Interval Valued m-polar Fuzzy BCK/BCI-Algebras

G. Muhiuddin; D. Al-Kadi

doi:10.2991/ijcis.d.210223.003

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Volume 14, Issue 1, 2021, Pages 1014 - 1021

Interval Valued m-polar Fuzzy BCK/BCI-Algebras

Authors

G. Muhiuddin¹^{, *}^,, D. Al-Kadi²

¹Department of Mathematics, University of Tabuk, Tabuk, 71491, Saudi Arabia

²Department of Mathematics and Statistic, College of Science, Taif University, Taif, 21944, Saudi Arabia

^*Corresponding author. Email: chishtygm@gmail.com

Corresponding Author

G. Muhiuddin

Received 6 December 2020, Accepted 9 February 2021, Available Online 10 March 2021.

DOI: 10.2991/ijcis.d.210223.003 How to use a DOI?
Keywords: Interval-valued m-polar fuzzy sets; Interval-valued m-polar fuzzy subalgebras; Interval-valued m-polar fuzzy (commutative) ideals
Abstract: The notion of interval-valued m-polar fuzzy sets (abbreviated IVmPF) is much wider than the notion of m-polar fuzzy sets. In this paper, we apply the theory of IVmPF on BCK/BCI-algebras. We introduce the concepts of IVmPF subalgebras, IVmPF ideals and IVmPF commutative ideals and some essential properties are discussed. We characterize IVmPF subalgebras in terms of fuzzy subalgebras and subalgebras of BCK/BCI-algebras. We show that in BCK-algebra, IVmPF ideals are IVmPF subalgebras and that the converse is not valid. We provide a condition under which an IVmPF subalgebra becomes an IVmPF ideal. Further, we characterize IVmPF ideals in terms of fuzzy ideals and ideals of BCK/BCI-algebras. Moreover, we prove that in any BCK-algebra, an IVmPF commutative ideal is an IVmPF fuzzy ideal but not the converse. Also, we provide conditions under which an IVmPF ideal becomes an IVmPF commutative ideal.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In 1966, Imai and Iséki introduced the concept of BCK/BCI-algebras, which is a generalization of propositional calculus and the set-theoretic difference. The literature on the theory of BCK/BCI-algebras has been developed since then, and more focus has been placed on the ideal theory of BCK/BCI-algebras in particular. In BCK/BCI-algebras and other related algebraic structures, different kinds of concepts were investigated in various ways (see, e.g., [1–8]).

The fuzzy set theory proposed by Zadeh [9] has been extended to a lot of areas. In addition, a variety of extensions and generalizations of fuzzy sets have been introduced such as the following well known sets: bipolar fuzzy sets, hesitant fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets and fuzzy multisets, etc. The interval-valued fuzzy set introduced by Zadeh takes the values of the membership functions as intervals instead of numbers. The study of interval-valued fuzzy algebraic structures started in [10] by introducing the concept of interval-valued fuzzy subgroups. Jun [11] extended the concept of interval-valued fuzzy sets to BCK/BCI-algebras and introduced the notions of interval-valued subalgebras and ideals. After that, the notion of interval-valued fuzzy sets in BCK/BCI-algebras with different aspects has been studied by several authors, for example, see [12–14].

Zhang introduced the notion of bipolar fuzzy sets which permits the membership degree of an element over two intervals [−1, 0] and [0, 1], that is, every element assigns negative and positive degree of memberships. By applying the notion of bipolar fuzzy sets to BCK/BCI-algebras, Lee [15] introduced the notions of bipolar fuzzy subalgebra and bipolar fuzzy ideal of BCK/BCI-algebras. Using (α,β)-bipolar fuzzy generalized bi-ideals, Ibrar et al. [16] characterized regular ordered semigroups whereas Bashir et al. [17] characterized the regular ordered ternary semigroups. For more related concepts on bipolar fuzzy sets, we refer to [18–22].

As in many problems, information often comes from several variables and there are often multi-attribute data that cannot be handled using current theories, a lot of approaches have been done to solve this problem. For example, Chen et al. [23] presented the m-polar fuzzy set, an expansion of the bipolar fuzzy set and as a new approach Akram et al. [24] introduced a technique in decision making based on m-polar fuzzy sets.

The m-polar fuzzy algebraic structures study began with the concept of m-polar fuzzy Lie subalgebras [25]. After that, the theory of m-polar fuzzy Lie ideals was studied in Lie algebras [26]. The concept of the m-polar fuzzy groups was given in [27]. Moreover, m-polar fuzzy matroids have been studied in [28]. Further, m-polar fuzzy sets have been studied in different areas (see [29–33]). Recently, Al-Masarwah and Ahmad introduced the notion of m-polar fuzzy (commutative) ideals [34] and m-polar (α,β)-fuzzy ideals [35] in BCK/BCI-algebras. As a continues work they introduced a new form of generalized m-polar fuzzy ideals in [36] and studied normalization of m-polar fuzzy subalgebras in [37]. A new advanced extensions are formed by merging two fuzzy information in one set as neutrosophic bipolar fuzzy sets, bipolar valued hesitant fuzzy sets and interval- valued m-polar fuzzy sets (IVmPF). For some recent work on these extensions, we refer the reader to [38–43].

The power of the theory of IVmPF as an advanced extension with all the work done on different algebraic structure motivated the authors to apply the theory of IVmPF on BCK/BCI-algebras. The novelty in this study lies in using the proposed model on BCK/BCI-algebras. The authors introduced and investigated the notions of interval-valued m-polar fuzzy subalgebras, interval-valued m-polar fuzzy ideals and interval-valued m-polar fuzzy commutative ideals in Sections 3, 4, 5, respectively. A summary of proposed and future work were given in Section 6.

2. PRELIMINARIES

An algebra (X;∗,0) of type (2,0) is called a BCI-algebra if ∀ν,κ,ℏ∈X, it satisfies

(K1)((ν∗ℏ)∗(ν∗κ))∗(κ∗ℏ)=0,

(K2)(ν∗(ν∗ℏ))∗ℏ=0,

(K3)ν∗ν=0,

(K4) ν∗ℏ=0 and ℏ∗ν=0⇒ν=ℏ,

If a BCI-algebra X satisfies

(K5)0∗ν=0∀ ν∈X

then X is a BCK-algebra.

The following conditions hold in any BCK/BCI-algebra X and for all ν,κ,ℏ∈X:

(P1) ν∗0=ν,

(P2) (ν∗ℏ)∗κ=(ν∗κ)∗ℏ,

(P3) ν≤ℏ⇒ν∗κ≤ℏ∗κ and κ∗ℏ≤κ∗ν,

(P4) 0∗(ν∗ℏ)=(0∗ν)∗(0∗ℏ),

(P5) 0∗(0∗(ν∗ℏ))=0∗(ℏ∗ν),

(P6) (ν∗κ)∗(ℏ∗κ)≤(ν∗ℏ),

(P7) ν∗(ν∗(ν∗ℏ))=ν∗ℏ,

(P8) 0∗(0∗((ν∗κ)∗(ℏ∗κ)))=(0∗ℏ)∗(0∗ν),

(P9) 0∗(0∗(ν∗ℏ)=(0∗ℏ)∗(0∗ν),

where ν≤κ⇔ν∗κ=0∀ ν,κ∈X. Clearly, (X,≤) is a partially ordered set.

A nonempty subset B of X is called a subalgebra of X if ν∗κ∈B∀ ν,κ∈B.

A nonempty subset L of X is called an ideal of X if

(L1) 0∈L,

(L2) ∀ ν,κ∈X,ν∗κ∈L and κ∈L⇒ν∈L.

Let X be a BCK/BCI-algebra. A fuzzy set of X is a mapping ξ:X→[0,1]. A fuzzy set ξ is called a fuzzy subalgebra if (∀ ν,κ∈X)ξ(ν∗κ)≥ξ(ν)∧ξ(κ) and it is called a fuzzy ideal if ξ(0)≥ξ(ν) and ξ(ν)≥ξ(ν∗κ)∧ξ(κ) for all ν,κ∈X. Further, ξ is called a fuzzy commutative ideal if ξ(0)≥ξ(ν) and ξ(ν∗(κ∗(κ∗ν)))≥ξ((ν∗κ)∗ℏ)∧ξ(ℏ).

By the interval number n~, we mean an interval denoted as [n−,n+], where 0≤n−≤n+≤1. We write S[0,1] to denote the set of all interval numbers. The interval [n,n] is indicated by the number n∈[0,1] for whatever follows. For the interval numbers n~i=[ni−,ni+], m~i=[mi−,mi+]∈S[0,1],i∈I, we describe

n~i∧m~i=[ni−∧mi−,ni+∧mi+];
n~i∨m~i=[ni−∨mi−,ni+∨mi+];
n~1≤n~2⇔n1−≤n2− and n1+≤n2+;
n~1=n~2⇔n1−=n2− and n1+=n2+.

Let X be a BCK/BCI-algebra. A mapping G~:X→S[0,1] is an interval-valued fuzzy set (briefly, IVF set) of X, where for all ν∈X, G~(ν)=[G~−(ν),G~+(ν)],G~− and G~+ are fuzzy sets of X with G~−(ν)≤G~+(ν).

An IVF set is called an IVF subalgebra if (∀ ν,κ∈X)G~(ν∗κ)≥G~(ν)∧G~(κ) and it is called an IVF ideal if G~(0)≥G~(ν) and G~(ν)≥G~(ν∗κ)∧G~(κ)∀ ν,κ∈X. Moreover, G~ is called an IVF commutative ideal if G~(0)≥G~(ν) and G~(ν∗(κ∗(κ∗ν)))≥G~((ν∗κ)∗ℏ)∧G~(ℏ)∀ ν,κ,ℏ∈X.

3. INTERVAL-VALUED m-POLAR FUZZY SUBALGEBRAS

The notion of an IVmPF subalgebra in BCK/BCI-algebras is introduced and characterized in terms of subalgebra and fuzzy subalgebra of BCK/BCI-algebras.

Definition 3.1.

Let X be a nonempty set. An IVmPF set of X is a mapping G~:X→S[0,1]m defined as

G~(ν)=(π1~∘G~(ν),π2~∘G~(ν),…,πm~∘G~(ν))

where for i∈{1,2,…,m}, πi~∘G~:X→S[0,1] is the ith-projection mapping.

That is,

G~(ν)=([G~1−(ν),G~1+(ν)], [G~2−(ν),G~2+(ν)],…,[G~m−(ν), G~m+(ν)])

for all ν∈X, G~i− and G~i+ are fuzzy sets of X with G~i−(ν)≤G~i+(ν) for all ν∈X and i∈{1,2,…,m}.

We define an order relation on S[0,1]m as pointwise, that is,

ν≤κ⇔πi~(ν)≤πi~(κ)

where πi~:S[0,1]m→S[0,1] is the ith-projection mapping and i∈{1,2,…,m}. For an element [α,β]~∈S[0,1]m, we mean that ([α,β],[α,β],…,[α,β]), while the element [α,β]^=([α1,β1],[α2,β2],…,[αm,βm]) represents an arbitrary element of S[0,1]m. Clearly, the elements [0,0]~ and [1,1]~ are the smallest and largest elements in S[0,1]m.

Definition 3.2.

An IVmPF set G~ of X is called an IVmPF subalgebra if

(∀ν,κ∈X) G~(ν∗κ) ≥ G~(ν)∧G~(κ),

that is,

(∀ ν,κ∈X,i∈{1,2,…,m}) πi~∘G~(ν∗κ)≥πi~∘G~(ν)∧πi~∘G~(κ).

Example 1.

Consider a BCK-algebra in which X={0,℘,κ,ℓ} and ∗ is given by the following table:

∗0℘κℓ00000℘℘00℘κκ℘0κℓℓℓℓ0

Let [ω,φ]^ = ([ω1,φ1],[ω2,φ2],…,[ωm,φm]),[α,β]^ = ([α1,β1],[α2,β2],…,[αm,βm]∈S[0,1]m such that [ω,φ]^≥[α,β]^. Now define an IVmPF set G~ on X as

G~(ν)={ ([ω1,φ1], [ω2,φ2],…,[ωm,φm]) if ν=0,([α1,β1], [α2,β2],…,[αm,βm]) if ν=℘,([0,0], [0,0], …, [0,0]) if ν∈{κ,l}.

It is easy to verify that G~ is an IVmPF subalgebra.

Lemma 3.3.

If G~ is an IVmPF subalgebra of X, then

G~(0)≥G~(ν)∀ν∈X.

Proof.

Let ν∈X. Then, we have

πi~∘G~(0)=πi~∘G~(ν∗ν) ≥πi~∘G~(ν)∧πi~∘G~(ν) =πi~∘G~(ν),

as required.

Theorem 3.4.

An IVmPF set G~=([G~1−,G~1+],[G~2−,G~2+],…,[G~m−,G~m+] is an IVmPF subalgebra of X ⇔G~i− and G~i+ are fuzzy subalgebras of X for all i's.

Proof.

(⇒) Assume that G~=([G~1−,G~1+], [G~2−,G~2+],…,[G~m−,G~m+] is an IVmPF subalgebra of X. Then for any ν,κ∈X,

πi~∘G~(ν∗κ)≥πi~∘G~(ν)∧πi~∘G~(κ)∀i∈{1,2,…,m},

implies

[G~i−(ν∗κ),G~i+(ν∗κ)]≥[G~i−(ν),G~i+(ν)]∧[G~i−(κ),G~i+(κ)] =[G~i−(ν)∧G~i−(κ),G~i+(ν)∧G~i+(κ)].

Therefore, G~i−(ν∗κ)≥G~i−(ν)∧G~i−(κ) and G~i+(ν∗κ)≥G~i+(ν)∧G~i+(κ). Hence, G~i− and G~i+ are fuzzy subalgebras of X for all i∈{1,2,…,m}.

(⇐) For the converse, suppose that G~i− and G~i+ are fuzzy subalgebras of X for all i's. So for any ν,κ∈X, we have

πi~∘G~(ν∗κ)=[G~i−(ν∗κ),G~i+(ν∗κ)] ≥[G~i−(ν)∧G~i−(κ),G~i+(ν)∧G~i+(κ)] =[G~i−(ν),G~i+(ν)]∧[G~i−(κ),G~i+(κ)] =πi~∘G~(ν)∧πi~∘G~(κ).

Hence, G~ is an IVmPF subalgebra of X.

Definition 3.5.

Let G~ be any IVmPF set. For [α,β]^=([α1,β1],[α2,β2],…,[αm,βm])∈S[0,1]m define a level subset U(G~;[α,β]^) as follows:

U(G~;[α,β]^)={x∈X|πi~∘G~(x)≥[αi,βi] for all i∈{1,2,…,m}}.

Theorem 3.6.

An IVmPF set G~ is an IVmPF subalgebra of X⇔ each nonempty level subset U(G~;[α,β]^) is a subalgebra of X∀[α,β]^=([α1,β1],[α2,β2],…,[αm,βm])∈S[0,1]m.

Proof.

(⇒) Take any ν,κ∈U(G~;[α,β]^). Therefore, πi~∘G~(ν)≥[αi,βi] and πi~∘G~(κ)≥[αi,βi] for all i∈{1,2,…,m}. Having G~ an IVmPF subalgebra of X, implies

πi~∘G~(ν∗κ)≥πi~∘G~(ν)∧πi~∘G~(κ) ≥[αi,βi]∧[αi,βi] =[αi,βi].

Therefore, ν∗κ∈U(G~;[α,β]^).

(⇐) Assume that U(G~;[α,β]^) is a subalgebra of X∀[α,β]^∈S[0,1]m. On contrary, let πi~∘G~(ν∗κ)<πi~∘G~(ν)∧πi~∘G~(κ) for some ν,κ∈X. So ∃[γ,λ]^=([γ1,λ1],[γ2,λ2],…,[γm,λm])∈S[0,1]m such that πi~∘G~(ν∗κ)<[γi,λi]≤πi~∘G~(ν)∧πi~∘G~(κ) for each i∈{1,2,…,m} implies ν,κ∈U(G~;[γ,λ]^) but ν∗κ∉U(G~;[γ,λ]^), which is a contradiction. Therefore, πi~∘G~(ν∗κ)≥πi~∘G~(ν)∧πi~∘G~(κ) for all i∈{1,2,…,m} and ν,κ∈X.

Example 2.

Consider a BCK-algebra in which X={0,℘,J,κ,ℓ} and ∗ is defined by the following table:

∗0℘Jκℓ000000℘℘0℘00JJJ0J0κκκκ00ℓℓℓκJ0

Now define an IVmPF set G~ on X as

G~(ν)={[0.8,0.8]~=([0.8,0.8],[0.8,0.8],…,[0.8,0.8]) if ν=0,[0.4,0.4]~=([0.4,0.4],[0.4,0.4],…,[0.4,0.4]) if ν=℘,[0.5,0.5]~=([0.5,0.5],[0.5,0.5],…,[0.5,0.5]) if ν=J,[0.7,0.7]~=([0.7,0.7],[0.7,0.7],…,[0.7,0.7]) if ν=κ,[0.3,0.3]~=([0.3,0.3],[0.3,0.3],…,[0.3,0.3]) if ν=ℓ.

Therefore,

U(G~;[α,β]^)={X, if [0,0]~<[α,β]^≤ [0.3,0.3]~; {0,℘,κ}, if [0.3,0.3]~<[α,β]^≤ [0.4,0.4]~; {0,κ}, if [0.4,0.4]~<[α,β]^≤ [0.5,0.5]~; {0,κ}, if [0.5,0.5]~<[α,β]^≤ [0.7,0.7]~; {0}, if [0.7,0.7]~<[α,β]^≤ [0.8,0.8]~; ∅, if [0.8,0.8]~<[α,β]^≤ [1,1]~.

Since for all [α,β]^∈S[0,1]m, U(G~;[α,β]^) is a subalgebra of X. Therefore by Theorem 3.6, G~ is an IVmPF subalgebra.

4. INTERVAL-VALUED m-POLAR FUZZY IDEALS

The notion of an IVmPF ideal in BCK/BCI-algebras is introduced and associated properties of IVmPF ideals and IVmPF subalgebras are considered.

Definition 4.1.

An IVmPF set G~ is called an IVmPF ideal if the following conditions satisfy for all ν,κ∈X:

G~(0)≥G~(ν),
G~(ν)≥G~(ν∗κ)∧G~(κ),

that is,

πi~∘G~(0)≥πi~∘G~(ν),
πi~∘G~(ν)≥πi~∘G~(ν∗κ)∧πi~∘G~(κ),

∀i∈{1,2,…,m}.

Example 3.

Consider a BCI-algebra in which X={0,1,℘,κ,ℓ} and ∗ is defined by the following table:

∗01℘κℓ000℘κℓ110℘κℓ℘℘℘0ℓκκκκℓ0℘ℓℓℓκ℘0

Now define an IV5PF set G~ on X as

G~(ν)={([0.6,0.7],[0.5,0.8],[0.3,0.4],[0.7,0.8],[0.6,0.7]) if ν=0,([0.5,0.6],[0.3,0.5],[0.2,0.3],[0.5,0.6],[0.4,0.6]) if ν=1,([0.2,0.4],[0.1,0.2],[0.1,0.2],[0.2,0.3],[0.2,0.3]) if ν∈{℘,ℓ},([0.3,0.4],[0.2,0.3],[0.1,0.2],[0.3,0.5],[0.4,0.5]) if ν=κ,

It is routine to verify that G~is an IV5PF ideal.

Lemma 4.2.

Let G~ be an IVmPF ideal of X and ν,κ∈X such that ν≤κ. Then,

G~(ν)≥G~(κ).

Proof.

Let ν,κ∈X such that ν≤κ. Then,

πi~∘G~(ν)≥πi~∘G~(ν∗κ)∧πi~∘G~(κ) =πi~∘G~(0)∧πi~∘G~(κ) =πi~∘G~(κ).

Hence, G~(ν)≥G~(κ).

Lemma 4.3.

Let G~ be an IVmPF ideal of X and ν,κ,ℏ∈X such that ν∗κ≤ℏ. Then,

G~(ν)≥G~(κ)∧G~(ℏ).

Proof.

Let ν,κ,ℏ∈X such that ν∗κ≤ℏ. Then by Lemma 4.2, we have

πi~∘G~(ν∗κ)≥πi~∘G~(ℏ).

As G~ is an IVmPF ideal of X, so

πi~∘G~(ν)≥πi~∘G~(ν∗κ)∧πi~∘G~(κ) ≥πi~∘G~(ℏ)∧πi~∘G~(κ).

It follows that G~(ν)≥G~(κ)∧G~(ℏ).

Theorem 4.4.

In a BCK4-algebra X, every IVmPF ideal of X is an IVmPF subalgebra.

Proof.

Suppose that G~ is any IVmPF ideal and let ν,κ∈X. As ν∗κ≤ν in X, so by Lemma 4.2, πi~∘G~(ν)≤πi~∘G~(ν∗κ). Since G~ is an IVmPF ideal of X, we have

πi~∘G~(ν∗κ)≥πi~∘G~(ν) ≥πi~∘G~(ν∗κ)∧πi~∘G~(κ) =πi~∘G~(ν)∧πi~∘G~(κ).

Hence, G~ is an IVmPF subalgebra.

Remark 1.

The converse of Theorem 4.4 is not true in general.

Example 4.

Consider a BCK-algebra in which X={0,℘,κ,ℓ} and ∗ is described by the following table:

∗0℘κℓ00000℘℘0℘0κκκ00ℓℓℓℓ0

Now define an IV3PF set G~ on X as:

G~(ν)={([0.7,0.8],[0.3,0.5],[0.2,0.3]) if ν=0,([0.5,0.6],[0.1,0.3],[0.1,0.2]) if ν=℘,([0.3,0.4],[0.1,0.1],[0.1,0.1]) if ν=κ,([0.6,0.7],[0.2,0.4],[0.1,0.3]) if ν=ℓ.

By routine calculation one can verify that G~ is an IV3PF subalgebra but not an IV3PF ideal because [0.5,0.6]=π1~∘G~(℘)≱π1~∘G~(℘∗ℓ)∧π1~∘G~(ℓ)=[0.6,0.7].

The following result provides a condition for an IVmPF subalgebra to be an IVmPF ideal.

Theorem 5.2.

Let G~ be an IVmPF subalgebra of X . Then G~ is an IVmPF ideal ⇔ for all ν,κ,ℏ∈X such that ν∗κ≤ℏ implies G~(ν)≥G~(κ)∧G~(ℏ).

Proof.

(⇒) Follows from Lemma 4.3.

(⇐) Let G~ be an IVmPF subalgebra such that for all ν,κ,ℏ∈X with ν∗κ≤ℏ implies G~(ν)≥G~(κ)∧G~(ℏ). As ν∗(ν∗κ)≤κ, so by hypothesis

G~(ν)≥G~(ν∗κ)∧G~(κ).

Hence, G~ is an IVmPF ideal of X.

In the following result, we give a relation between an IVmPF ideal and fuzzy ideals of X.

Theorem 4.6.

An IVmPF set G~=([G~1 −,G~1 +],[G~2 −,G~2 +],…,[G~m −,G~m +]) in X is an IVmPF ideal of X ⇔G~i− and G~i+ are fuzzy ideals of X for all i's.

Proof.

(⇒) Assume that G~(ν)=([G~1 −(ν),G~1 +(ν)],[G~2 −(ν),G~2 +(ν)],…,[G~m −(ν),G~m +(ν)]) in X is an IVmPF ideal of X. For any ν∈X, we have

πi~∘G~(0)≥πi~∘G~(ν)∀i∈{1,2,…,m},

implies that

[G~i−(0),G~i+(0)]≥[G~i−(ν),G~i+(ν)].

It follows that G~i−(0)≥G~i−(ν) and G~i+(0)≥G~i+(ν). Take any ν,κ∈X. By hypothesis, we have

πi~∘G~(ν)≥πi~∘G~(ν∗κ)∧πi~∘G~(κ)∀i∈{1,2,…,m},

implies that

[G~i−(ν),G~i+(ν)]≥[G~i−(ν∗κ),G~i+(ν∗κ)]∧[G~i−(κ),G~i+(κ)] =[G~i−(ν∗κ)∧G~i−(κ),G~i+(ν∗κ)∧G~i+(κ)].

Therefore, G~i−(ν)≥G~i−(ν∗κ)∧G~i−(κ) and G~i+(ν)≥G~i+(ν∗κ)∧G~i+(κ). Hence, G~i− and G~i+ are fuzzy ideals of X.

(⇐) For the converse, suppose that G~i− and G~i+are fuzzy ideals of X. Then for all ν,κ∈X,

πi~∘G~(0)=[G~i−(0),G~i+(0)] ≥[G~i−(ν),G~i+(ν)] =πi~∘G~(ν)

and

πi~∘G~(ν)=[G~i−(ν),G~i+(ν)] ≥[G~i−(ν∗κ)∧G~i−(κ),G~i+(ν∗κ)∧G~i+(κ)] =[G~i−(ν∗κ),G~i+(ν∗κ)]∧[G~i−(κ),G~i+(κ)] =πi~∘G~(ν∗κ)∧πi~∘G~(κ).

Hence, G~ is an IVmPF ideal.

The following result provides a correspondence between an IVmPF ideal of X and an ideal of X.

Theorem 4.7.

An IVmPF set G~ is an IVmPF ideal of X⇔ each nonempty level subset U(G~;[α,β]^) is an ideal of X∀[α,β]^=([α1,β1],[α2,β2],…,[αm,βm])∈S[0,1]m.

Proof.

(⇒) Let us suppose that G~ is an IVmPF ideal of X and ν∈U(G~;[α,β]^). Then πi~∘G~(ν)≥[αi,βi]. By hypothesis, πi~∘G~(0)≥πi~∘G~(ν)≥[αi,βi]. Thus, 0∈U(G~;[α,β]^). Next, take any ν∗κ∈U(G~;[α,β]^) and κ∈U(G~;[α,β]^). Therefore, πi~∘G~(ν∗κ)≥[αi,βi] and πi~∘G~(κ)≥[αi,βi]. As G~ is an IVmPF ideal, so

πi~∘G~(ν)≥πi~∘G~(ν∗κ)∧πi~∘G~(κ) ≥[αi,βi]∧[αi,βi] =[αi,βi].

It follows that ν∈U(G~;[α,β]^). Hence, U(G~;[α,β]^) is an ideal.

(⇐) Now let U(G~;[α,β]^) be an ideal of X∀[α,β]^∈S[0,1]m. If πi~∘G~(0)<πi~∘G~(ν) for some ν∈X. So ∃[δ,γ]^=([δ1,γ1],[δ2,γ2],…,[δm,γm])∈S[0,1]m such that πi~∘G~(0)<[δi,γi]≤πi~∘G~(ν) for all i∈{1,2,…,m} implies ν∈U(G~;[δ,γ]^) but 0∉U(G~;[δ,γ]^), which is a contradiction. Therefore, πi~∘G~(0)≥πi~∘G~(ν) for all ν∈X and i∈{1,2,…,m}. Again, if πi~∘G~(ν)<πi~∘G~(ν∗κ)∧πi~∘G~(κ) for some ν,κ∈X. So ∃[δ,γ]^=([δ1,γ1],[δ2,γ2],…,[δm,γm])∈S[0,1]m such that πi~∘G~(ν)<[δi,γi]≤πi~∘G~(ν∗κ)∧πi~∘G~(κ) for all i∈{1,2,…,m} implies ν∗κ∈U(G~;[δ,γ]^) and κ∈U(G~;[δ,γ]^) but ν∉U(G~;[δ,γ]^), which is again a contradiction. Therefore, πi~∘G~(ν)≥πi~∘G~(ν∗κ)∧πi~∘G~(κ) for all ν,κ∈X and i∈{1,2,…,m}.

5. INTERVAL-VALUED m-POLAR COMMUTATIVE IDEALS

The notion of an IVmPF commutative ideal of BCK/BCI-algebras is defined. Relations among the IVmPF subalgebras, IVmPF ideals and IVmPF commutative ideals are discussed.

Definition 5.1.

An IVmPF set G~ is called an IVmPF commutative ideal if the following conditions satisfy for all ν,κ,ℏ∈X:

G~(0)≥G~(ν),
G~(ν∗(κ∗(κ∗ν)))≥G~((ν∗κ)∗ℏ)∧G~(ℏ),

that is,

πi~∘G~(0)≥πi~∘G~(ν),
πi~∘G~(ν∗(κ∗(κ∗ν)))≥πi~∘G~((ν∗κ)∗ℏ)∧πi~∘G~(ℏ),

∀i∈{1,2,…,m}.

Example 5.

Consider the BCK-algebra X of Example 1. Let [ω,φ]^=([ω1,φ1],[ω2,φ2],…,[ωm,φm]),[α,β]^ = [α1,β1],[α2,β2],…,[αm,βm],[δ,γ]^=([δ1,γ1],[δ2,γ2],…,[δm,γm])∈S[0,1]m such that [ω,φ]^≥[α,β]^≥[δ,γ]^. Now define an IVmPF set G~ on X as:

G~(ν)={[ω,φ]^=([ω1,φ1],[ω2,φ2],…,[ωm,φm]) if ν=0,[α,β]^=([α1,β1],[α2,β2],…,[αm,βm]) if ν∈{℘,κ},[δ,γ]^=([δ1,γ1],[δ2,γ2],…,[δm,γm]) if ν=ℓ.

It is easy to verify that G~ is an IVmPF commutative ideal.

Theorem 5.2.

In any BCK-algebra X, every IVmPF commutative ideal of X is an IVmPF ideal.

Proof.

For any IVmPF commutative ideal G~ of X and ν,κ,ℏ∈X, we have

πi~∘G~(ν)=πi~∘G~(ν∗(0∗(0∗ν))) ≥πi~∘G~((ν∗0)∗κ)∧πi~∘G~(κ) =πi~∘G~(ν∗κ)∧πi~∘G~(κ).

Hence, G~ is an IVmPF ideal.

Corollary 5.3.

Every IVmPF commutative ideal of X is an IVmPF subalgebra of X.

Remark 2.

In general, the converse of Theorem 5.2 is not true as shown next.

Example 6.

Consider a BCK-algebra in which X={0,℘,,κ,ℓ} and ∗ is described with the following table:

∗0℘Jκℓ000000℘℘0℘00JJJ000κκκκ00ℓℓℓκJ0

Let [θ,λ]^ = ([θ1,λ1],[θ2,λ2],…,[θm,λm]),[ψ,ϕ]^ = [ψ1,ϕ1],[ψ2,ϕ2],…,[ψm,ϕm],[ρ,σ]^=([ρ1,σ1],[ρ2,σ2],…,[ρm,σm]∈S[0,1]m such that [θ,λ]^≥[ψ,ϕ]^≥[ρ,σ]^. Now define an IVmPF set G~ on X as:

G~(ν)={[θ,λ]^=([θ1,λ1],[θ2,λ2],…,[θm,λm]) if ν=0,[ψ,ϕ]^=([ψ1,ϕ1],[ψ2,ϕ2],…,[ψm,ϕm]) if ν=℘,[ρ,σ]^=([ρ1,σ1],[ρ2,σ2],…,[ρm,σm]) if ν∈{j,κ,ℓ}.

It can be shown that G~ is an IVmPF ideal but not an IVmPF commutative ideal because [ρ1,σ1]=π1~∘G~(J)=π1~∘G~(J∗(κ∗(κ∗J)))≱π1~∘G~((J∗κ)∗0)∧π1~∘G~(0)=π1~∘G~(0)=[θ1,λ1].

Following two results provide conditions for an IVmPF ideal to be an IVmPF commutative ideal.

Theorem 5.4.

Let G~ be an IVmPF ideal of X. Then G~ is an IVmPF commutative ideal ⇔ for all ν,κ∈X,

G~(ν∗(κ∗(κ∗ν)))≥G~(ν∗κ).

Proof.

(⇒) Let G~ be an IVmPF commutative ideal. Then for all ν,κ,ℏ∈X, we have

G~(ν∗(κ∗(κ∗ν)))≥G~((ν∗κ)∗ℏ)∧G~(ℏ).

Taking ℏ=0, we get

G~(ν∗(κ∗(κ∗ν)))≥G~((ν∗κ)∗0)∧G~(0) =G~(ν∗κ)∧G~(0) =G~(ν∗κ).

(⇐) Let G~ be an IVmPF ideal such that G~(ν∗(κ∗(κ∗ν)))≥G~(ν∗κ) for all ν,κ∈X. By assumption, we have for all ν,κ,ℏ∈X

G~(ν∗κ)≥G~((ν∗κ)∗ℏ)∧G~(ℏ).

Therefore, G~(ν∗(κ∗(κ∗ν)))≥G~((ν∗κ)∗ℏ)∧G~(ℏ), as required.

Theorem 5.5.

Let X be a commutative BCK-algebra. Then every IVmPF ideal of X is an IVmPF commutative ideal.

Proof.

Suppose that G~ is an IVmPF ideal of X. Then for all ν,κ,ℏ∈X,

((ν∗(κ∗(κ∗ν)))∗((ν∗κ)∗ℏ))∗ℏ =((ν∗(κ∗(κ∗ν)))∗ℏ)∗((ν∗κ)∗ℏ) ≤(ν∗(κ∗(κ∗ν)))∗(ν∗κ) =(ν∗(ν∗κ))∗(κ∗(κ∗ν)) =0

It follows that ((ν∗(κ∗(κ∗ν)))∗((ν∗κ)∗ℏ))≤ℏ. As G~ is an IVmPF ideal of X, then by Lemma 4.3, G~(ν∗(κ∗(κ∗ν)))≥G~((ν∗κ)∗ℏ)∧G~(ℏ).

6. CONCLUSION

In this paper, by applying the theory of IVmPF on BCK/BCI-algebra, the notions of interval-valued m-polar fuzzy subalgebras, interval-valuedm-polar fuzzy ideals and interval-valued m-polar fuzzy commutative ideals are introduced and some essential properties are discussed. Characterizations of interval-valued m-polar fuzzy subalgebras and interval-valued m-polar fuzzy ideals are considered. Moreover, the relations among interval-valued m-polar fuzzy subalgebras, interval-valued m-polar fuzzy ideals and interval-valued m-polar fuzzy commutative ideals are obtained. This work can be a basis for further analysis of the interval-valued m-polar fuzzy structures in related algebraic structures. For future study, this concept may be applied to study some application fields like decision-making, knowledge base system, data analysis, and so on. 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, Q-algebras, and so on.

CONFLICTS OF INTEREST

Authors declare that they have no conflicts of interest.

AUTHORS' CONTRIBUTIONS

All authors have contributed to the manuscript equally.

ACKNOWLEDGMENTS

The authors are grateful to the anonymous referee(s) for a careful checking of the details and for helpful comments that improved this paper. This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 1014 - 1021
Publication Date: 2021/03/10
ISSN (Online): 1875-6883
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TY  - JOUR
AU  - G. Muhiuddin
AU  - D. Al-Kadi
PY  - 2021
DA  - 2021/03/10
TI  - Interval Valued m-polar Fuzzy BCK/BCI-Algebras
JO  - International Journal of Computational Intelligence Systems
SP  - 1014
EP  - 1021
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210223.003
DO  - 10.2991/ijcis.d.210223.003
ID  - Muhiuddin2021
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