Abstract

The concepts of interval-valued∈,∈∨κ∗˜,qκ˜-fuzzy subalgebras, interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy ideals, and interval-valued ∈∨κ∗˜,qκ˜,∈∨κ∗˜,qκ˜-fuzzy ideals are introduced, and related properties are studied. Many examples are given in support of these new notions. Furthermore, interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy commutative ideals are defined, and some important properties are discussed. For a BCK-algebra X, it is proved that every interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy commutative ideal of BCK-algebra X is an interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy ideal of X, but the converse need not be true, in general, and then a counterexample is constructed.


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## Body

## 1. Introduction

As an extension of fuzzy sets, Zadeh defined fuzzy sets with an interval-valued membership function proposing the concept of interval-valued fuzzy sets. This concept has been studied from various points of view in different algebraic structures as BCK-algebras and some of its generalization (see, for example, [1–5]), groups (see for example, [6–10]), and rings (see, for example, [11–13]). Moreover, as novel approaches in decision-making, theoretical models were introduced based on (fuzzy) soft sets in [14–19]. In BCK/BCI-algebras and other related algebraic structures, different kinds of related concepts were investigated in various ways (see, for example, [20–33]). Jun [34] studied interval-valued fuzzy ideals in BCI-algebras. Zhan et al. [35, 36] studied ∈,∈∨q-fuzzy ideals of BCI-algebras. The concept of “quasi-coincidence” of an interval-valued fuzzy point together with “belongingness” within an interval-valued fuzzy set was used in the studies made by Ma et al. in [37, 38] where they discussed properties of some types of ∈,∈∨q-interval-valued fuzzy ideals of BCI-algebras.It is natural to introduce the general form of the existing interval-valued fuzzy ideals of BCK/BCI-algebras. For this purpose, we first recall in Section 2 some elementary notions used in the sequel. Then, in Section 3, we introduce the concepts: interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy subalgebras, interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy ideals, interval-valued ∈,∈∨q-fuzzy ideals, and interval-valued ∈∨κ∗˜,qκ˜,∈∨κ∗˜,qκ˜-fuzzy ideals, and related properties are studied. In Section 4, interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy commutative ideals are introduced, some properties are studied, and their relation with interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy ideals is investigated.


## 2. Preliminaries

An algebraX=X;∗,0 of type 2,0 is a BCI-algebra if for all υ,p,ℏ∈X,(1)
υ∗p∗υ∗ℏ∗ℏ∗p=0(2)
υ∗υ∗p∗p=0(3)
υ∗υ=0(4)
υ∗p=0 and p∗υ=0⇒υ=pIfX satisfies (1)–(4) and (5) 0∗υ=0, then X is a BCK-algebra.Any BCK/BCI-algebraX satisfies(1)
υ∗0=υ(2)
υ∗p∗ℏ=υ∗ℏ∗pFrom now on, letX be a BCK/BCI-algebra unless otherwise specified.We define a partially ordered setX,≤, where υ≤κ⇔υ∗p=0.A nonempty subsetP of X is said to be a subalgebra of X if ℏ∗υ∈P for all ℏ,υ∈P.A nonempty subsetI of X is said to be an ideal of X ifI10∈II2∀ℏ,υ∈I,ℏ∗υ∈I,andυ∈I⇒ℏ∈IBy an interval numbera˜, we mean an interval, denoted by a−,a+, where 0≤a−≤a+≤1. The set of all interval numbers is denoted by D0,1. In whatever follows, the interval a,a is identified by the number a∈0,1. For the interval numbers a˜i=ai−,ai+ and b˜i=bi−,bi+∈D0,1,i∈I, we define(1)mina˜i,b˜i=minai−,bi−,minai+,bi+,maxa˜i,b˜i=maxai−,bi−,maxai+,bi+,a˜1≤a˜2⇔a1−≤a2−anda1+≤a2+,a˜1=a˜2⇔a1−=a2−anda1+=a2+,κa˜=κa−,κa+,whenever0≤κ≤1,κ˜∗=κ∗−,κ∗+,whenever0<κ∗≤1.A mappingU˜:X⟶D0,1 is called an interval-valued fuzzy subset (briefly, IVFS) of X, where U˜x=U˜−x,U˜+x for all x∈X, U˜− and U˜+ are fuzzy sets of X with U˜−x≤U˜+x for all x∈X.


## 3. Interval-Valued∈,∈∨κ∗˜,qκ˜-Fuzzy Ideals

Definition 1.
Leta∈X and ℓ˜∈D0,1. An interval-valued ordered fuzzy point (briefly, IVOFP) aℓ˜ of X is defined as:(2)aℓ˜x=ℓ˜,ifx∈a,0,0,ifx∉a,for all x∈X.
Clearly,aℓ˜ is an IVFS of X. For any IVFS U˜ of X, we denote aℓ˜⊆U˜ as aℓ˜∈U˜ in the sequel. So, aℓ˜∈U˜⇔U˜a≥ℓ˜.Definition 2.
Letℏl˜ be an IVOFP of X and κ∗˜∈D0,1. Then, ℏl˜ is called κ∗˜,q-quasi-coincident with an IVFS U˜ of X, represented as ℏl˜κ∗˜,qU˜, if U˜ℏ+l˜>κ∗˜.
Assume0,0≤κ˜<κ∗˜≤1,1. For an IVOFP ℏℓ˜, we write(1)
ℏℓ˜κ∗˜,qκ˜U˜ if U˜ℏ+ℓ˜+κ˜>κ∗˜(2)
ℏℓ˜∈∨κ∗˜,qκ˜U˜ if ℏℓ˜∈U˜ or ℏℓ˜κ∗˜,qκ˜U˜(3)
ℏℓ˜α¯U˜ if ℏℓ˜αU˜ does not hold for α∈κ∗˜,qκ˜,∈∨κ∗˜,qκ˜Definition 3.
An IVFSU˜ of X is called an interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy subalgebra (in short, IV ∈,∈∨κ∗˜,qκ˜-FS) of X if ℏℓ˜∈U˜ and υj˜∈U˜ imply ℏ∗υminℓ˜,j˜∈∨κ∗˜,qκ˜U˜ for all ℓ˜,j˜∈D0,1 and ℏ,υ∈X.Theorem 1.
An IVFSU˜ of X is an IV ∈,∈∨κ∗˜,qκ˜-FS of X⇔(3)U˜ℏ∗υ≥minU˜ℏ,U˜υ,κ∗−−κ−2,κ∗+−κ+2,for all ℏ,υ∈X.Proof.
⇒ On the contrary, suppose that U˜ℏ∗υ<minU˜ℏ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2, for some ℏ,υ∈X. Choose ω∈D0,1 such that(4)U˜ℏ∗υ<ω≤minU˜ℏ,U˜υ,κ∗−−κ−2,κ∗+−κ+2.
Then,ℏω∈U˜ and υω∈U˜, but ℏ∗υω∈∨κ∗˜,qκ˜¯U˜, which is impossible. Hence, U˜ℏ∗υ≥minU˜ℏ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2.
⇐ Assume that U˜ℏ∗υ≥minU˜ℏ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2 for all ℏ,υ∈X. Let ℏℓ˜∈U˜ and υj˜∈U˜ for all ℓ˜,j˜∈D0,1. Then, U˜ℏ≥ℓ˜ and U˜υ≥j˜. So, U˜ℏ∗υ≥minU˜ℏ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2≥minℓ˜,j˜,κ∗−−κ−/2,κ∗+−κ+/2. If minℓ˜,j˜≤κ∗−−κ−/2,κ∗+−κ+/2, then U˜ℏ∗υ≥minℓ˜,j˜ implies that ℏ∗υminℓ˜,j˜∈U˜. If minℓ˜,j˜>κ∗−−κ−/2,κ∗+−κ+/2, then U˜ℏ∗υ≥κ∗−−κ−/2,κ∗+−κ+/2. So, U˜ℏ∗υ+minℓ˜,j˜>κ∗−−κ−/2,κ∗+−κ+/2+κ∗−−κ−/2,κ∗+−κ+/2=κ∗−−κ−,κ∗+−κ+ implies that ℏ∗υminℓ˜,j˜κ∗˜,qκ˜U˜. Hence, ℏ∗υminℓ˜,j˜∈∨κ∗˜,qκ˜U˜. Therefore, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FS of X.Definition 4.
An IVFSU˜ of X is called an interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy ideal (in short, IV ∈,∈∨κ∗˜,qκ˜-FI) of X if(1)
ℏℓ˜∈U˜ implies 0ℓ˜∈∨κ∗˜,qκ˜U˜(2)
ℏ∗υℓ˜∈U˜ and υj˜∈U˜ imply ℏminℓ˜,j˜∈∨κ∗˜,qκ˜U˜for all ℏ,υ∈X and ℓ˜,j˜∈D0,1.Example 1.
Consider a BCI-algebraX=0,1,2,3 with the binary operation ∗ as defined in Table 1.
DefineU˜:X⟶D0,1 by(5)U˜ℏ=0.9,1,ifℏ=0,0.3,0.4,ifℏ∈1,2,3.
Chooseκ∗˜=0.2,0.3 and κ˜=0.1,0.2. Then, with direct computation, we find that U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X.Table 1
Cayley Table of the binary operation∗.
∗012300003110032220333330Definition 5.
An IVFSU˜ of X is called an interval-valued ∈,∈∨q-fuzzy ideal (in short, IV ∈,∈∨q-FI) of X if(1)
ℏℓ˜∈U˜ implies 0ℓ˜∈∨qU˜(2)
ℏ∗υℓ˜∈U˜ and υj˜∈U˜ imply ℏminℓ˜,j˜∈∨qU˜for all ℏ,υ∈X and ℓ˜,j˜∈D0,1.Theorem 2.
InX, every IV ∈,∈∨q-FI is an IV ∈,∈∨κ∗˜,qκ˜-FI.Proof.
LetU˜ be any IV ∈,∈∨q-FI of X. Take any ℏℓ˜∈U˜ for ℏ∈X and ℓ˜∈D0,1. Then, by hypothesis, 0ℓ˜∈∨qU˜. It follows that U˜0≥ℓ˜ or U˜0+ℓ˜≥1,1, and so, U˜0≥ℓ˜ or U˜0+κ˜+ℓ˜≥κ∗˜. Therefore, 0ℓ˜∈∨κ∗˜,qκ˜U˜. Next, let ℏ∗υℓ˜∈U˜ and υj˜∈U˜. So, ℏminℓ˜,j˜∈∨qU˜ implies U˜ℏ≥minℓ˜,j˜ or U˜ℏ+minℓ˜,j˜>1,1. Therefore, U˜ℏ≥minℓ˜,j˜ or U˜ℏ+κ˜+minℓ˜,j˜>κ∗˜. Thus, ℏℓ˜∈∨κ∗˜,qκ˜U˜. Hence, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X.Example 2.
Consider aBCK-algebra X=0,1,2,3,4 with the binary operation ∗ as defined in Table 2.
DefineU˜:X⟶D0,1 by(6)U˜ℏ=0.4,0.5,ifℏ=0,0.2,0.3,ifℏ∈1,2,0.5,0.6,ifℏ=3,0.1,0.2,ifℏ=4.
Chooseκ∗˜=0.2,0.3 and κ˜=0.1,0.2. Then, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X but is not an IV ∈,∈∨q-FI of X as 2ℓ˜=0.6,0.6=4∗2ℓ˜=0.6,0.6∈U˜ and 2j˜=0.6,0.6∈U˜ but 4minℓ˜,j˜=0.6,0.6∈∨q¯U˜.Table 2
Cayley Table of the binary operation∗.
∗01234000000110101222020331303444240Definition 6.
An IVFSU˜ of X is called an interval-valued ∈∨κ∗˜,qκ˜,∈∨κ∗˜,qκ˜-fuzzy ideal (in short, IV ∈∨κ∗˜,qκ˜,∈∨κ∗˜,qκ˜-FI) of X if(1)
ℏℓ˜∈∨κ∗˜,qκ˜U˜ implies 0ℓ˜∈∨κ∗˜,qκ˜U˜(2)
ℏ∗υℓ˜∈∨κ∗˜,qκ˜U˜ and υj˜∈∨κ∗˜,qκ˜U˜ imply ℏminℓ˜,j˜∈∨κ∗˜,qκ˜U˜for all ℏ,υ∈X and ℓ˜,j˜∈D0,1.Theorem 3.
InX, every IV ∈∨κ∗˜,qκ˜,∈∨κ∗˜,qκ˜-FI is an IV ∈,∈∨κ∗˜,qκ˜-FI.Proof.
LetU˜ be any IV ∈∨κ∗˜,qκ˜,∈∨κ∗˜,qκ˜-FI of X. Take any ℏℓ˜∈U˜ for ℏ∈X and ℓ˜∈D0,1. Then, ℏℓ˜∈∨κ∗˜,qκ˜U˜. So, by hypothesis, 0ℓ˜∈∨κ∗˜,qκ˜U˜. Suppose that ℏ∗ℓℓ˜∈U˜ and ℓj˜∈U˜. Then, ℏ∗ℓℓ˜∈∨κ∗˜,qκ˜U˜ and ℓj˜∈∨κ∗˜,qκ˜U˜. Therefore, by hypothesis, ℏminℓ˜,j˜∈∨κ∗˜,qκ˜U˜. Hence, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X.Example 3.
Consider aBCK-algebra of Example 2. Define U˜:X⟶D0,1 by(7)U˜ℏ=0.4,0.5,ifℏ=0,0.6,0.7,ifℏ∈1,3,0.1,0.2,ifℏ=2,4.
Chooseκ˜=0,0 and κ∗˜=0.7,0.9. Then, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X but is not an IV ∈∨κ∗˜,qκ˜,∈∨κ∗˜,qκ˜-FI of X as 2ℓ˜=0.95,0.95=2∗1ℓ˜=0.95,0.95∈∨κ∗˜,qκ˜U˜ and 1j˜=0.5,0.6∈∨κ∗˜,qκ˜U˜ but 2minℓ˜,j˜=0.5,0.6∈∨κ∗˜,qκ˜¯U˜.Lemma 1.
LetU˜ be an IVFS of X. Then, ℏℓ˜∈U˜implies0ℓ˜∈∨κ∗˜,qκ˜U˜⇔∀ℏ∈X,U˜0≥minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2.Proof.
⇒ On the contrary, suppose that, for some ℏ∈X, U˜0<minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2. Take ℓ˜∈D0,κ∗−k/2 such that(8)U˜0<ℓ˜≤minU˜ℏ,κ∗−−κ−2,κ∗+−κ+2.
Then,ℏℓ˜∈U˜, but 0ℓ˜∈∨κ∗˜,qκ˜¯U˜, a contradiction. Hence, U˜0≥minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2.
⇐ Let ℏ∈X such that ℏℓ˜∈U˜. Then, U˜ℏ≥ℓ˜. So,(9)U˜0≥minU˜ℏ,κ∗−−κ−2,κ∗+−κ+2≥minℓ˜,κ∗−−κ−2,κ∗+−κ+2.
Now, ifℓ˜≤κ∗−−κ−/2,κ∗+−κ+/2, then U˜0≥ℓ˜. Therefore, 0ℓ˜∈U˜. On the contrary, if ℓ˜>κ∗−−κ−/2,κ∗+−κ+/2, then U˜0≥κ∗−−κ−/2,κ∗+−κ+/2. So, U˜0+ℓ˜>κ∗−−κ−/2,κ∗+−κ+/2+κ∗−−κ−/2,κ∗+−κ+/2=κ∗−−κ−,κ∗+−κ+. This implies that 0ℓ˜κ∗˜,qκ˜U˜. Hence, 0ℓ˜∈∨κ∗˜,qκ˜U˜.Lemma 2.
LetU˜ be an IVFS of X. Then, ℏ∗υℓ˜∈U˜ and υj˜∈U˜ imply ℏminℓ˜,j˜∈∨κ∗˜,qκ˜U˜⇔U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2.Proof.
⇒ On the contrary, suppose that U˜ℏ<minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2 for some ℏ,υ∈X. Choose ℓ˜∈D0,κ∗−k/2 such that U˜ℏ<ℓ˜≤minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2. Then, ℏ∗υℓ˜∈U˜ and υℓ˜∈U˜, but ℏℓ˜∈∨κ∗˜,qκ˜¯U˜, which is not possible. Thus, we have shown that(10)U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−2,κ∗+−κ+2.
⇐ Let ℏ∗υℓ˜∈U˜ and υj˜∈U˜ for all ℓ˜,j˜∈D0,1. Then, U˜ℏ∗υ≥ℓ˜ and U˜υ≥j˜. Thus,(11)U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−2,κ∗+−κ+2≥minℓ˜,j˜,κ∗−−κ−2,κ∗+−κ+2.
Now, ifminℓ˜,j˜≤κ∗−−κ−/2,κ∗+−κ+/2, then U˜ℏ≥minℓ˜,j˜ and ℏminℓ˜,j˜∈U˜; otherwise, i.e., when minℓ˜,j˜>κ∗−−κ−/2,κ∗+−κ+/2, then U˜ℏ≥κ∗−−κ−/2,κ∗+−κ+/2. So, we have(12)U˜ℏ+minℓ˜,j˜>κ∗−−κ−2,κ∗+−κ+2+κ∗−−κ−2,κ∗+−κ+2=κ∗−−κ−,κ∗+−κ+.
This implies thatℏminℓ˜,j˜κ∗˜,qκ˜U˜. Hence, ℏminℓ˜,j˜∈∨κ∗˜,qκ˜U˜, as required.
By combining Lemmas1 and 2, we have the following theorem.Theorem 4.
An IVFSU˜ of X is an IV ∈,∈∨κ∗˜,qκ˜-FI of X⇔(1)
U˜0≥minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2 and(2)
U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2for all ℏ,υ∈X.Lemma 3.
LetU˜ be an IV ∈,∈∨κ∗˜,qκ˜-FI of X such that ℏ≤υ. Then, U˜ℏ≥minU˜υ,κ∗−−κ−/2,κ∗+−κ+/2.Proof.
Letℏ≤υ for ℏ,υ∈X. Then, ℏ∗υ=0. By hypothesis, we have(13)U˜ℏ≥minU˜ℏ∗υ,Uυ,κ∗−−κ−2,κ∗+−κ+2=minU˜0,Uυ,κ∗−−κ−2,κ∗+−κ+2≥minU˜υ,κ∗−−κ−2,κ∗+−κ+2.Lemma 4.
LetU˜ be an IV ∈,∈∨κ∗˜,qκ˜-FI of X. Then, for any ℏ,υ,z∈X, ℏ∗υ≤z⇒U˜ℏ≥minU˜υ,U˜z,κ∗−−κ−/2,κ∗+−κ+/2.Proof.
Suppose thatℏ∗υ≤z for ℏ,υ,z∈X. Then, we have(14)U˜ℏ≥minU˜ℏ∗υ,Uυ,κ∗−−κ−2,κ∗+−κ+2≥minminU˜z,κ∗−−κ−2,κ∗+−κ+2,U˜υ,κ∗−−κ−2,κ∗+−κ+2=minU˜υ,Uz,κ∗−−κ−2,κ∗+−κ+2.Theorem 5.
Every IV∈,∈∨κ∗˜,qκ˜-FI of BCK-algebra X is an IV ∈,∈∨κ∗˜,qκ˜-FS of X.Proof.
LetU˜ be an IV ∈,∈∨κ∗˜,qκ˜-FI of X and υ,ℏ∈X. As υ∗ℏ≤υ in X, by Lemma 3, we have(15)U˜υ∗ℏ≥minU˜υ,κ∗−−κ−2,κ∗+−κ+2.
SinceU˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X, we have(16)U˜υ∗ℏ≥minU˜υ,κ∗−−κ−2,κ∗+−κ+2≥minminU˜υ∗ℏ,Uℏ,κ∗−−κ−2,κ∗+−κ+2,κ∗−−κ−2,κ∗+−κ+2≥minU˜υ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2.
Hence,U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FS of X.Example 4.
Consider a BCK-algebraX=0,1,2,3 with the binary operation ∗ as defined in Table 3.
Consider the IV∈,∈∨κ∗˜,qκ˜-FS U˜ of X, where U˜:X⟶D0,1 is defined by(17)U˜ℏ=0.3,0.4,ifℏ=0,0.1,0.2,ifℏ∈1,2,0.2,0.3,ifℏ=3.
Chooseκ∗˜=0.8,0.9 and κ˜=0.1,0.2. Then, U˜ is not an IV ∈,∈∨κ∗˜,qκ˜-FI of X as 0ℓ˜=0.3,0.3=1∗3ℓ˜=0.3,0.3∈U˜ and 3j˜=0.2,0.2∈U˜ but 1minℓ˜,j˜=0.2,0.2∈∨κ∗˜,qκ˜¯U˜.Table 3
Cayley Table of the binary operation∗.
∗012300000110102220033330Theorem 6.
LetU˜ be an IV ∈,∈∨κ∗˜,qκ˜-FS of X. Then, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI ⇔ for all υ,ℏ,z∈X such that υ∗ℏ≤z implies U˜υ≥minU˜ℏ,U˜z,κ∗−−κ−/2,κ∗+−κ+/2.Proof.
⇒ It follows from Lemma 4.
⇐ Let U˜ be an IV ∈,∈∨κ∗˜,qκ˜-FS such that for all υ,ℏ,z∈X with υ∗ℏ≤z imply U˜υ≥minU˜ℏ,U˜z,κ∗−−κ−/2,κ∗+−κ+/2. As υ∗υ∗ℏ≤ℏ, by hypothesis,(18)U˜υ≥minU˜υ∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2.
Hence,U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X.Theorem 7.
LetU˜ be an IVFS of X. Then, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X⇔ the set U˜ℓ˜≠∅ is an ideal of X for each ℓ˜∈D0,κ∗−κ/2.Proof.
⇒ Let ℓ˜∈D0,κ∗−κ/2 such that U˜ℓ˜≠∅. By Theorem 4, we have(19)U˜0≥minU˜ℏ,κ∗−−κ−2,κ∗+−κ+2,with ℏ∈U˜ℓ˜. It follows that U˜0≥minℓ˜,κ∗−−κ−/2,κ∗+−κ+/2=ℓ˜. Therefore, 0∈U˜ℓ˜.
Next, suppose thatℏ∗υ∈U˜ℓ˜ and υ∈U˜ℓ˜. Then, U˜ℏ∗υ≥ℓ˜ and U˜υ≥ℓ˜. Again, by Theorem 4, we have(20)U˜ℏ≥minU˜ℏ∗υ,Uυ,κ∗−−κ−2,κ∗+−κ+2≥minℓ˜,κ∗−−κ−2,κ∗+−κ+2=ℓ˜.
Therefore,ℏ∈U˜ℓ˜. Hence, U˜ℓ˜ is an ideal of X.
⇐ Suppose that U˜ℓ˜ is an ideal of X for all ℓ˜∈D0,κ∗−k/2. If U˜0<minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2 for some ℏ∈X, then ∃ℓ˜∈D0,κ∗−k/2 such that U˜0<ℓ˜≤minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2. It follows that ℏ∈U˜ℓ˜ but 0∉U˜ℓ˜, a contradiction. Therefore, U˜0≥minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2. Also, if U˜ℏ<minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2 for some ℏ,υ∈X, then ∃ℓ˜∈D0,κ∗−k/2 such that(21)U˜ℏ<ℓ˜≤minU˜ℏ∗υ,U˜υ,κ∗−−κ−2,κ∗+−κ+2.
It follows thatℏ∗υ∈U˜ℓ˜ and υ∈U˜ℓ˜ but ℏ∉U˜ℓ˜, which is again a contradiction. Therefore, U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2. Hence, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X.Definition 7.
LetU˜ be an IVFS of X. The set(22)U˜ℓ˜=ℏ∈X|ℏℓ˜∈∨κ∗˜,qκ˜U˜,whereℓ˜∈D0,1,is called an ∈∨κ∗˜,qκ˜-level subset of U˜.Theorem 8.
LetU˜ be an IVFS of X. Then, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X⇔ the ∈∨κ∗˜,qκ˜-level subset U˜ℓ˜ of U˜ is an ideal of X for each ℓ˜∈D0,1.Proof.
⇒ Suppose U˜ is an interval-valued ∈,∈∨κ∗˜,qκ˜-FI of X. Take any ℏ∈U˜ℓ˜. Then, ℏℓ˜∈∨κ∗˜,qk˜U˜. So, U˜ℏ≥ℓ˜ or U˜ℏ+ℓ˜>κ∗−−κ−,κ∗+−κ+. Now, by Theorem 4, we have U˜0≥minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2. Thus, U˜0≥minℓ˜,κ∗−−κ−/2,κ∗+−κ+/2 when U˜ℏ≥ℓ˜. If ℓ˜>κ∗−−κ−/2,κ∗+−κ+/2, then U˜0≥κ∗−−κ−/2,κ∗+−κ+/2 implies 0∈U˜ℓ˜. Also, if ℓ˜≤κ∗−−κ−/2,κ∗+−κ+/2, then U˜0≥ℓ˜ implies 0∈U˜ℓ˜. Similarly, 0∈U˜ℓ˜ when U˜ℏ+ℓ˜>κ∗−−κ−,κ∗+−κ+.
Next, take anyℏ∗υ∈U˜ℓ˜ and υ∈U˜ℓ˜. Then, ℏ∗υℓ˜∈∨κ∗˜,qκ˜U˜ and υℓ˜∈∨κ∗˜,qκ˜U˜, i.e., either U˜ℏ∗υ≥ℓ˜ or U˜ℏ∗υ+ℓ˜>κ∗−−κ−,κ∗+−κ+ and either U˜υ≥ℓ˜ or U˜υ+ℓ˜>κ∗−−κ−,κ∗+−κ+. By assumption, U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2. Thus, the following cases arise.Case 1.
LetU˜ℏ∗υ≥ℓ˜ and U˜υ≥ℓ˜. If ℓ˜>κ∗−−κ−/2,κ∗+−κ+/2, then(23)U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−2,κ∗+−κ+2≥minℓ˜,κ∗−−κ−2,κ∗+−κ+2=κ∗−−κ−2,κ∗+−κ+2,and so, ℏℓ˜κ∗˜,qk˜U˜. If ℓ˜≤κ∗−−κ−/2,κ∗+−κ+/2, then(24)U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−2,κ∗+−κ+2≥minℓ˜,κ∗−−κ−2,κ∗+−κ+2=ℓ˜.
So,ℏℓ˜∈U˜. Hence, ℏℓ˜∈∨κ∗˜,qk˜U˜.Case 2.
LetU˜ℏ∗υ≥ℓ˜ and U˜υ+ℓ˜≥κ∗−−κ−,κ∗+−κ+. If ℓ˜>κ∗−−κ−/2,κ∗+−κ+/2, then(25)U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−2,κ∗+−κ+2≥minℓ˜,κ∗−−κ−,κ∗+−κ+−ℓ˜,κ∗−−κ−2,κ∗+−κ+2=κ∗−−κ−,κ∗+−κ+−ℓ˜,i.e., U˜ℏ+ℓ˜>κ∗−−κ−,κ∗+−κ+, and thus, ℏℓ˜κ∗˜,qk˜U˜. If ℓ˜≤κ∗−−κ−/2,κ∗+−κ+/2, then(26)U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−2,κ∗+−κ+2≥minℓ˜,κ∗−−κ−,κ∗+−κ+−ℓ˜,κ∗−−κ−2,κ∗+−κ+2=ℓ˜,and so, ℏℓ˜∈U˜. Hence, ℏℓ˜∈∨κ∗˜,qk˜U˜.
Similarly, in other two cases, i.e., whenU˜ℏ∗υ+ℓ˜>κ∗−−κ−,κ∗+−κ+, U˜υ≥ℓ˜, and U˜ℏ∗υ+ℓ˜>κ∗−−κ−,κ∗+−κ+, U˜υ+ℓ˜>κ∗−−κ−,κ∗+−κ+, we have ℏℓ˜∈∨κ∗˜,qk˜U˜. Hence, in each case, ℏℓ˜∈∨κ∗˜,qk˜U˜, and thus, ℏ∈U˜ℓ˜.
⇐ Let U˜ℓ˜ be an ideal of X for all ℓ˜∈D0,1. On the contrary, let(27)U˜0<minU˜ℏ,κ∗−−κ−2,κ∗+−κ+2,with ℏ∈X. Then, ∃ℓ˜∈D0,1 such that U˜0<ℓ˜≤minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2. It follows that ℏ∈U˜ℓ˜, but 0∉U˜ℓ˜, which is not possible. Therefore,(28)U˜0≥minU˜ℏ,κ∗−−κ−2,κ∗+−κ+2.
Also, ifU˜ℏ<minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2 for some ℏ,υ∈X, then ∃ℓ˜∈D0,1 such that(29)U˜ℏ<ℓ˜≤minU˜ℏ∗υ,U˜υ,κ∗−−κ−2,κ∗+−κ+2.
It follows thatℏ∗υ∈U˜ℓ˜ and υ∈U˜ℓ˜ but ℏ∉U˜ℓ˜, which is again a contradiction. Therefore, U˜ℏ≥minU˜ℏ∗υ,U˜υ,κ∗−−κ−/2,κ∗+−κ+/2. Hence, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X.


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

Throughout the following sections,X denotes BCK-algebras unless stated otherwise.Definition 8.
LetX be a BCK-algebra. An IVFS U˜ is called an interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy commutative ideal (in short, IV ∈,∈∨κ∗˜,qκ˜-FCI) if for all υ,p,ℏ∈X:(1)
ℏℓ˜∈U˜ implies 0ℓ˜∈∨κ∗˜,qκ˜U˜(2)
υ∗p∗ℏℓ˜∈U˜ and ℏj˜∈U˜ imply υ∗p∗p∗υminℓ˜,j˜∈∨κ∗˜,qκ˜U˜Example 5.
Consider aBCK-algebra X=0,1,2,3 with the binary operation ∗ as defined in Table 4.
DefineU˜:X⟶D0,1 by(30)U˜ℏ=0.5,0.6,ifℏ=0,0.3,0.4,ifℏ∈1,2,0.1,0.2,ifℏ=3.
Chooseκ∗˜=0.2,0.3 and κ˜=0.1,0.2. Then, it is easy to see that U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FCI of X.Table 4
Cayley Table of the binary operation∗.
∗012300000110012210233330Theorem 9.
An IVFSU˜ of X is an IV ∈,∈∨κ∗˜,qκ˜-FCI of X⇔(1)
U˜0≥minU˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2 and(2)
U˜υ∗p∗p∗υ≥minU˜υ∗p∗ℏ,U˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2for all υ,p,ℏ∈X.Proof.
⇒ Condition (1) follows from Lemma 1. To show that (2) holds in X, suppose, on the contrary, that (2) does not hold in X, so we have(31)U˜υ∗p∗p∗υ<minU˜υ∗p∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2,for some υ,p,ℏ∈X. Choose ℓ˜∈D0,1 such that U˜υ∗p∗p∗υ<ℓ˜≤minU˜υ∗p∗ℏ,U˜ℏ,κ∗−−κ−/2,κ∗+−κ+/2. Then, υ∗p∗ℏℓ˜∈U˜ and ℏℓ˜∈U˜, but υ∗p∗p∗υℓ˜∈∨κ∗˜,qκ˜¯U˜, which is not possible. Thus,(32)U˜υ∗p∗p∗υ≥minU˜υ∗p∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2,for each υ,p,ℏ∈X.
⇐ Suppose that conditions (1) and (2) hold in X. It follows from condition (1) and Lemma 1 that ℏℓ˜∈U˜ implies 0ℓ˜∈∨κ∗˜,qκ˜U˜. Let υ∗p∗ℏℓ˜∈U˜ and ℏj˜∈U˜ for all ℓ˜,j˜∈D0,1. Then, U˜υ∗p∗ℏ≥ℓ˜ and U˜ℏ≥j˜. Thus,(33)U˜υ∗p∗p∗υ≥minU˜υ∗p∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2≥minℓ˜,j˜,κ∗−−κ−2,κ∗+−κ+2.
Now, ifminℓ˜,j˜≤κ∗−−κ−/2,κ∗+−κ+/2, then U˜υ∗p∗p∗υ≥minℓ˜,j˜ implies υ∗p∗p∗υminℓ˜,j˜∈U˜; otherwise, i.e., when minℓ˜,j˜>κ∗−−κ−/2,κ∗+−κ+/2, then U˜υ∗p∗p∗υ≥κ∗−−κ−/2,κ∗+−κ+/2. So, we have(34)U˜υ∗p∗p∗υ+minℓ˜,j˜≥κ∗−−κ−2,κ∗+−κ+2+κ∗−−κ−2,κ∗+−κ+2=κ∗−−κ−,κ∗+−κ+.
This implies thatυ∗p∗p∗υminℓ˜,j˜κ∗˜,qκ˜U˜. Therefore, υ∗p∗p∗υminℓ˜,j˜∈∨κ∗˜,qκ˜U˜. Hence, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FCI of X.Theorem 10.
Every IV∈,∈∨κ∗˜,qκ˜-FCI of BCK-algebra X is an IV ∈,∈∨κ∗˜,qκ˜-FI of X.Proof.
LetU˜ be any IV ∈,∈∨κ∗˜,qκ˜-FCI of X and υ,ℏ∈X. Then, we have(35)U˜υ=U˜υ∗0∗0∗υ≥minU˜υ∗0∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2=minU˜υ∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2.
Hence,U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X.Example 6.
Consider aBCK-algebra X=0,1,2,3,4 with the binary operation ∗ as defined in Table 5.
Chooseκ∗˜=0.7,0.8 and κ˜=0.2,0.3. Then, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FI of X but is not an IV ∈,∈∨κ∗˜,qκ˜-FCI of X as 0ℓ˜=0.2,0.2=2∗3∗3∗2ℓ˜=0.2,0.2∈U˜ and 0j˜=0.3,0.3∈U˜ but 2∗3∗3∗2minℓ˜,j˜=0.2,0.2=2ℓ˜=0.2,0.2∈∨κ∗˜,qκ˜¯U˜.Table 5
Cayley Table of the binary operation∗.
∗01234000000110100222000333300444320(36)U˜ℏ=0.5,0.6,ifℏ=0,0.4,0.5,ifℏ=1,0.0,0.1,ifℏ∈2,3,4.Theorem 11.
LetU˜ be an IV ∈,∈∨κ∗˜,qκ˜-FI of BCK-algebra X. Then, U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FCI of X⇔ for all υ,p∈X,(37)U˜υ∗p∗p∗υ≥minU˜υ∗p,κ∗−−κ−2,κ∗+−κ+2.Proof.
⇒ Let U˜ be an IV ∈,∈∨κ∗˜,qκ˜-FCI of X. Then, for all υ,p,ℏ∈X, we have(38)U˜υ∗p∗p∗υ≥minU˜υ∗p∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2.
By takingℏ=0, we have(39)U˜υ∗p∗p∗υ≥minU˜υ∗p∗0,U˜0,κ∗−−κ−2,κ∗+−κ+2=minU˜υ∗p,κ∗−−κ−2,κ∗+−κ+2.
⇐ Let υ,p,ℏ∈X. By assumption, we have(40)U˜υ∗p≥minU˜υ∗p∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2.
By assumption and equation (40), we have(41)U˜υ∗p∗p∗υ≥minU˜υ∗p,κ∗−−κ−2,κ∗+−κ+2≥minU˜υ∗p∗ℏ,U˜ℏ,κ∗−−κ−2,κ∗+−κ+2.
Hence,U˜ is an IV ∈,∈∨κ∗˜,qκ˜-FCI of X.


## 5. Conclusion

The aim of this paper is to present a general form of interval-valued fuzzy ideals ofBCK/BCI-algebras. In fact, we introduced the concepts of interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy (subalgebras) ideals and interval-valued ∈∨κ∗˜,qκ˜,∈∨κ∗˜,qκ˜-fuzzy ideals of BCK/BCI-algebras. In addition, interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy commutative ideals were defined, and some essential properties were discussed. Moreover, the relationship between ∈,∈∨κ∗˜,qκ˜-fuzzy ideals and interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy commutative ideals is considered. In this present study, we conclude the following cases:(1)
If we takeκ∗˜=1,1 and κ˜=0,0, then interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy subalgebras and interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy ideals reduce to the concepts of interval-valued ∈,∈∨q-fuzzy subalgebras and interval-valued ∈,∈∨q-fuzzy ideals of X as introduced in [34](2)
If we takeκ∗˜=1,1 and κ˜=κ˜, then interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy subalgebras and interval-valued ∈,∈∨κ∗˜,qκ˜-fuzzy ideals reduce to the concepts of interval-valued ∈,∈∨qκ˜-fuzzy subalgebras and interval-valued ∈,∈∨qκ˜-fuzzy ideals of X as introduced in [37]Consequently, the notions introduced in this paper, i.e.,∈,∈∨κ∗˜,qκ˜-fuzzy subalgebras and ∈,∈∨κ∗˜,qκ˜-fuzzy ideals, are more general than ∈,∈∨qκ˜-fuzzy subalgebras and ∈,∈∨qκ˜-fuzzy ideals. In future work, one may extend these concepts to various algebraic structures such as rings, hemirings, LA-semigroups, semi-hypergroups, semi-hyperrings, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, EQ-algebras, and lattice implication algebras.



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