# Interval-Valuedm-Polar Fuzzy Positive Implicative Ideals in BCK-Algebras

Authors: G. Muhiuddin; D. Al-Kadi; A. Mahboob; Amjad Albjedi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the notion of interval-valuedm-polar fuzzy positive implicative ideals in BCK-algebras is presented. Then, the relationships between interval-valued m-polar fuzzy positive implicative ideals and interval-valued m-polar fuzzy ideals are investigated. After that, the concepts of interval-valued m-polar ∈,∈∨qκ˜-fuzzy positive implicative ideals and interval-valued m-polar ∈,∈∨qκ˜-fuzzy ideals are defined and some equivalent conditions are provided. Furthermore, we show that interval-valued m-polar ∈,∈∨qκ˜-fuzzy positive implicative ideals are interval-valued m-polar ∈,∈∨qκ˜-fuzzy ideals, but the converse need not be true in general and an example is given in this aim.


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## Body

## 1. Introduction

As an extension of fuzzy sets, Zadeh [1] defined fuzzy sets with an interval-valued membership function proposing the concept 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, [2–7]), groups (see, for example, [8–10]), and rings (see, for example, [11–13]). Jun [14] studied interval-valued fuzzy ideals in BCI-algebras. Zhan et al. [15, 16] 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 were used in the studies made by Ma et al. in [17, 18], where they discussed properties of some types of ∈,∈∨q-interval-valued fuzzy ideals of BCI-algebras. Also, in [19–24], some more general ideas on bipolar fuzzy sets’ related ideals were considered.Them-polar fuzzy set, an extension of the bipolar fuzzy set, was introduced by Chen et al. [25] in 2014. When more than one agreement has to work with the m-polar fuzzy model, it offers the system more accuracy, flexibility, and compatibility. The investigation of m-polar fuzzy algebraic structures started with the idea of textitm-pF lie subalgebras proposed by Akram et al. [26]. Following that, Akram and Farooq [27] in lie subalgebras introduced the theory of m-pF lie ideal. A concept proposed by [28] for the m-pF subgroups. The notions of m-pF ideals and m-pF commutative ideals on BCK/BCI-algebras were introduced by Al-Masarwah and Ahmad [29]. The concepts of ∈,∈∨q-fuzzy ideals and ∈,∈∨q-fuzzy commutative ideals have been considered by Al-Masarwah and Ahmad in [30]. In [31], Muhiuddin et al. introduced and characterized the notion of m-polar ∈,∈-fuzzy q-ideal in BCI-algebras. Takallo et al. [32] proposed the notion of ∈,∈-fuzzy p-ideal in BCI-algebras and studied related properties of m-polar ∈,∈-fuzzy ideals and m-polar ∈,∈-fuzzy p-ideals in BCI-algebras. Recently, by generalizing the concept of m-polar fuzzy positive implicative ideals of BCK-algebras, Al-Masarwah et al. [33] introduced the notions of ∈,∈∨q-fuzzy positive implicative ideals and ∈¯,∈¯∨q¯-fuzzy positive implicative ideals in BCK-algebras. Also, different kinds of concepts, related to this study, were investigated in various ways (see, for example, [34–40]).In this paper, the notion of interval-valuedm-polar fuzzy positive implicative ideals in BCK-algebras is presented. We prove that every interval-valued m-polar fuzzy positive implicative ideal of BCK-algebras is an interval-valued m-polar fuzzy ideal but the converse statement is not true in general and an example is given in this aim. Moreover, the concepts of interval-valued m-polar ∈,∈∨qκ˜-fuzzy positive implicative ideals and interval-valued m-polar ∈,∈∨qκ˜-fuzzy ideals are defined and some equivalent conditions are provided. Furthermore, we show that interval-valued m-polar ∈,∈∨qκ˜-fuzzy positive implicative ideals are interval-valued m-polar ∈,∈∨qκ˜-fuzzy ideals, but converse need not be true in general and an example is given in this aim.


## 2. Preliminaries

An algebraZ˜;∗,0 of type (2, 0) is called a BCK-algebra if, for all ϑ,ω,ℏ∈Z˜,(i)
ϑ∗ω∗ϑ∗ℏ≤ℏ∗ω.(ii)
ϑ∗ϑ∗ω≤ω.(iii)
ϑ∗ϑ=0.(iv)
0∗ϑ=0.(v)
ϑ≤ω and ω≤ϑ imply ϑ=ω, where ≤ can be presented by ϑ≤ω⇔ϑ∗ω=0. Every BCK-algebra Z˜ satisfies the following axioms, for all ϑ,ω,ℏ∈Z˜:(1)
ϑ∗0=ϑ.(2)
ϑ∗ω∗ℏ=ϑ∗ℏ∗ω.A subsetø≠A of Z˜ is called a subalgebra if, for all ϑ,ω∈Z˜, ϑ∗ω∈A and is called an ideal of Z˜ if 0∈A and, for all ϑ,ω∈Z˜,ϑ∗ω∈A,ω∈A implies ϑ∈A.Definition 1.
(see [33]). A subset ∅≠P of Z˜ is called a positive implicative ideal of Z˜ if ∀ϑ,ω,ℏ∈Z˜:(i)
0∈P(ii)
ϑ∗ω∗ℏ∈P and ω∗ℏ∈P imply ϑ∗ℏ∈P
The interval numbert˜ is the interval t−,t+, where 0≤t−≤t+≤1, and D0,1 is the set of all interval numbers. For the interval numbers t˜i=ti−,ti+, d˜i=di−,di+∈D0,1,i∈I, we describe(a)
rmint˜i,d˜i=minti−,di−,minti+,di+(b)
rmaxt˜i,d˜i=minti−,di−,minti+,di+(c)
t˜1≤t˜2⇔t1−≤t2− and t1+≤t2+(d)
t˜1=t˜2⇔t1−=t2− and t1+=t2+
A mappingUP˜:Z˜⟶D0,1 is called an interval-valued fuzzy set of Z˜, where UP˜ϑ=UP−ϑ,UP+ϑ, for all ϑ∈Z˜, where UP− and UP+ are fuzzy sets of Z˜ with UP−ϑ≤UP+ϑ, for all ϑ∈Z˜.Definition 2.
A mappingUP˜:Z˜⟶D0,1m is called an interval-valued m-polar fuzzy set (briefly, IVmPF set) of Z˜ and is defined as(1)UP˜ϑ=q1˜∘UP˜ϑ,q2˜∘UP˜ϑ,…,qm˜∘UP˜ϑ,where qi˜:D0,1m⟶D0,1 is the ith projection mapping for i∈1,2,…,m. That is,(2)UP˜ϑ=U1P−ϑ,U1P+ϑ,U2P−ϑ,U2P+ϑ,…,UmP−ϑ,UmP+ϑ,for all ϑ∈Z˜, where UiP− and UiP+ are fuzzy sets of Z˜ with UiP−ϑ≤UiP+ϑ, for all ϑ∈Z˜ and i∈1,2,…,m.
Theith projection map qi˜ is order preserving and vice versa, i.e.,(3)ϑ≤ω⇔qi˜ϑ≤qi˜ω,∀i∈1,2,…,m.Definition 3.
(see [40]). An IVmPF set UP˜ of Z˜ is called an IVmPF ideal of Z˜ if, for any ϑ,ω∈Z˜,(1)
UP˜0≥UP˜ϑ(2)
UP˜ϑ≥rminUP˜ϑ∗ω,UP˜ω
That is,(1)
qi˜∘UP˜0≥qi˜∘UP˜ϑ(2)
qi˜∘UP˜ϑ≥rminqi˜∘UP˜ϑ∗ω,qi˜∘UP˜ω, ∀i=1,2,…,mDefinition 4.
(see [40]). The set UP˜α,β˜=ϑ∈Z˜|UP˜ϑ≥α,β˜, where UP˜ is an IVmPF set of Z˜ is called the level cut subset of UP˜, ∀α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m.Lemma 1.
(see [40]). Every IVmPF ideal UP˜ of Z˜ satisfies the following assertion, ∀ϑ,ω∈Z˜:(4)ϑ≤ω⇒UP˜ϑ≥UP˜ω.


## 3. Interval-Valuedm-Polar Fuzzy Positive Implicative Ideals

Definition 5.
AnIVmPF set UP˜ of Z˜ is called an IVmPFPI ideal of Z˜ if, for any ϑ,ω,ℏ∈Z˜,(1)
UP˜0≥UP˜ϑ(2)
UP˜ϑ∗ℏ≥rminUP˜ϑ∗ω∗ℏ,UP˜ω∗ℏ
That is,(1)
qi˜∘UP˜0≥qi˜∘UP˜ϑ(2)
qi˜∘UP˜ϑ∗ℏ≥rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ, ∀i=1,2,…,mExample 1.
Consider aBCK-algebra Z˜=0,1,2,3,4 with the Cayley table (Table 1).
LetUP˜ be an IV4PF set defined as(5)UP˜ϑ=0.7,0.8,0.4,0.5,0.9,01,0.7,0.8,ifϑ=0,0.6,0.7,0.3,0.4,0.8,0.9,0.6,0.7,ifϑ=1,0.5,0.6,0.2,0.3,0.6,0.7,0.5,0.6,ifϑ=2,0.4,0.5,0.1,0.2,0.3,0.4,0.4,0.5,ifϑ=3,0.3,0.4,0.2,0.3,0.5,0.6,0.3,0.4,ifϑ=4.
It is straightforward to check thatUP˜ is an IV4PFPI ideal of Z˜.Table 1
Cayley table of the binary operation∗.
∗01234000000110000222002333303444440Theorem 1.
EveryIV4PFPI ideal of Z˜ is an IVmPF ideal of Z˜.Proof.
LetUP˜ be an IV4PFPI ideal of Z˜. Then, condition (1) of Definition 5 holds. By assumption, we have(6)qi˜∘UP˜ϑ∗ℏ≥rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ.
Putℏ=0, so(7)qi˜∘UP˜ϑ≥rminqi˜∘UP˜ϑ∗ω,qi˜∘UP˜ω.
Hence,UP˜ is an IVmPF ideal of Z˜.
As shown by the following example, the converse of the preceding Theorem1 is not valid in general.Example 2.
Consider aBCK-algebra Z˜=0,1,2,3 with the Cayley table (Table 2).
Now, define anIV3PF set UP˜ as follows:(8)UP˜ϑ=0.6,0.7,0.6,0.7,0.9,0.9,ifϑ=0,0.5,0.6,0.5,0.6,0.8,0.8,ifϑ=1,2,0.3,0.3,0.3,0.3,0.3,0.3,ifϑ=3.
It is straightforward to check thatUP˜ is an IV3PF ideal of Z˜, but it is not an IV3PFPI ideal of Z˜ since q1˜∘UP˜2∗1=q1˜∘UP˜1=0.5,0.6<rminq1˜∘UP˜2∗1∗1,q1˜∘UP˜1∗1=rminq1˜∘UP˜0,q1˜∘UP˜0=q1˜∘UP˜0=0.6,0.7.Table 2
Cayley table of the binary operation∗.
∗012300000110012210233330Theorem 2.
AnIVmPF set of Z˜ is an IVmPFPI ideal of Z˜⇔; it is an IVmPF ideal of Z˜ and UP˜ϑ∗ω≥UP˜ϑ∗ω∗ω∀ϑ,ω∈Z˜.Proof.
(⇒) Suppose UP˜ is an IVmPFPI ideal of Z˜. By Theorem 1, UP˜ is an IVmPF ideal of Z˜. By assumption, we have(9)qi˜∘UP˜ϑ∗ℏ≥rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ.
Now, replaceℏ by ω; then,(10)qi˜∘UP˜ϑ∗ω≥rminqi˜∘UP˜ϑ∗ω∗ω,qi˜∘UP˜ω∗ω=rminqi˜∘UP˜ϑ∗ω∗ω,qi˜∘UP˜0=qi˜∘UP˜ϑ∗ω∗ω,∀ϑ,ω∈Z˜.
(⇐) Suppose that UP˜ is an IVmPF ideal of Z˜. Then, condition (1) of Definition 5 holds. As ϑ∗ℏ∗ℏ∗ω∗ℏ≤ϑ∗ℏ∗ω=ϑ∗ω∗ℏ∀ϑ,ω∈Z˜, so by Lemma 1, we have(11)qi˜∘UP˜ϑ∗ℏ∗ℏ∗ω∗ℏ≥qi˜∘UP˜ϑ∗ω∗ℏ.
Now, by assumption,(12)qi˜∘UP˜ϑ∗ℏ≥qi˜∘UP˜ϑ∗ℏ∗ℏ≥rminqi˜∘UP˜ϑ∗ℏ∗ℏ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ≥rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ.
Hence,UP˜ is an IVmPFPI ideal of Z˜.Theorem 3.
AnIVmPF set UP˜ of Z˜ is an IVmPFPI ideal of Z˜⇔UP˜α,β˜≠ϕ is a positive implicative ideal of Z˜, ∀α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m.Proof.
(⇒) Suppose that UP˜ is an IVmPFPI ideal of Z˜. Let α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m be such that ϑ∈UP˜α,β˜. Then, qi˜∘UP˜0≥qi˜∘UP˜ϑ≥αi,βi, and we have 0∈UP˜α,β˜. Let ϑ,ω,ℏ∈Z˜ be such that ϑ∗ω∗ℏ∈UP˜α,β˜ and ω∗ℏ∈UP˜α,β˜. Then, qi˜∘UP˜ϑ∗ω∗ℏ≥αi,βi and qi˜∘UP˜ω∗ℏ≥αi,βi. It follows from Definition 5 (2) that(13)qi˜∘UP˜ϑ∗ℏ≥rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ≥αi,βi.
Thus,ϑ∗ℏ∈UP˜α,β˜. Hence, UP˜α,β˜ is a positive implicative ideal of Z˜.
(⇐) Assume that UP˜α,β˜ is a positive implicative ideal of Z˜, ∀α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m. If qi˜∘UP˜0<qi˜∘UP˜t for some t∈Z˜, then ∃θ,λ˜=θ1,λ1,θ2,λ2,…,θm,λm∈D0,1m such that qi˜∘UP˜0<θi,λi≤qi˜∘UP˜t. It implies that 0∈¯UP˜θ,λ˜, a contradiction. Thus, UP˜0≥UP˜ϑ, ∀ϑ∈Z˜. Again, if qi˜∘UP˜ϑ∗ℏ<rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ, for some ϑ,ω,ℏ∈Z˜; then,(14)qi˜∘UP˜ϑ∗ℏ<ρi,σi≤rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ,for some ρ,σ˜=ρ1,σ1,ρ2,σ2,…,ρm,σm∈D0,1m. It follows that ϑ∗ω∗ℏ∈UP˜ρ,σ˜ and ω∗ℏ∈UP˜ρ,σ˜, but ϑ∗ℏ∈¯UP˜ρ,σ˜. This is a contradiction. Thus, UP˜ϑ∗ℏ≥rminUP˜ϑ∗ω∗ℏ,UP˜ω∗ℏ∀ϑ,ω,ℏ∈Z˜. Hence, UP˜ is an IVmPFPI ideal of Z˜.


## 4.m-Polar ∈,∈∨qk˜-Fuzzy Positive Implicative Ideals

AnIVmPF set UP˜ of Z˜ of the form(15)UP˜ℏ=α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m,ifℏ=ϑ,0˜=0,0,0,0,…,0,0,ifℏ≠ϑ,is called an IVmPF point, denoted as ϑα,β˜, with support ϑ and value α,β˜=α1,β1,α2,β2,…,αm,βm. An IVmPF point ϑα,β˜(1)
Belongs toUP˜, written as ϑα,β˜∈UP˜, if UP˜ϑ≥α,β˜, i.e., ∀i=1,2,…,m, qi˜∘UP˜ϑ≥αi,βi(2)
Is quasi-coincidence withUP˜, written as ϑα,β˜qk˜UP˜, if UP˜ϑ+α,β˜+k˜>1˜, i.e., ∀i=1,2,…,m, qi˜∘UP˜ϑ+αi,βi+κi+,κi−>1, where k˜=κ1,κ2,…,κm and 1˜=1,1,1,1,…,1,1 in which κi=κi+,κi− and 1=1,1Assume0˜≤κ˜<1˜. We write(1)
ϑα,β˜γ¯UP˜ if ϑα,β˜γUP˜ does not hold(2)
ϑα,β˜∈∨qk˜UP˜ (resp. ϑα,β˜∈∧qk˜UP˜) if ϑα,β˜∈UP˜ or ϑα,β˜qk˜UP˜ (resp. ϑα,β˜∈UP˜ and ϑα,β˜qk˜UP˜)Definition 6.
AnIVmPF set UP˜ of a BCK-algebra Z˜ is called an IVmP∈,∈∨qk˜‐F ideal of Z˜ if(1)
ϑα,β˜∈UP˜implies0α,β˜∈∨qk˜UP˜(2)
ϑ∗ℏα,β˜∈UP˜ and ℏρ,σ˜∈UP˜implyϑrminα,β˜,ρ,σ˜∈∨qk˜UP˜, ∀ϑ,ω,ℏ∈Z˜, and α,β˜,ρ,σ˜∈D0,1mExample 3.
Consider aBCK-algebra Z˜=0,1,2,3 with the Cayley table (Table 3).
Define anIV5PF set UP˜:Z˜⟶D0,15 as(16)Uℏ=0.9,0.8,0.8,0.7,0.7,0.6,0.6,0.5,0.5,0.4,ifℏ=0,0.6,0.5,0.5,0.4,0.4,0.3,0.3,0.2,0.2,0.1,ifℏ∈1,2,3.
Chooseκ=0.9,0.9. Then, with direct computation, we find that UP˜ is an IV5P∈,∈∨qk˜‐F ideal of X.Table 3
Cayley table of the binary operation∗.
∗012300003110032220333330Theorem 4.
AnIVmPF set UP˜ of Z˜ is an IVmP∈,∈∨qk˜−F ideal of Z˜⇔.(1)
UP˜0≥rminUP˜ϑ,1˜−k˜/2(2)
UP˜ϑ≥rminUP˜ϑ∗ℏ,UP˜ℏ,1˜−k˜/2, ∀ϑ,ω,ℏ∈Z˜Proof.
(⇒) Suppose, on the contrary, that qi˜∘UP˜0<rminqi˜∘UP˜ϑ,1−ki/2; then, qi˜∘UP˜0<αi,βi≤rminqi˜∘UP˜ϑ,1−ki/2 for some α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m and 1≤i≤m. This implies that ϑα,β˜∈UP˜, but 0α,β˜∈¯UP˜, a contradiction. Thus, UP˜0≥rminUP˜ϑ,1˜−k˜/2.
Again, suppose the contrary thatqi˜∘UP˜ϑ<rminqi˜∘UP˜ϑ∗ℏ,qi˜∘UP˜ℏ,1−ki/2. Then, qi˜∘UP˜ϑ<αi,βi≤rminqi˜∘UP˜ϑ∗ℏ,qi˜∘UP˜ℏ,1−ki/2 for some α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m. This implies that ϑ∗ℏα,β˜∈UP˜ and ℏα,β˜∈UP˜, but ϑα,β˜∈∨qk˜¯UP˜, a contradiction. Hence, UP˜ϑ≥rminUP˜ϑ∗ℏ,UP˜ℏ,1˜−k˜/2.
(⇐) Suppose that ℏ∈Z˜ such that ℏα,β˜∈UP˜. Then, qi˜∘UP˜ℏ≥αi,βi. So,(17)qi˜∘UP˜0≥minqi˜∘UP˜ℏ,1−κi2≥minαi,βi,1−κi2.
Now, ifαi,βi≤1−κi/2, then qi˜∘UP˜0≥αi,βi. Therefore, 0α,β˜∈UP˜. On the contrary, if αi,βi>1−κi/2, then qi˜∘UP˜0≥1−κi/2. So, qi˜∘UP˜0+αi,βi>1−κi/2+1−κi/2=1−κi. This implies that 0α,β˜qκUP˜.
Hence,0α,β˜∈∨qκUP˜. Let ϑ∗ℏθ,λ˜∈UP˜ and ℏρ,σ˜∈UP˜, ∀θ,λ˜=θ1,λ1,θ2,λ2,…,θm,λm, ρ,σ˜=ρ1,σ1,ρ2,σ2,…,ρm,σm∈D0,1m. Then, qi˜∘UP˜ϑ∗ℏ≥θi,λi and qi˜∘UP˜ℏ≥ρi,σi. Thus,(18)qi˜∘UP˜ϑ≥rminqi˜∘UP˜ϑ∗ℏ,qi˜∘UP˜ℏ,1−κi2,≥rminθi,λi,ρi,σi,1−κi2.
Now, ifrminθi,λi,ρi,σi≤1−κi/2, then qi˜∘UP˜ϑ≥rminθi,λi,ρi,σi implies ϑrminθ,λ˜,ρ,σ˜∈UP˜; otherwise, when rminθi,λi,ρi,σi>1−κi/2, then qi˜∘UP˜ϑ≥1−κi/2. So, we have(19)qi˜∘UP˜ϑ+rminθi,λi,ρi,σi>1−κi2+1−κi2=1−κi.
This implies thatϑrminθ,λ˜,ρ,σ˜qκUP˜. Hence, ϑrminθ,λ˜,ρ,σ˜∈∨qκUP˜, as required.Lemma 2.
LetUP˜ be an IVmP∈,∈∨qk˜−F ideal of Z˜ and ϑ,ω∈Z˜ such that ϑ≤ω. Then,(20)ϑ≤ω⟹UP˜ϑ≥rminUP˜ω,1˜−k˜2.Proof.
Letϑ,ω∈Z˜ such that ϑ≤ω. Then, we have(21)qi˜∘UP˜ϑ≥rminqi˜∘UP˜ϑ∗ω,qi˜∘UP˜ω,1−ki2,=rminqi˜∘UP˜0,qi˜∘UP˜ω,1−ki2=rminqi˜∘UP˜ω,1−ki2.
Hence,ϑ≤ω⇒UP˜ϑ≥rminUP˜ω,1˜−k˜/2Definition 7.
AnIVmPF set UP˜ of a BCK-algebra Z˜ is called an IVmP∈,∈∨qk˜−FPI ideal of Z˜ if(1)
ϑα,β˜∈UP˜implies0α,β˜∈∨qk˜UP˜(2)
ϑ∗ω∗ℏα,β˜∈UP˜ and ω∗ℏρ,σ˜∈UP˜implyϑ∗ℏrminα,β˜,ρ,σ˜∈∨qk˜UP˜, ∀ϑ,ω,ℏ∈Z˜ and α,β˜,ρ,σ˜∈D0,1mExample 4.
Consider aBCK-algebra Z˜=0,1,2,3 which is given in Example 2. Let UP˜ be an IVmPF set defined as(22)UP˜ϑ=0.5,0.5,0.5,0.5,…,0.5,0.5,ifϑ=0,0.4,0.4,0.4,0.4,…,0.4,0.4,ifϑ=1,2,3.
Chooseκ=0.4,0.3. Then, UP˜ is an IVmP∈,∈∨qk˜−FPI ideal of Z˜.Theorem 5.
AnIVmPF set UP˜ of Z˜ is an IVmP∈,∈∨qk˜−FPI ideal of Z˜⇔, ∀ϑ,ω,ℏ∈Z˜:(1)
UP˜0≥rminUP˜ϑ,1˜−k˜/2(2)
UP˜ϑ∗ℏ≥rminUP˜ϑ∗ω∗ℏ,UP˜ω∗ℏ,1˜−k˜/2Proof.
(⇒) Suppose that UP˜ is an IVmP∈,∈∨qk˜−FPI ideal of Z˜. If qi˜∘UP˜0<rminqi˜∘UP˜ϑ,1−ki/2, then ∃α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m such that qi˜∘UP˜0<αi,βi≤rminqi˜∘UP˜ϑ,1−ki/2. This implies that ϑα,β˜∈UP˜, but 0α,β˜∈¯UP˜, a contradiction. Hence, UP˜0≥rminUP˜ϑ,1˜−k˜/2.
If we assume thatqi˜∘UP˜ϑ∗ℏ<rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ,1−ki/2, then ∃α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1m such that qi˜∘UP˜ϑ∗ℏ<αi,βi≤rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ,1−ki/2. This implies that ϑ∗ω∗ℏα,β˜∈UP˜ and ω∗ℏα,β˜∈UP˜, but ϑ∗ℏα,β˜∈∨qk˜¯UP˜, a contradiction. Hence, UP˜ϑ∗ℏ≥rminUP˜ϑ∗ω∗ℏ,UP˜ω∗ℏ,1˜−k˜/2.
(⇐) Let ℏ∈Z˜ such that ℏα,β˜∈UP˜. Then, qi˜∘UP˜ℏ≥αi,βi. So,(23)qi˜∘UP˜0≥rminqi˜∘UP˜ℏ,1−κi2≥rminαi,βi,1−κi2.
Now, ifαi,βi≤1−κi/2, then qi˜∘UP˜0≥αi,βi. Therefore, 0α,β˜∈UP˜. On the contrary, if αi,βi>1−κi/2, then qi˜∘UP˜0≥1−κi/2. So, qi˜∘UP˜0+αi,βi>1−κi/2+1−κi/2=1−κi. This implies that 0α,β˜qκUP˜. Hence, 0α,β˜∈∨qκUP˜.
Letϑ∗ω∗ℏθ,λ˜∈U and ω∗ℏρ,σ˜∈U, ∀θ,λ˜=θ1,λ1,θ2,λ2,…,θm,λm, and ρ,σ˜=ρ1,σ1,ρ2,σ2,…,ρm,σm∈D0,1m. Then, qi˜∘UP˜ϑ∗ω∗ℏ≥θi,λi and qi˜∘UP˜ω∗ℏ≥ρi,σi. Thus,(24)qi˜∘UP˜ϑ∗ℏ≥rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ,1−κi2,≥rminθi,λi,ρi,σi,1−κi2.
Now, ifrminθi,λi,ρi,σi≤1−κi/2, then qi˜∘UP˜ϑ∗ℏ≥rminθi,λi,ρi,σi and ϑ∗ℏrminθ,λ˜,ρ,σ˜∈U; otherwise, when rminθi,λi,ρi,σi>1−κi/2, then qi˜∘UP˜ϑ∗ℏ≥1−κi/2. So, we have(25)qi˜∘UP˜ϑ∗ℏ+rminθi,λi,ρi,σi>1−κi2+1−κi2=1−κi.
This implies thatϑ∗ℏrminθ,λ˜,ρ,σ˜qκUP˜. Hence, ϑ∗ℏrminθ,λ˜,ρ,σ˜∈∨qκUP˜, as required.Theorem 6.
EveryIVmP∈,∈∨qk˜FPI ideal of Z˜ is an IVmP∈,∈∨qk˜−F ideal of Z˜.Proof.
LetIVmP∈,∈∨qk˜FPI be an IVmP∈,∈∨qk˜FPI ideal of Z˜. Then, condition (1) of Definition 6 holds. By assumption, we have(26)ϑ∗ω∗ℏθ,λ˜∈UP˜andω∗ℏρ,σ˜∈UP˜implyϑ∗ℏrminθ,λ˜,ρ,σ˜∈∨qk˜UP˜.
Putℏ=0, so,(27)ϑ∗ω∗ℏθ,λ˜∈UP˜andω∗0ρ,σ˜∈UP˜implyϑ∗0rminθ,λ˜,ρ,σ˜∈∨qk˜UP˜.
Thus,(28)ϑ∗ωθ,λ˜∈UP˜andωρ,σ˜∈UP˜implyϑ∗0rminθ,λ˜,ρ,σ˜∈∨qk˜UP˜.
Hence,UP˜ is an IVmP∈,∈∨qk˜−F ideal of Z˜.
As shown by the following example, the converse of the preceding Theorem6 is not valid in general.Example 5.
Reconsider theBCK-algebras Z˜ given in Example 2. Define an IV3PF set UP˜ as(29)UP˜ϑ=0.8,0.7,0.7,0.6,0.6,0.5,ifϑ=0,0.5,0.4,0.4,0.3,0.3,0.2,ifϑ=1,2,0.4,0.3,0.3,0.2,0.2,0.1,ifϑ=3.
Chooseκ=0.1,0.1. Clearly, UP˜ is an IV3P∈,∈∨qk˜−FI of Z˜, but is not an IV3P∈,∈∨qk˜−FPI ideal of Z˜ because q1˜∘UP˜2∗1=q1˜∘UP˜1=0.4,0.3<rminq1˜∘UP˜2∗1∗1,q1˜∘UP˜1∗1,1−k/2 = rminq1˜∘UP˜0,q1˜∘UP˜0,1−k/2=0.45,0.45.Theorem 7.
LetUP˜ be an IVmP∈,∈∨qk˜−F ideal of Z˜. Then, UP˜ is an IVmP∈,∈∨qk˜−FPI ideal of Z˜⇔UP˜ϑ∗ω≥rminUP˜ϑ∗ω∗ω,1−k/2, ∀ϑ,ω∈Z˜.Proof.
(⇒) Assume UP˜ is an IVmP∈,∈∨qk˜−FPI ideal of Z˜. Now, replace ℏ by ω in Theorem 5 (2); then,(30)UP˜ϑ∗ω≥rminUP˜ϑ∗ω∗ω,UP˜ω∗ω,1˜−k˜2,=rminUP˜ϑ∗ω∗ω,UP˜0,1˜−k˜2,=rminUP˜ϑ∗ω∗ω,1˜−k˜2,∀ϑ,ω∈Z˜. (⇐) Let UP˜ be an IVmP∈,∈∨qk˜−F ideal of Z˜. Then, condition (1) holds. As ϑ∗ℏ∗ℏ∗ω∗ℏ≤ϑ∗ℏ∗ω=ϑ∗ω∗ℏ, ∀ϑ,ω∈Z˜. By Lemma 2, we have(31)UP˜ϑ∗ℏ∗ℏ∗ω∗ℏ≥rminUP˜ϑ∗ω∗ℏ,1˜−k˜2.
SinceUP˜ is an IVmP∈,∈∨qk˜−F ideal, so(32)UP˜ϑ∗ℏ≥rminUP˜ϑ∗ℏ∗ℏ,1˜−k˜2,≥rminUP˜ϑ∗ℏ∗ℏ∗ω∗ℏ,UP˜ω∗ℏ,1˜−k˜2,1˜−k˜2,≥rminUP˜ϑ∗ℏ∗ℏ,UP˜ω∗ℏ,1˜−k˜2.
Hence,UP˜ is an IVmP∈,∈∨qk˜FPI ideal of Z˜.Theorem 8.
AnIVmPF set UP˜ of Z˜ is an IVmP∈,∈∨qk˜−F ideal of a BCK-algebra Z˜⇔UP˜α,β˜≠ϕ is a positive implicative ideal of Z˜, ∀α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1−k/2m.Proof.
(⇒) Let ϑ∈UP˜α,β˜ for α,β˜=α1,β1,α2,β2,…,αm,βm∈D0,1−k/2m. Then, qi˜∘UP˜ϑ≥αi,βi. It follows from Theorem 5 (i) that(33)qi˜∘UP˜0≥rminqi˜∘UP˜ϑ,1−ki2=αi,βi.
Thus,0∈UP˜α,β˜.
Next, suppose thatϑ∗ω∗ℏ∈UP˜α,β˜ and ω∗ℏ∈UP˜α,β˜. Then, qi˜∘UP˜ϑ∗ω∗ℏ≥αi,βi and qi˜∘UP˜ω∗ℏ≥αi,βi. Again, it follows from Theorem 5 (ii) that(34)qi˜∘UP˜ϑ∗ℏ≥rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ,1˜−k˜2≥rminαi,βi,αi,βi,1−ki2=αi,βi.
Therefore,ϑ∗ℏ∈UP˜α,β˜. Hence, UP˜α,β˜ is a positive implicative ideal of Z˜.
(⇐) Suppose, on the contrary, that qi˜∘UP˜0<rminqi˜∘UP˜ϑ,1−k/2 for some ϑ∈Z˜. Choose θ,λ˜=θ1,λ1,θ2,λ2,…,θm,λm∈D0,1−k/2m such that(35)qi˜∘UP˜0<θi,λi≤rminqi˜∘UP˜ϑ,1−ki2.
It follows thatϑ∈UP˜θ,λ˜, but 0∈¯UP˜θ,λ˜, a contradiction. Therefore, UP˜0≥rminUP˜ϑ,1˜−k˜/2 ,∀ϑ∈Z˜. Suppose that(36)qi˜∘UP˜ϑ∗ω<rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ϑ∗ω,1˜−k˜2,for some ϑ,ω,ℏ∈Z˜. Then, ∃ρ,σ˜=ρ1,σ1,ρ2,σ2,…,ρm,σm∈D0,1−k/2m such that qi˜∘UP˜ϑ∗ω<ρi,σi≤rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ϑ∗ω,1−ki/2 implies that ϑ∗ω∗ℏ∈UP˜ρ,σ˜ and ϑ∗ω∈UP˜ρ,σ˜, but ϑ∗ω∈¯UP˜ρ,σ˜, which is not possible. Thus,(37)qi˜∘UP˜ϑ∗ℏ≥rminqi˜∘UP˜ϑ∗ω∗ℏ,qi˜∘UP˜ω∗ℏ,1−ki2,∀ϑ,ω,ℏ∈Z˜.
Hence, by Theorem5, UP˜ is an IVmP∈,∈∨qk˜−F ideal of Z˜.


## 5. Conclusion

We applied the theory of interval-valued fuzzy sets on positive implication ideals ofBCK-algebras. In this aim, the concept of interval-valued m-polar fuzzy positive implicative ideals in BCK-algebras is introduced. The related propertied of interval-valued m-polar fuzzy positive implicative ideals and interval-valued m-polar fuzzy ideals are investigated. In addition, the concepts of interval-valued m-polar ∈,∈∨qκ˜-fuzzy positive implicative ideals and interval-valued m-polar ∈,∈∨qκ˜-fuzzy ideals are defined and characterized. Furthermore, we have shown that interval-valued m-polar ∈,∈∨qκ˜-fuzzy positive implicative ideals are interval-valued m-polar ∈,∈∨qκ˜-fuzzy ideals, but converse is not valid and an illustration is provided in this support.In future work, one may extend these concepts to various algebraic structures such as rings, hemirings,LA-semigroups, semihypergroups, semihyperrings, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, EQ-algebras, and lattice implication algebras.



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