m-Polar Picture Fuzzy Ideal of a BCK Algebra

Shovan Dogra; Madhumangal Pal

doi:10.2991/ijcis.d.200330.001

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Volume 13, Issue 1, 2020, Pages 409 - 420

m-Polar Picture Fuzzy Ideal of a BCK Algebra

Authors

Shovan Dogra, Madhumangal Pal^*

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India

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

Corresponding Author

Madhumangal Pal

Received 17 December 2019, Accepted 26 March 2020, Available Online 14 April 2020.

DOI: 10.2991/ijcis.d.200330.001 How to use a DOI?
Keywords: m-polar picture fuzzy ideal; Homomorphism of m-polar picture fuzzy ideal; m-polar picture fuzzy implicative ideal; m-polar picture fuzzy commutative ideal
Abstract: In this paper, the notions of m-polar picture fuzzy subalgebra (PFSA), m-polar picture fuzzy ideal (PFI) and m-polar picture fuzzy implicative ideal (PFII) of BCK algebra are introduced and some related basic results are presented. A relation between m-polar PFI and m-polar PFII is established. It is shown that an m-polar PFII of a BCK algebra is an m-polar PFI. But the converse of the proposition is not necessarily true. Converse is true only in implicative BCK algebra. The concept of m-polar picture fuzzy commutative ideal (PFCI) is also explored here and some related results are investigated.
Copyright: © 2020 The Authors. Published by Atlantis Press SARL.
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

After the initiation of fuzzy set (FS) by Zadeh [1] in 1965, the notion of intuitionistic fuzzy set (IFS) was propounded by Atanassov [2] in 1986. IFS includes both the degree of membership (DMS) and the degree of non-membership (DNonMS), whereas fuzzy set includes only the DMS. The concept of BCI/BCK algebra was presented by Iseki and co-workers [3–5]. Merging the concepts of FS and BCK algebra, fuzzy BCK algebra was initiated by Xi [6]. In 1993, the idea of FS was connected with BCI algebra by Ahmad [7]. Later on a lot of works on BCK/BCI algebra and ideals in fuzzy set environment were done by several researchers [8–12]. Intuitionistic fuzzy subalgebra and intuitionistic fuzzy ideal (IFI) in BCK algebra were presented by Jun and Kim [13] in 2000 as an extension of FS concept in BCK algebra. As the time goes, BCK/BCI algebra and ideals were studied by Senapati et al. [14,15] in context of intuitionistic in various directions. Bipolar fuzzy set (BFS) [16] is the generalization of FS which involves the degree of positive membership (DPMS) and the degree of negative membership (DNegMS) of an element. Bipolar fuzzy environment can be realized by an example. The serials broadcasted in Television have both good effect and bad effect on young generation. Good effect can be treated as positive effect and bad effect can be treated as negative effect. Extension work on BFS was given by Chen [17] in the form of m-polar FS. In 2013, including the measure of neutral membership and generalizing the notion of IFS, the concept of picture fuzzy set (PFS) was initiated by Cuong [18]. After the initiation of PFS, different types of research works in context of PFS were performed by several researchers [19–21]. In this paper, we introduce the concept of m-polar picture fuzzy subalgebra (PFSA), m-polar picture fuzzy ideal (PFI) and m-polar picture fuzzy implicative ideal (PFII), m-polar picture fuzzy commutative ideal (PFII) of BCK algebra and explore some results related to these. Also, we develop relationships of m-polar PFI with m-polar PFII and m-polar PFCI of BCK algebra.

2. LIST OF ABBREVIATIONS

FS - Fuzzy set

IFS - Intuitionistic fuzzy set

BFS - Bipolar fuzzy Set

PFS - Picture fuzzy set

DMS - Degree of membership

DNonMS - Degree of non-membership

DPMS - Degree of positive membership

DNegMS - Degree of negative membership

DNeuMS - Degree of neutral membership-

FI - Fuzzy ideal

IFI - Intuitionistic fuzzy ideal

PFSA - Picture fuzzy subalgebra

PFI - Picture fuzzy ideal

PFII - Picture fuzzy implicative ideal

PFPII - Picture fuzzy positive implicative ideal

PFCI - Picture fuzzy commutative ideal

3. PRELIMINARIES

Here, we recapitulate some basic concepts of FS, IFS, BCK/BCI algebra, FI, IFI, BFS, m-polar FS and PFS. We define m-polar PFS, some basic operations on m-polar PFSs, (θ,ϕ,ψ)-cut of m-polar PFS, image and inverse of m-polar PFS.

Definition 1.

Let A be the set of universe. Then a FS [1] P over A is defined as P={(a,μP(a)):a∈A}, where μP:A→[0,1]. Here, μP(a) is DMS of a in P.

The DNonMS was missing in FS. Including this type of uncertainty, Atanassov defined IFS in 1986.

Definition 2.

Let A be the set of universe. An IFS [2] P over A is defined as P={(a,μP(a),vP(a)):a∈A}, where μP(a)∈[0,1] is the DMS of a in P and vP(a)∈[0,1] is the DNonMS of a in P with the condition 0⩽μP(a)+vP(a)⩽1 for all a∈A.

Here, sP(a)=1−(μP(a)+vP(a)) is the measure of suspicion of a in P, which excludes the DMS and the DNonMS.

Iseki introduced a special type of algebra namely BCI algebra in 1980.

Definition 3.

An algebra (A,◇,0) is said to be BCI algebra [4] if for any a,b,c∈A, the below stated conditions are meet.

[(a◇b)◇(a◇c)]◇(c◇b)=0
[a◇(a◇b)]◇b=0
a◇a=0
a◇b=0 and b◇a=0⇒a=b

A BCI algebra with the condition 0◇a=0 for all a∈A is called BCK algebra.

A relation “⩽” on A is defined as a⩽b iff a◇b=0.

Proposition 1.

In a BCK algebra (A,◇,0) the followings hold.

0◇a=0
a◇0=a
a◇(a◇b)⩽b
a◇b⩽a
(a◇b)◇c=(a◇c)◇b
(a◇(a◇(a◇b)))=a◇b for all a,b,c∈A

Definition 4.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be a FS in A. Then P is said to be FI [6] of A if

μP(0)⩾μP(a)
μP(a)⩾μP(a◇b)∧μP(b) for all a,b∈A and for l=1,2,…,m

Definition 5.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an IFS in A. Then P is said to IFI [13] of A if

μP(0)⩾μP(a) and vP(0)⩽vP(a)
μP(a)⩾μP(a◇b)∧μP(b) and vP(a)⩽vP(a◇b)∨vP(b) for all a,b∈A

Definition 6.

A BFS [16] P is defined as P=(a,μP(a),vP(a)):a∈A, where μP(a)∈(0,1] measures how much a particular property is satisfied by an element and vP(a)∈[−1,0) measures how much its anti property is satisfied by that element. DMS 0 means the element has no relevancy to the property.

Definition 7.

An m-polar FS [17] P over the set of universe A is an object of the form P={(a,μP(a)):a∈A}, where μP:A→[0,1]m (m is a natural number). Here, [0,1]m is the poset with respect to partial order relation “” which is defined as: a⩽b iff pl(a)⩽pl(b) for l=1,2,…,m; where pl:[0,1]m→[0,1] is called l-th projection mapping.

Including more possible types of uncertainity, Cuong defined PFS in 2013 generalizing the concepts of FS and IFS.

Definition 8.

Let A be the set of universe. Then a PFS [18] P over the universe A is defined as P={(a,μP(a),ηP(a),vP(a)):a∈A}, where μP(a)∈[0,1] is the DPMS of a in P, ηP(a)∈[0,1] is the degree of neutral membership (DNeuMS) of a in P and vP(a)∈[0,1] is the DNegMS of a in P with the condition 0⩽μP(a)+ηP(a)+vP(a)⩽1 for all a∈A. For all a∈A,1−(μP(a)+ηP(a)+vP(a)) is the measure of denial membership a in P. Sometimes, (μP(a),ηP(a),vP(a)) is called picture fuzzy value for a∈A.

Motivated by this definition, below we define m-polar PFS.

Definition 9.

An m-polar PFS P over the set of universe A is an object of the form P={(a,μP(a),ηP(a),vP(a)):a∈A}, where μP:A→[0,1]m, ηP:A→[0,1]m and vP:A→[0,1]m (m is a natural number) with the condition 0⩽pl∘μP(a)+pl∘ηP(a)+pl∘vP(a)⩽1 for all a∈A and for l=1,2,…,m. For a∈A, each of μP(a),ηP(a) and vP(a) is an m-tuple fuzzy value. Here, pl∘μP(a), pl∘ηP(a) and pl∘vP(a) represent l-th components of μP(a), ηP(a) and vP(a) respectively for l=1,2,…,m.

The basic operations on m-polar PFSs consisting of equality, union and intersection are defined below.

Definition 10.

Let P={(a,μP(a),ηP(a),vP(a)):a∈A} and Q={(a,μQ(a),ηQ(a),vQ(a)):a∈A} be two m-polar PFSs over the universe A. Then

P⊆Q iff pl∘μP(a)⩽pl∘μQ(a), pl∘ηP(a)⩽pl∘ηQ(a) and pl∘vP(a)⩾pl∘vQ(a) for all a∈A and for l=1,2,…,m.
P=Q iff pl∘μP(a)=pl∘μQ(a),pl∘ηP(a)=pl∘ηQ(a) and pl∘vP(a)=pl∘vQ(a) for all a∈A and for l=1,2,‥,m.
pl∘(P∪Q)={(a,max(pl∘μP(a),pl∘μQ(a)),min(pl∘ηP(a),pl∘ηQ(a)),min(pl∘vP(a),pl∘vQ(a))):a∈A} for l = 1, 2, … , m.
pl∘(P∩Q)={(a,min(pl∘μP(a),pl∘μQ(a)),min(pl∘ηP(a),pl∘ηQ(a)),max(pl∘vP(a),pl∘vQ(a))):a∈A} for l = 1, 2, … , m.

Definition 11.

Let P={(a,μP,ηP,vP):a∈A} be an m-polar PFS over the universe A. Then (θ,ϕ,ψ)-cut of P is the crisp set in A denoted by Cθ,ϕ,ψ(P) and is defined as Cθ,ϕ,ψ(P)={a∈A:pl∘μP(a)⩾pl∘θ,pl∘ηP(a)⩾pl∘ϕ,pl∘vP(a)⩽pl∘ψ for l=1,2,…,m}, where pl∘θ∈[0,1],pl∘ϕ∈[0,1],pl∘ψ∈[0,1] with the condition 0⩽pl∘θ+pl∘ϕ+pl∘ψ⩽1 for l=1,2,…,m. The mentionable fact is that each of θ, ϕ and ψ is an m-polar fuzzy value. Here, pl∘θ, pl∘ϕ and pl∘ψ represent l-th components of the m-polar fuzzy values θ, ϕ and ψ for l=1,2,…,m

Definition 12.

Let A1 and A2 be two sets of universe. Let h:A1→A2 be a surjective mapping and P={(a1,μP(a1),ηP(a1),vP(a1)):a1∈A1} be an m-polar PFS in A1. Then the image of P under the map h is the m-polar PFS h(P)={(a2,μh(P)(a2),ηh(P)(a2),vh(P)(a2)):a2∈A2}, where pl∘μh(P)(a2)=∨a1∈h−1(a2)pl∘μP(a1), pl∘ηh(P)(a2)=∧a1∈h−1(a2)pl∘ηP(a1) and pl∘vh(P)(a2)=∧a1∈h−1(a2)pl∘vP(a1) for all a2∈A2 and for l=1,2,…,m.

Definition 13.

Let A1 and A2 be two sets of universe. Let h:A1→A2 be a mapping and Q={(a2,μQ(a2),ηQ(a2),vQ(a2)):a2∈A2} be an m-polar PFS in A2. Then the inverse image of Q under the map h is the m-polar PFS h−1(Q)={(a1,μh−1(Q)(a1),ηh−1(Q)(a1),vh−1(Q)(a1)):a1∈A1}, where pl∘μh−1(Q)(a1)=pl∘μQ(h(a1)), pl∘ηh−1(Q)(a1)=pl∘ηQ(h(a1)) and pl∘vh−1(Q)(a1)=pl∘vQ(h(a1)) for all a1∈A1 and for l=1,2,‥,m.

Definition 14.

Let P={(a1,μP(a1),ηP(a1),vP(a1)):a1∈A1} and Q={(a2,μQ(a2),ηQ(a2),vQ(a2)):a2∈A2} be two m-polar PFSs over the sets of universe A1 and A2 respectively. Then the Cartesian product of P and Q is the m-polar PFS P×Q={((a,b),μP×Q((a,b)),ηP×Q((a,b)),vP×Q((a,b))):(a,b)∈A1×A2}, where pl∘μP×Q((a,b))=pl∘μP(a)∧pl∘μQ(b), pl∘ηP×Q((a,b))=pl∘ηP(a)∧pl∘ηQ(b) and pl∘vP×Q((a,b))=pl∘vP(a)∨pl∘vQ(b) for all (a,b)∈A1×A2 and for l=1,2,…,m.

4. m-POLAR PFI

Let us first define m-polar PFSA of a BCK algebra.

Definition 15.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an m-polar PFS in A. Then P is said to be m-polar PFSA of A if pl∘μP(a◇b)⩾pl∘μP(a)∧pl∘μP(b), pl∘ηP(a◇b)⩾pl∘ηP(a)∧pl∘ηP(b) and pl∘vP(a◇b)⩽pl∘vP(a)∨pl∘vP(b) for all a,b∈A and for l=1,2,…,m.

Example 1.

Consider a BCK algebra (A,◇,0) defined in the following tabular form:

◇	0	p	q	r
0	0	0	0	0
p	p	0	0	p
q	q	p	0	q
r	r	r	r	0

Now, let us consider a 3-polar PFS P as follows:

μP(a)=(0.25,0.35,0.4),if a=0(0.15,0.25,0.35),if a=p(0.1,0.15,0.25),if a=q(0.15,0.25,0.4),if a=r

ηP(a)=(0.2,0.3,0.4),if a=0(0.1,0.2,0.3),if a=p(0.05,0.1,0.2),if a=q(0.1,0.2,0.35),if a=r

and

vP(a)=(0.1,0.15,0.2),if a=0(0.15,0.2,0.3),if a=p(0.2,0.3,0.4),if a=q(0.15,0.2,0.25),if a=r

It is easy to show that P is a 3-polar PFSA of A.

Proposition 2.

Let P=(μP,ηP,vP) be an m-polar PFSA of a BCK algebra A. Then pl∘μP(0)⩾pl∘μP(a), pl∘ηP(0)⩾pl∘ηP(a) and pl∘vP(0)⩽pl∘vP(a) for all a∈A and for l=1,2,…,m.

Proof.

It is observed that

pl∘μP(0)=pl∘μP(a◇a) ⩾pl∘μP(a)∧pl∘μP(a) [because P is an m-polar PFSA of A] =pl∘μP(a),

pl∘ηP(0)=pl∘ηP(a◇a) ⩾pl∘ηP(a)∧pl∘ηP(a) [because P is an m-polar PFSA of A] =pl∘ηP(a)

and pl∘vP(0)=pl∘vP(a◇a) ⩽pl∘vP(a)∨pl∘vP(a) [because P is an m-polar PFSA of A] =pl∘vP(a) for all a∈A and for l=1,2,…,m.

Thus, pl∘μP(0)⩾pl∘μP(a), pl∘ηP(0)⩾pl∘ηP(a) and pl∘vP(0)⩽pl∘vP(a) for all a∈A and for l=1,2,…,m.

Now, let us define m-polar PFI of a BCK algebra.

Definition 16.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an m-polar PFS in A. Then P is said to m-polar PFI of A if

pl∘μP(0)⩾pl∘μP(a), pl∘ηP(0)⩾pl∘ηP(a) and pl∘vP(0)⩽pl∘vP(a)
pl∘μP(a)⩾pl∘μP(a◇b)∧pl∘μP(b), pl∘ηP(a)⩾pl∘ηP(a◇b)∧pl∘ηP(b) and pl∘vP(a)⩽pl∘vP(a◇b)∨pl∘vP(b) for all a,b∈A and for l=1,2,…,m

Now, we are going to investigate some important results on m-polar PFI of a BCK algebra.

Proposition 3.

Let P=(μP,ηP,vP) be an m-polar PFI of a BCK algebra (A,◇,0). Then pl∘μP(a)⩾pl∘μP(b), pl∘ηP(a)⩾pl∘ηP(b) and pl∘vP(a)⩽pl∘vP(b) for a,b∈A with a⩽b and for l=1,2,…,m.

Proof.

Let a,b∈A such that a⩽b. Then a◇b=0.

Now,pl∘μP(a) ⩾pl∘μP(a◇b)∧pl∘μP(b) [as P is an m-polar PFI of A] =pl∘μP(0)∧pl∘μP(b) =pl∘μP(b) [as P is an m-polar PFI of A],

pl∘ηP(a) ⩾pl∘ηP(a◇b)∧pl∘ηP(b) [as P is an m-polar PFI of A] =pl∘ηP(0)∧pl∘ηP(b) =pl∘ηP(b) [as P is an m-polar PFI of A]

and pl∘vP(a) ⩽pl∘vP(a◇b)∨pl∘vP(b) [as P is an m-polar PFI of A] =pl∘vP(0)∨pl∘vP(b) =pl∘vP(b) [as P is an m-polar PFI of A] for l=1,2,…,m.

Thus, pl∘μP(a)⩾pl∘μP(b), pl∘ηP(a)⩾pl∘ηP(b) and pl∘vP(a)⩽pl∘vP(b) for a,b∈A with a⩽b and for l=1,2,…,m.

Proposition 4.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an m-polar PFI of A. Then pl∘μP(a)⩾pl∘μP(b)∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP(b)∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP(b)∨pl∘vP(c) for a,b,c∈A with a◇b⩽c.

Proof.

Let a,b,c∈A with a◇b⩽c. Then (a◇b)◇c=0.

Now,pl∘μP(a) ⩾pl∘μP(a◇b)∧pl∘μP(b) [because P is an m-polar PFI of A] ⩾pl∘μP((a◇b)◇c)∧pl∘μP(c)∧pl∘μP(b) [because P is an m-polar PFI of A] =pl∘μP(0)∧pl∘μP(c)∧pl∘μP(b) =pl∘μP(b)∧pl∘μP(c) [because P is an m-polar PFI of A],

pl∘ηP(a) ⩾pl∘ηP(a◇b)∧pl∘ηP(b) [because P is an m-polar PFI of A] ⩾pl∘ηP((a◇b)◇c)∧pl∘ηP(c)∧pl∘ηP(b) [because P is an m-polar PFI of A] =pl∘ηP(0)∧pl∘ηP(c)∧pl∘ηP(b) =pl∘ηP(b)∧pl∘ηP(c) [because P is an m-polar PFI of A]

and pl∘vP(a) ⩽pl∘vP(a◇b)∨pl∘vP(b) [because P is an m-polar PFI of A] ⩽pl∘vP((a◇b)◇c)∨pl∘vP(c)∨pl∘vP(b) [because P is an m-polar PFI of A] =pl∘vP(0)∨pl∘vP(c)∨pl∘vP(b) =pl∘vP(b)∨pl∘vP(c) [because P is an m-polar PFI of A] for l=1,2,…,m.

Thus, it is obtained that pl∘μP(a)⩾pl∘μP(b)∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP(b)∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP(b)∨pl∘vP(c) for a,b,c∈A with a◇b⩽c.

Proposition 5.

Every m-polar PFI of a BCK algebra is an m-polar PFSA.

Proof.

Let (A,◇,0) be a BCK algebra and A is an m-PFI of A. Since P is an m-polar PFI, therefore,

pl∘μP(a◇b) ⩾pl∘μP((a◇b)◇a)∧pl∘μP(a) =pl∘μP((a◇a)◇b)∧pl∘μP(a) [by Proposition 1] =pl∘μP(0◇b)∧pl∘μP(a) =pl∘μP(0)◇pl∘μP(a) [by Proposition 1] ⩾pl∘μP(a)◇pl∘μP(b),

pl∘ηP(a◇b) ⩾pl∘ηP((a◇b)◇a)∧pl∘ηP(a) =pl∘ηP((a◇a)◇b)∧pl∘ηP(a) [by Proposition 1] =pl∘ηP(0◇b)∧pl∘ηP(a) =pl∘ηP(0)∧pl∘ηP(a) [by Proposition 1] ⩾pl∘ηP(a)∧pl∘ηP(b) and pl∘vP(a◇b) ⩽pl∘vP((a◇b)◇a)∨pl∘vP(a) =pl∘vP((a◇a)◇b)∨pl∘vP(a) [by Proposition 1] =pl∘vP(0◇b)∨pl∘vP(a) =pl∘vP(0)∨pl∘vP(a) [by Proposition 1] ⩽pl∘vP(a)∨pl∘vP(b), for all a,b∈A and for l=1,2,…,m.

Hence, P is an m-polar PFSA of A.

But, the converse of the above proposition is not true in general which is shown in following example. Proposition 6 states under which condition an m-polar PFSA is an m-polar PFI.

Example 2.

Let us suppose the BCK algebra given in Example 1 and a 3-polar PFS P as follows:

μP(a)=(0.2,0.3,0.4),if a=0,q(0.1,0.2,0.3),if a=p,r

ηP(a)=(0.25,0.35,0.45),if a=0,q(0.15,0.25,0.3),if a=p,r

and

vP(a)=(0.3,0.2,0.1),if a=0,q(0.4,0.3,0.2),if a=p,r

Here, (0.1,0.2,0.3)=μP(p)≱μP(p◇q)∧μP(q)=(0.2,0.3,0.4), (0.15,0.25,0.3)=ηP(p)≱ηP(p◇q)∧ηP(q)=(0.25,0.35,0.45) and (0.4,0.3,0.2)=vP(p)≰vP(p◇q)∨vP(q)=(0.3,0.2,0.1). So, P is not a 3-polar PFI of A although it is a 3-polar PFSA.

Proposition 6.

Let P=(μP,ηP,vP) be an m-polar PFSA of a BCK algebra (A,◇,0). Then P is an m-polar PFI of A if for all a,b,c∈A, a◇b⩽c⇒pl∘μP(a)⩾pl∘μP(b)∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP(b)∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP(b)∨pl∘vP(c) for l=1,2,…,m.

Proof.

By given conditions, for all a,b,c∈A, a◇b⩽c⇒pl∘μP(a)⩾pl∘μP(b)∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP(b)∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP(b)∨pl∘vP(c). Since A is a BCK algebra therefore by Proposition 1, a◇(a◇b)⩽b. So, it is obtained that

pl∘μP(a)⩾pl∘μP(a◇b)∧pl∘μP(b) pl∘ηP(a)⩾pl∘μP(a◇b)∧pl∘μP(b) and pl∘vP(a)⩽pl∘vP(a◇b)∨pl∘vP(b) for l=1,2,…,m

Thus, P is an m-polar PFI of A.

Proposition 7.

Let (A,◇,0) a BCK algebra and P=(μp,ηP,vP), Q=(μQ,ηQ,vQ) be two m-polar PFIs of A. Then P∩Q is an m-polar PFI of A.

Proof.

Let P∩Q=R=(μR,ηR,vR). Then pl∘μR(a)=pl∘μP(a)∧pl∘μQ(a), pl∘ηR(a)=pl∘ηP(a)∧pl∘ηQ(a) and pl∘vR(a)=pl∘vP(a)∨pl∘vQ(a),∀a∈A and for l=1,2,…,m.

Now,pl∘μR(0)=pl∘μP(0)∧pl∘μQ(0)⩾pl∘μP(a)∧pl∘μQ(a) [as P,Q are m-polar PFIs of A]=pl∘μR(a)

pl∘ηR(0) =pl∘ηP(0)∧pl∘ηQ(0) ⩾pl∘ηP(a)∧pl∘ηQ(a) [as P,Q are m-polar PFIs of A] =pl∘ηR(a)

and pl∘vR(0) =pl∘vP(0)∨pl∘vQ(0) ⩽pl∘vP(a)∨pl∘vQ(a) [as P,Q are m-polar PFIs of A] =pl∘vR(a),∀a∈A and l=1,2,…,m.

Also,pl∘μR(a) =pl∘μP(a)∧pl∘μQ(a) ⩾(pl∘μP(a◇b)∧pl∘μP(b))∧(pl∘μQ(a◇b)∧pl∘μQ(b)) [as P,Q are m-polar PFIs of A] =(pl∘μP(a◇b)∧pl∘μQ(a◇b))∧(pl∘μP(b)∧pl∘μQ(b)) =pl∘μR(a◇b)∧pl∘μR(b),

pl∘ηR(a) =pl∘ηP(a)∧pl∘ηQ(a) ⩾(pl∘ηP(a◇b)∧pl∘ηP(b))∧(pl∘ηQ(a◇b)∧pl∘ηQ(b)) [as P,Q are m-polar PFIs of A] =(pl∘ηP(a◇b)∧pl∘ηQ(a◇b))∧(pl∘ηP(b)∧pl∘ηQ(b)) =pl∘ηR(a◇b)∧pl∘ηR(b)

and pl∘vR(a) =pl∘vP(a)∨pl∘vQ(a) ⩽(pl∘vP(a◇b)∨pl∘vP(b))∨(pl∘vQ(a◇b)∨pl∘vQ(b)) [as P,Q are m-polar PFIs of A] =(pl∘vP(a◇b)∨pl∘vQ(a◇b))∨(pl∘vP(b)∨pl∘vQ(b)) =pl∘vR(a◇b)∨pl∘vR(b),∀a,b∈A and l=1,2,…,m.

Thus, pl∘μR(a)⩾pl∘μR(a◇b)∧pl∘μR(b), pl∘ηR(a)⩾pl∘ηR(a◇b)∧pl∘ηR(b) and pl∘vR(a◇b)∨pl∘vR(b),∀a,b∈A and for l=1,2,…,m. Consequently, R=P∩Q is an m-polar PFI of A.

Proposition 8.

Let P=(μP,ηP,vP) and Q=(μQ,ηQ,vQ) be two m-polar PFIs of a BCK algebra (A,◇,0). Then P×Q is an m-polar PFI of A×A.

Proof.

Proof is same as Proposition 7. So, it is omitted.

Proposition 9.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an m-polar PFI of A. Then Cθ,ϕ,ψ(P) is a crisp ideal of A, provided that Pl∘μP(0)⩾pl∘θ, pl∘ηP(0)⩾pl∘ϕ and pl∘vP(0)⩽pl∘ψ for l=1,2,…,m.

Proof.

Clearly, Cθ,ϕ,ψ(P) contains at least one element. Let a◇b, b∈Cθ,ϕ,ψ(P). Then pl∘μP(a◇b)⩾pl∘θ, pl∘ηP(a◇b)⩾pl∘ϕ, pl∘vP(a◇b)⩽ψ and pl∘μP(b)⩾pl∘θ, pl∘ηP(b)⩾pl∘ϕ, pl∘vP(b)⩽pl∘ψ for l=1,2,…,m.

Now,pl∘μP(a) ⩾pl∘μP(a◇b)∧pl∘μP(b) [because P is an m-polar PFI of A] ⩾pl∘θ∧pl∘θ=pl∘θ,

pl∘ηP(a) ⩾pl∘ηP(a◇b)∧pl∘ηP(b) [because P is an m-polar PFI of A] ⩾pl∘ϕ∧pl∘ϕ=pl∘ϕ

and pl∘vP(a) ⩽pl∘vP(a◇b)∨pl∘vP(b) [because P is an m-polar PFI of A] ⩽pl∘ψ∨pl∘ψ=pl∘ψ for l=1,2,‥‥m

Thus, a◇b, b∈Cθ,ϕ,ψ(P)⇒a∈Cθ,ϕ,ψ(P). So, Cθ,ϕ,ψ(P) is a crisp ideal of A.

Proposition 10.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an m-polar PFS in A. Then P is an m-polar PFI of A if all (θ,ϕ,ψ)-cuts of P are crisp ideals of A.

Proof.

Let a,b∈A. Let pl∘μP(a◇b)∧pl∘μP(b)=pl∘θ, pl∘ηP(a◇b)∧pl∘ηP(b)=pl∘ϕ and pl∘vP(a◇b)∨pl∘vP(b)=pl∘ψ for l=1,2,…,m. Clearly, pl∘θ∈[0,1], pl∘ϕ∈[0,1] and pl∘ψ∈[0,1] with 0⩽pl∘θ+pl∘ϕ+pl∘ψ⩽1 for l=1,2,…,m.

Now,pl∘μP(a◇b)⩾pl∘μP(a◇b)∧pl∘μP(b)=pl∘θ, pl∘ηP(a◇b)⩾pl∘μP(a◇b)∧pl∘μP(b)=pl∘ϕ and pl∘vP(a◇b)⩽pl∘vP(a◇b)∨pl∘vP(b)=pl∘ψ for l=1,2,…,m.

Also,pl∘μP(b)⩾pl∘μP(a◇b)∧pl∘μP(b)=pl∘θ, pl∘ηP(b)⩾pl∘μP(a◇b)∧pl∘μP(b)=pl∘ϕ and pl∘vP(b)⩽pl∘vP(a◇b)∨pl∘vP(b)=pl∘ψ for l=1,2,…,m.

Thus, a◇b and b∈Cθ,ϕ,ψ(P). Since Cθ,ϕ,ψ(P) is a crisp ideal of A therefore a◇b∈Cθ,ϕ,ψ(P) and b∈Cθ,ϕ,ψ(P)⇒a∈Cθ,ϕ,ψ(P).

Therefore, pl∘μP(a)⩾pl∘θ=pl∘μP(a◇b)∧pl∘μP(b), pl∘ηP(a)⩾pl∘ϕ=pl∘ηP(a◇b)∧pl∘ηP(b) and pl∘vP(a)⩽pl∘ψ=pl∘vP(a◇b)∨pl∘vP(b) for l=1,2,‥,m.

Since a,b are arbitrary elements of A therefore pl∘μP(a)⩾pl∘μP(a◇b)∧pl∘μP(b), pl∘ηP(a)⩾pl∘ηP(a◇b)∧pl∘ηP(b) and pl∘vP(a)⩽pl∘vP(a◇b)∨pl∘vP(b) for all a,b∈A and for l=1,2,…,m. Hence, P is an m-polar PFI of A.

5. PRE-IMAGE AND IMAGE PFI UNDER HOMOMORPHISM OF BCK ALGEBRA

In the current section, we explore some properties of m-polar PFI of BCK algebra under homomorphism of BCK algebra.

Definition 17.

Let (A1,◇,0) and (A2,∗,0) be two BCK algebras. Then a mapping h:A1→A2 is said to be homomorphism if h(a◇b)=h(a)∗h(b) for all a,b∈A1.

It is observed that h(a)∗h(a)=0 i.e. h(a◇a)=0 i.e. h(0)=0.

Proposition 11.

Let (A1,◇,0) and (A2,∗,0) be two BCK algebras and Q=(μQ,ηQ,vQ) be an m-polar PFI of A2. Then for a BCK algebra homomorphism h:A1→A2, h−1(Q) is an m-polar PFI of A1.

Proof.

Let h−1(Q)=(μh−1(Q),ηh−1(Q),vh−1(Q)), where μh−1(Q)=μQ(h(a)), ηh−1(Q)(a)=ηQ(h(a)) and vh−1(Q)(a)=vQ(h(a)) for all a∈A1.

Now,pl∘μh−1(Q)(0) =pl∘μQ(h(0)) =pl∘μQ(0) [as h(0)=0] ⩾pl∘μQ(h(a)) [because Q is an m-polar PFI of A2] =pl∘μh−1(Q)(a),

pl∘ηh−1(Q)(0) =pl∘ηQ(h(0)) =pl∘ηQ(0) [as h(0)=0] ⩾pl∘ηQ(h(a)) [because Q is an m-polar PFI of A2] =pl∘ηh−1(Q)(a)

and pl∘vh−1(Q)(0) =pl∘vQ(h(0)) =pl∘vQ(0) [as h(0)=0] ⩽pl∘vQ(h(a)) [because Q is an m-polar PFI of A2] =pl∘vh−1(Q)(a) for all a∈A1 and for l=1,2,…,m.

Thus, pl∘μh−1(Q)(0)⩾pl∘μh−1(Q)(a), pl∘ηh−1(Q)(0)⩾pl∘ηh−1(Q)(a) and pl∘vh−1(Q)(0)⩽pl∘vh−1(Q)(a) for all a∈A1 and for l=1,2,…,m.

Also,pl∘μh−1(Q)(a) =pl∘μQ(h(a)) ⩾pl∘μQ(h(a)∗h(b))∧pl∘μQ(h(b)) [because Q is an m-polar PFI of A2] =pl∘μQ(h(a◇b))∧pl∘μQ(h(b)) [because h is a homomorphism] =pl∘μh−1(Q)(a◇b)∧pl∘μh−1(Q)(b),

pl∘ηh−1(Q)(a) =pl∘ηQ(h(a)) ⩾pl∘ηQ(h(a)∗h(b))∧pl∘ηQ(h(b)) [because Q is an m-polar PFI of A2] =pl∘ηQ(h(a◇b))∧pl∘ηQ(h(b)) [because h is a homomorphism] =pl∘ηh−1(Q)(a◇b)∧pl∘ηh−1(Q)(b)

and pl∘vh−1(Q)(a) =pl∘vQ(h(a)) ⩽pl∘vQ(h(a)∗h(b))∨pl∘vQ(h(b)) [because Qis an m-polar PFI of A2] =pl∘vQ(h(a◇b))∨pl∘vQ(h(b)) [because h is a homomorphism] =pl∘vh−1(Q)(a◇b)∨pl∘vh−1(Q)(b) for all a,b∈A1 and for l=1,2,…,m.

Thus, pl∘μh−1(Q)(a)⩾pl∘μh−1(Q)(a◇b)∧pl∘μh−1(Q)(b), pl∘ηh−1(Q)(a)⩾pl∘ηh−1(a◇b)∧pl∘ηh−1(Q)(b) and pl∘vh−1(Q)(a)⩽pl∘vh−1(Q)(a◇b)∨pl∘vh−1(Q)(b) for all a,b∈A1 and for l=1,2,…,m. Hence, h−1(Q) is an m-polar PFI of A1.

Proposition 12.

Let (A1,◇) and (A2,∗) be two BCK algebras and P=(μP,ηP,vP) be an m-polar PFI of A1. Then for a bijective homomorphism h:A1→A2, h(P) is an m-polar PFI of A2.

Proof.

Let h(P)=(μh(P),ηh(P),vh(P)). Now, let b∈A2.

Then pl∘μh(P)(b)=∨a∈h−1(b)pl∘μP(a), pl∘ηh(P)(b)=∧a∈h−1(b)pl∘ηP(a) and pl∘vh(P)(b)=∧a∈h−1(b)pl∘vP(a) for l=1,2,‥,m.

Since h is bijective therefore h−1(b) must be a singleton set. So, for b∈A2, there exists an unique a∈A1 such that a=h−1(b) i.e. h(a)=b. Thus, in this case, pl∘μh(P)(b)=pl∘μh(P)(h(a))=pl∘μP(a), pl∘ηh(P)(b)=pl∘ηh(P)(h(a))=pl∘ηP(a) and pl∘vh(P)(b)=pl∘vh(P)(h(a))=pl∘vP(a) for l=1,2,…,m.

Now,pl∘μh(P)(0) =pl∘μh(P)(h(0)) [as h(0)=0] =pl∘μP(0) ⩾pl∘μP(a) =pl∘μh(P)(h(a)) =pl∘μh(P)(b),

pl∘ηh(P)(0) =pl∘ηh(P)(h(0)) [as h(0)=0] =pl∘ηP(0) ⩾pl∘ηP(a) =pl∘ηh(P)(h(a)) =pl∘ηh(P)(b)

and pl∘vh(P)(0) =pl∘vh(P)(h(0)) [as h(0)=0] =pl∘vP(0) ⩽pl∘vP(a) =pl∘vh(P)(h(a)) =pl∘vh(P)(b) for l=1,2,…,m.

Since b is an arbitrary element of A2 therefore pl∘μh(P)(0)⩾pl∘μh(P)(b), pl∘ηh(P)(0)⩾pl∘ηh(P)(b) and pl∘vh(P)(0)⩽pl∘vh(P)(b) for all b∈A2 and for l=1,2,…,m.

Also,pl∘μh(P)(b) =pl∘μh(P)(h(a)) [where b=h(a) for unique a∈A1] =pl∘μP(a) ⩾pl∘μP(a◇c)∧pl∘μP(c) [as P is an m-polar PFI of A1] =pl∘μh(P)(h(a◇c))∧pl∘μh(P)(h(c)) =pl∘μh(P)(h(a)∗h(c))∧pl∘μh(P)(h(c)) [as h is a homomorphism] =pl∘μh(P)(b∗h(c))∧pl∘μh(P)(h(c)),

pl∘ηh(P)(b) =pl∘ηh(P)(h(a)) [where b=h(a) for unique a∈A1] =pl∘ηP(a) ⩾pl∘ηP(a◇c)∧pl∘ηP(c) [as P is an m-polar PFI of A1] =pl∘ηh(P)(h(a◇c))∧pl∘ηh(P)(h(c)) =pl∘ηh(P)(h(a)∗h(c))∧pl∘ηh(P)(h(c)) [as h is a homomorphism] =pl∘ηh(P)(b∗h(c))∧pl∘ηh(P)(h(c))

and pl∘vh(P)(b) =pl∘vh(P)(h(a)) [where b=h(a) for unique a∈A1] =pl∘vP(a) ⩽pl∘vP(a◇c)∨pl∘vP(c) [as P is an m-polar PFI of A1] =pl∘vh(P)(h(a◇c))∨pl∘vh(P)(h(c)) =pl∘vh(P)(h(a)∗h(c))∨pl∘vh(P)(h(c)) [as h is a homomorphism] =pl∘vh(P)(b∗h(c))∨pl∘vh(P)(h(c)) for all c∈A1 and for l=1,2,…,m.

Thus, pl∘μh(P)(b)⩾pl∘μh(P)(b∗h(c))∧pl∘μh(P)(h(c)), pl∘ηh(P)(b)⩾pl∘ηh(P)(b∗h(c))∧pl∘ηh(P)(h(c)) and pl∘vh(P)(b)⩽pl∘vh(P)(b∗h(c))∨pl∘vh(P)(h(c)) for all c∈A1 and for l=1,2,…,m. Since h is bijective therefore h(A1)=A2. So, for all c∈A1, h(c) can capture all the elements of A2. Letting d=h(c), it is observed that the inequalities hold for all d∈A2. Since b is arbitrary therefore we obtain that pl∘μh(P)(b)⩾pl∘μh(P)(b◇d)∧pl∘μh(P)(d), pl∘ηh(P)(b)⩾pl∘ηh(P)(b◇d)∧pl∘ηh(P)(d) and pl∘vh(P)(b)⩽pl∘vh(P)(b◇d)∨pl∘vh(P)(d) for all b,d∈A2 and for l=1,2,…,m. Hence, h(P) is an m-polar PFI of A2.

6. m-POLAR PFII

The current section introduces the concept of implicative BCK algebra, m-polar PFII of a BCK algebra and studies some properties related to these. We also investigate a relationship between m-polar PFI and m-polar PFII of a BCK algebra.

Definition 18.

A BCK algebra (A,◇,0) is said to be implicative if a=(a◇b)◇a for all a,b∈A.

Proposition 13.

An m-polar PFS P=(μP,ηP,vP) in a BCK algebra (A,◇,0) is said to be m-polar PFII of A if the below stated conditions are meet.

pl∘μP(0)⩾pl∘μP(a), pl∘ηP(0)⩾pl∘ηP(a) and pl∘vP(0)⩽pl∘vP(a)
pl∘μP(a)⩾pl∘μP{(a◇(b◇a))◇c}∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP{(a◇(b◇a))◇c}∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP{(a◇(b◇a))◇c}∨pl∘vP(c) for all a,b,c∈A and for l=1,2,…,m

Example 3.

Let us consider the BCK algebra (A,◇) as follows:

◇	0	p	q	r	s
0	0	0	0	0	0
p	p	0	p	0	0
q	q	q	0	0	0
r	r	r	r	0	0
s	s	r	s	p	0

Let us consider a 3-polar PFS P=(μP,ηP,vP) as follows:

μP(a)=(0.39,0.41,0.42),if a=0,p,q(0.25,0.27,0.3),if a=r,s

ηP(a)=(0.37,0.39,0.4),if a=0,p,q(0.29,0.33,0.35),if a=r,s

and

vP(a)=(0.14,0.17,0.18),if a=0,p,q(0.3,0.32,0.35),if a=r,s

It can be easily shown that P is a 3-polar PFII of A.

Proposition 14.

Every m-polar PFII of a BCK algebra (A,◇,0) is an m-polar PFI of A.

Proof.

Let P=(μP,ηP,vP) be an m-polar PFII of A.

Then pl∘μP(a)⩾pl∘μP{(a◇(b◇a))◇c}∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP{(a◇(b◇a))◇c}∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP{(a◇(b◇a))◇c}∨pl∘vP(c) for all a,b,c∈A and for l=1,2,…,m.

Setting b=a, it is obtained that

pl∘μP(a)⩾pl∘μP{(a◇(a◇a))◇c}∧pl∘μP(c) =pl∘μP{(a◇0)◇c}∧pl∘μP(c) =pl∘μP(a◇c)∧pl∘μP(c) [by Proposition 1]

pl∘ηP(a)⩾pl∘ηP{(a◇(a◇a))◇c}∧pl∘ηP(c) =pl∘ηP{(a◇0)◇c}∧pl∘ηP(c) =pl∘ηP(a◇c)∧pl∘ηP(c) [by Proposition 1]

and pl∘vP(a)⩽pl∘vP{(a◇(a◇a))◇c}∨pl∘vP(c) =pl∘vP{(a◇0)◇c}∨pl∘vP(c) =pl∘vP(a◇c)∨pl∘vP(c) [by Proposition 1] for all a,c∈A and for l=1,2,…,m.

Therefore, P is an m-polar PFI of A.

The above proposition does not hold in reverse direction i.e. an m-polar PFI of a BCK algebra is not necessarily m-polar PFII which is clear from the following example. It is necessary to mention that in an implicative BCK algebra, the converse of the above proposition holds which is shown through Proposition 15.

Example 4.

Now, let us consider a 3-polar PFS P=(μP,ηP,vP) in BCK algebra A given in Example 3 as follows:

μP(a)=(0.42,0.43,0.45),if a=0,q(0.25,0.27,0.3),if a=p,r,s

ηP(a)=(0.3,0.33,0.35),if a=0,q(0.15,0.18,0.2),if a=p,r,s

and

vP(a)=(0.14,0.16,0.2),if a=0,q(0.45,0.48,0.5),if a=p,r,s

It is clear that (0.25,0.27,0.3)=μP(p)≱μP{(p◇(r◇p))◇q}∧μP(q)=(0.42,0.43,0.45)∧(0.42,0.43,0.45)=(0.42,0.43,0.45), (0.15,0.18,0.2)=ηP(p)≱ηP{(p◇(r◇p))◇q}∧ηP(q)=(0.3,0.33,0.35)∧(0.3,0.33,0.35)=(0.3,0.33,0.35) and (0.45,0.48,0.5)=vP(p)≰vP{(p◇(q◇p))◇q}∧vP(q)=(0.14,0.16,0.2)∨(0.14,0.16,0.2)=(0.14,0.16,0.2). Thus, P is not 3-polar PFII although it is a 3-polar PFI of A.

Proposition 15.

In an implicative BCK algebra, every m-polar PFI is m-polar PFII.

Proof.

Let (A,◇,0) be an implicative BCK algebra. Therefore, a=(a◇b)◇a for all a,b∈A. Let P=(μP,ηP,vP) be an m-polar PFI of A. Then

pl∘μP(a)⩾pl∘μP(a◇c)∧pl∘μP(c) =pl∘μP{((a◇b)◇a)◇c}∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP(a◇c)∧pl∘ηP(c) =pl∘ηP{((a◇b)◇a)◇c}∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP(a◇c)∨pl∘vP(c) =pl∘vP{((a◇b)◇a)◇c}∨pl∘vP(c) for all a,b,c∈A and for l=1,2,…,m.

Thus, P is an m-polar PFII of A.

Proposition 16.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an m-polar PFII of A. Then Cθ,ϕ,ψ(P) is an implicative ideal of A, provided that pl∘μP(0)⩾pl∘θ, pl∘ηP(0)⩾pl∘ϕ and pl∘vP(0)⩽pl∘ψ for l=1,2,‥‥m.

Proof.

Clearly, Cθ,ϕ,ψ(P) contains at least one element. Let ((a◇b)◇a)◇c, c∈Cθ,ϕ,ψ(P). Then pl∘μP{((a◇b)◇a)◇c}⩾pl∘θ, pl∘ηP{((a◇b)◇a)◇c}⩾pl∘ϕ, pl∘vP{((a◇b)◇a)◇c}⩽pl∘ψ and pl∘μP(c)⩾pl∘θ, pl∘ηP(c)⩾pl∘ϕ, pl∘vP(c)⩽pl∘ψ for l=1,2,…,m.

Now,pl∘μP(a)⩾pl∘μP{((a◇b)◇a)◇c}∧pl∘μP(c) [because P is an m-polar PFII of A] ⩾pl∘θ∧pl∘θ=pl∘θ,

pl∘ηP(a)⩾pl∘ηP{((a◇b)◇a)◇c}∧pl∘ηP(c) [because P is an m-polar PFII of A] ⩾pl∘ϕ∧pl∘ϕ=pl∘ϕ

and pl∘vP(a)⩽pl∘vP{((a◇b)◇a)◇c}∨pl∘vP(c) [because P is an m-polar PFII of A] ⩽pl∘ψ∨pl∘ψ=pl∘ψ for l=1,2,…,m.

Thus, it is observed that {((a◇b)◇a)◇c}, c∈Cθ,ϕ,ψ(P)⇒a∈Cθ,ϕ,ψ(P). So, Cθ,ϕ,ψ(P) is an implicative ideal of A.

Proposition 17.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an m-polar PFS in A. Then P is an m-polar PFII of A if all (θ,ϕ,ψ)-cuts of P are implicative ideals of A.

Proof.

Let a,b∈A. Let pl∘μP{((a◇b)◇a)◇c}∧pl∘μP(c)=pl∘θ, pl∘ηP{((a◇b)◇a)◇c}∧pl∘ηP(c)=pl∘ϕ and pl∘vP{((a◇b)◇a)◇c}∨pl∘vP(c)=pl∘ψ for l=1,2,…,m. Clearly, pl∘θ∈[0,1], pl∘ϕ∈[0,1] and pl∘ψ∈[0,1] with 0⩽pl∘θ+pl∘ϕ+pl∘ψ⩽1 for l=1,2,…,m.

Now,pl∘μP{((a◇b)◇a)◇c}⩾pl∘μP{((a◇b)◇a)◇c} ∧pl∘μP(c) =pl∘θ, pl∘ηP{((a◇b)◇a)◇c}⩾pl∘ηP{((a◇b)◇a)◇c} ∧pl∘ηP(c) =pl∘ϕ and pl∘vP{((a◇b)◇a)◇c}⩽pl∘vP{((a◇b)◇a)◇c} ∨pl∘vP(c) =pl∘ψ for l=1,2,…,m.

Also,pl∘μP(c)⩾pl∘μP{((a◇b)◇a)◇c} ∧pl∘μP(c)=pl∘θ, pl∘ηP(c)⩾pl∘ηP{((a◇b)◇a)◇c} ∧pl∘ηP(c)=pl∘ϕ and pl∘vP(c)⩽pl∘vP{((a◇b)◇a)◇c} ∨pl∘vP(c)=pl∘ψ for l=1,2,…,m.

Thus, ((a◇b)◇a)◇c and c∈Cθ,ϕ,ψ(P). Since Cθ,ϕ,ψ(P) is an implicative ideal of A therefore a∈Cθ,ϕ,ψ(P).

Therefore, pl∘μP(a)⩾pl∘θ=pl∘μP{((a◇b)◇a)◇c}∧pl∘μP(c), pl∘ηP(a)⩾pl∘ϕ=pl∘ηP{((a◇b)◇a)◇c}∧pl∘ηP(c) and pl∘vP(a)⩽pl∘ψ=pl∘vP{((a◇b)◇a)◇c}∨pl∘vP(c) for l=1,2,…,m.

Since a,b,c are arbitrary elements of A therefore pl∘μP(a)⩾pl∘μP{((a◇b)◇a)◇c}∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP{((a◇b)◇a)◇c}∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP{((a◇b)◇a)◇c}∨pl∘vP(c) for all a,b,c∈A and for l=1,2,…,m. Hence, P is an m-polar PFII of A.

Proposition 18.

Let S1 and S2 be two ideals of a BCK algebra (A,◇,0) such that S1⊆S2. If S1 is implicative then S2 also.

Proposition 19.

Let P1 and P2 be two m-polar PFIs of a BCK algebra (A,◇,0) with P1⊆P2. If P1 is m-polar PFII of A then P2 also.

Proof.

Let a∈Cθ,ϕ,ψ(P1). Then pl∘μP1(a)⩾pl∘θ, pl∘ηP1(a)⩾pl∘ϕ and pl∘vP1(a)⩽pl∘ψ for l=1,2,…,m. Now, P1⊆P2⇒pl∘μP1(a)⩽pl∘μP2(a),pl∘ηP1(a)⩽pl∘ηP2(a) and pl∘vP1(a)⩾pl∘vP2(a) for l=1,2,…,m. It follows that pl∘μP2(a)⩾pl∘θ, pl∘ηP2(a)⩾pl∘ϕ and pl∘vP2(a)⩽pl∘ψ for l=1,2,…,m. Thus, a∈Cθ,ϕ,ψ(P2). As a result, Cθ,ϕ,ψ(P1)⊆Cθ,ϕ,ψ(P2). Since P1 is an m-polar PFII of A therefore Cθ,ϕ,ψ(P1) is implicative ideal of A by Proposition 16. By Proposition 18, Cθ,ϕ,ψ(P2) is implicative ideal of A. Therefore, by Proposition 17, P2 is an m-polar PFII of A.

Proposition 20.

Let P=(μP,ηP,vP) be an m-polar PFI of a BCK algebra A. Then the below stated statements are equivalent.

P is m-polar PFII.
pl∘μP(a)⩾pl∘μP(a◇(b◇a)), pl∘ηP(a)⩾pl∘ηP(a◇(b◇a)) and pl∘vP(a)⩽pl∘vP(a◇(b◇a)) for all a,b∈A and for l=1,2,…,m.
pl∘μP(a)=pl∘μP(a◇(b◇a)), pl∘ηP(a)=pl∘ηP(a◇(b◇a)) and pl∘vP(a)=pl∘vP(a◇(b◇a)) for all a,b∈A and for l=1,2,…,m.

Proof.

(i)⇒(ii): Since P is an m-polar PFII of A, therefore,

pl∘μP(a)⩾pl∘μP{(a◇(b◇a))◇0}∧pl∘μP(0) =pl∘μP(a◇(b◇a))∧pl∘μP(0) [by Proposition 1] =pl∘μP(a◇(b◇a))

pl∘ηP(a)⩾pl∘ηP{(a◇(b◇a))◇0}∧pl∘ηP(0) =pl∘ηP(a◇(b◇a))∧pl∘ηP(0) [by Proposition 1] =pl∘ηP(a◇(b◇a)) and pl∘vP(a)⩽pl∘vP{(a◇(b◇a))◇0}∨pl∘vP(0) =pl∘vP(a◇(b◇a))∨pl∘vP(0) [by Proposition 1] =pl∘vP(a◇(b◇a)) for all a,b∈A and for l=1,2,…,m.

(ii)⇒(iii): It is known by Proposition 1 that a◇(b◇a)⩽a. Then by Proposition 3, pl∘μP(a)⩽pl∘μP(a◇(b◇a)), pl∘ηP(a)⩽pl∘ηP(a◇(b◇a)) and pl∘vP(a)⩾pl∘vP(a◇(b◇a)) for all a,b∈A and for l=1,2,…,m. By (ii), pl∘μP(a)⩾pl∘μP(a◇(b◇a)), pl∘ηP(a)⩾pl∘μP(a◇(b◇a)) and pl∘vP(a)⩽pl∘vP(a◇(b◇a)) for all a,b∈A and l=1,2,…,m. As a result, pl∘μP(a)=pl∘μP(a◇(b◇a)), pl∘ηP(a)=pl∘μP(a◇(b◇a)) and pl∘vP(a)=pl∘vP(a◇(b◇a)) for all a,b∈A and for l=1,2,…,m.

(iii)⇒(i): Since P is an m-polar PFI of A therefore pl∘μP(a◇(b◇a))⩾pl∘μP{a◇(b◇a)◇c}∧pl∘μP(c), pl∘ηP(a◇(b◇a))⩾pl∘ηP{a◇(b◇a)◇c}∧pl∘ηP(c) and pl∘vP(a◇(b◇a))⩽pl∘vP{a◇(b◇a)◇c}∨pl∘vP(c) for all a,b,c∈A and for l=1,2,…,m. By (iii), we have, pl∘μP(a)⩾pl∘μP{a◇(b◇a)◇c}∧pl∘μP(c), pl∘ηP(a)⩾pl∘ηP{a◇(b◇a)◇c}∧pl∘ηP(c) and pl∘vP(a)⩽pl∘vP{a◇(b◇a)◇c}∨pl∘vP(c) for all a,b,c∈A and for l=1,2,…,m. Thus, P is an m-polar PFII of A.

Definition 19.

Let P=(μP,ηP,vP) be an m-polar PFS in a BCK algebra (A,◇,0). Then P is said to be an m-polar picture fuzzy positive implicative ideal (PFPII) if the below stated conditions are meet.

pl∘ηP(0)⩾pl∘ηP(a) and pl∘vP(0)⩽pl∘vP(a)
pl∘ηP(a◇c)⩾pl∘ηP((a◇b)◇c)∧pl∘ηP(b◇c) and pl∘vP(a◇c)⩽pl∘vP((a◇b)◇c)∧pl∘vP(b◇c), ∀a,b∈A and l=1,2,…,m

Proposition 21.

An m-polar PFI P=(μP,ηP,vP) of a BCK algebra (A,◇,0) is an m-polar PFPII iff pl∘μP(a◇b)⩾pl∘μP((a◇b)◇b), pl∘ηP(a◇b)⩾pl∘ηP((a◇b)◇b) and pl∘vP(a◇b)⩽pl∘vP((a◇b)◇b), ∀a,b∈A and for l=1,2,…,m.

Proof.

The proof is easy. So, it is omitted here.

Since (a◇b)◇b⩽a◇b, it follows from Proposition 3 that pl∘μP(a◇b)⩽pl∘μP((a◇b)◇b), pl∘ηP(a◇b)⩽pl∘ηP((a◇b)◇b) and pl∘vP(a◇b)⩾pl∘vP((a◇b)◇b), ∀a,b∈A and l=1,2,…,m. So, the above Proposition can be modified in the following way:

Proposition 22.

An m-polar PFI P=(μP,ηP,vP) of a BCK algebra (A,◇,0) is a m-polar PFPII iff pl∘μP(a◇b)=pl∘μP((a◇b)◇b), pl∘ηP(a◇b)=pl∘ηP((a◇b)◇b) and pl∘vP(a◇b)=pl∘vP((a◇b)◇b), ∀a,b∈A and l=1,2,…,m.

7. m-POLAR PFCI

Definition 20.

Let (A,◇,0) be a BCK algebra and P=(μP,ηP,vP) be an m-polar PFS in A. Then P is said to be m-polar PFCI of A if the following conditions are met:

pl∘μP(0)⩾pl∘μP(a), pl∘ηP(0)⩾pl∘ηP(a) and pl∘vP(0)⩽pl∘vP(a)
pl∘μP(a◇(b◇(b◇a)))⩾pl∘μP((a◇b)◇c)∧pl∘μP(c), pl∘ηP(a◇(b◇(b◇a)))⩾pl∘ηP((a◇b)◇c)∧pl∘ηP(c) and pl∘vP(a◇(b◇(b◇a)))⩽pl∘vP((a◇b)◇c)∨pl∘vP(c) for all a,b∈A and for l=1,2,…,m

Example 5.

Let us consider the BCK algebra (A,◇) as follows:

◇	0	p	q	r
0	0	0	0	0
p	p	0	0	p
q	q	p	0	q
r	r	r	r	0

Now, let us suppose a 3-polar PFS P=(μP,ηP,vP) defined by

μP(a)=(0.34,0.36,0.37),if a=0(0.28,0.3,0.32),if a=p(0.17,0.18,0.18),if a=q,r

ηP(a)=(0.35,0.36,0.39),if a=0(0.25,0.27,0.3),if a=p(0.2,0.23,0.27),if a=q,r

and

ηP(a)=(0.1,0.15,0.17),if a=0(0.2,0.27,0.31),if a=p(0.55,0.57,0.58),if a=q,r

Clearly, P is a 3-polar PFCI of A.

Definition 21.

A BCK algebra (A,◇,0) is said to be commutative if b◇(b◇a)=a◇(a◇b) for all a,b∈A.

Proposition 23.

Every m-polar PFCI of a BCK algebra is an m-polar PFI.

Proof.

Let P=(μP,ηP,vP) is an m-polar PFCI of a BCK algebra (A,◇,0).

Now, (a◇(0◇(0◇a)))

=(a◇0) [by Proposition 1]

=a [by Proposition 1]

Now, pl∘μP(a)=pl∘μP(a◇(0◇(0◇a)))⩾pl∘μP((a◇0)◇c)∧pl∘μP(c)=pl∘μP(a◇c)∧pl∘μP(c), pl∘ηP(a)=pl∘ηP(a◇(0◇(0◇a)))⩾pl∘ηP((a◇0)◇c)∧pl∘ηP(c)=pl∘ηP(a◇c)∧pl∘ηP(c) and pl∘vP(a)=pl∘vP(a◇(0◇(0◇a)))⩽pl∘vP((a◇0)◇c)∨pl∘vP(c)=pl∘vP(a◇c)∨pl∘vP(c) for all a,c∈A and for l=1,2,…,m. Consequently, P is an m-polar PFI of A.

The above proposition is not true in reverse direction which is clear from following example. But the converse of the above proposition holds in commutative BCK algebra which is highlighted through Proposition 24.

Example 6.

Let us consider a BCK algebra (A,◇) as follows:

◇	0	p	q	r	s
0	0	0	0	0	0
p	p	0	p	0	0
q	q	q	0	0	0
r	r	r	r	0	0
s	s	s	s	r	0

Now, let us suppose a 3-polar PFS P=(μP,ηP,vP) defined by

μP(a)=(0.4,0.41,0.43),if a=0(0.3,0.32,0.33),if a=p(0.2,0.24,0.27),if a=q,r,s

ηP(a)=(0.43,0.45,0.47),if a=0(0.35,0.36,0.37),if a=p(0.21,0.22,0.23),if a=q,r,s

vP(a)=(0.08,0.09,0.1),if a=0(0.27,0.28,0.3),if a=p(0.45,0.47,0.5),if a=q,r,s

Clearly, P is a 3-polar PFI of A.

It is observed that

μP((q◇(r◇(r◇q))))=μP(q)=(0.2,0.24,0.27),

μP((q◇r)◇0)∧μP(0)=(0.4,0.41,0.43)

ηP((q◇(r◇(r◇q))))=ηP(q)=(0.21,0.22,0.23),

ηP((q◇r)◇0)∧ηP(0)=(0.43,0.45,0.47)

vP((q◇(r◇(r◇q))))=vP(q)=(0.45,0.47,0.5),

vP((q◇r)◇0)∨vP(0)=(0.08,0.09,0.1).

Here, μP((q◇(r◇(r◇q))))≱μP((q◇r)◇0)∧μP(0), ηP((q◇(r◇(r◇q))))≱ηP((q◇r)◇0)∧ηP(0) and vP((q◇(r◇(r◇q))))≰vP((q◇r)◇0)∨vP(0). Clearly, P is not a 3-polar PFCI of A.

Proposition 24.

In a commutative BCK algebra, every m-polar PFI is an m-polar PFCI.

Proof.

Let P=(μP,ηP,vP) be an m-polar PFI of a commutative BCK algebra (A,◇,0). We have, [((a◇(b◇(b◇a)))◇((a◇b)◇c))]◇c=((a◇(b◇(b◇a)))◇c)◇((a◇b)◇c) [by Proposition 1]

⩽(a◇(b◇(b◇a)))◇(a◇b) [by Proposition 1]

=(a◇(a◇(a◇b)))◇(a◇b) [as A is commutative therefore (a◇(a◇b))=(b◇(b◇a)) for all a,b∈A]

=(a◇b)◇(a◇b) [by Proposition 1]

=0

i.e. (a◇(b◇(b◇a)))◇((a◇b)◇c)⩽c.

Thus, by Proposition 4, it is obtained that pl∘μP((a◇(b◇(b◇a))))⩾pl∘μP(((a◇b)◇c))∧pl∘μP(c), pl∘ηP((a◇(b◇(b◇a))))⩾pl∘ηP(((a◇b)◇c))∧pl∘ηP(c) and pl∘vP((a◇(b◇(b◇a))))⩽pl∘vP(((a◇b)◇c))∨pl∘vP(c) for all a,b,c∈A and for l=1,2,…,m. Consequently, P is an m-polar PFCI of A.

Now, we are interested to develop a relationship between m-polar PFII and m-polar PFCI. Before that we state some propositions which are necessary in this regard.

Meng et al. [10] stated the following proposition:

Proposition 25.

The followings hold in a BCK algebra (A,◇,0).

((a◇c)◇c)◇(b◇c)⩽(a◇b)◇c.
(a◇c)◇(a◇(a◇c))=(a◇c)◇c.
(a◇(b◇(b◇a)))◇(b◇(a◇(b◇(b◇a))))⩽a◇b.

Proposition 26.

An m-polar PFI P=(μP,ηP,vP) of a BCK algebra (A,◇,0) is an m-polar PFCI iff

pl∘μP(a◇(b◇(b◇a)))⩾pl∘μP(a◇b), pl∘ηP(a◇(b◇(b◇a)))⩾pl∘ηP(a◇b) and pl∘vP(a◇(b◇(b◇a)))⩽pl∘vP(a◇b), ∀a,b∈A and l=1,2,…,m.

Proof.

The proof is easy. So, it is omitted here.

It is observed that a◇b⩽a◇(b◇(b◇a)) and using Proposition 3 we get, pl∘μP(a◇(b◇(b◇a)))⩽pl∘μP(a◇b), pl∘ηP(a◇(b◇(b◇a)))⩽pl∘ηP(a◇b) and pl∘vP(a◇(b◇(b◇a)))⩾pl∘vP(a◇b), ∀a,b∈A and l=1,2,…,m. So, above Proposition can be modified in the following way:

Proposition 27.

An m-polar PFI P=(μP,ηP,vP) of a BCK algebra (A,◇,0) is an m-polar PFCI iff pl∘μP(a◇(b◇(b◇a)))=pl∘μP(a◇b), pl∘ηP(a◇(b◇(b◇a)))=pl∘ηP(a◇b) and pl∘vP(a◇(b◇(b◇a)))=pl∘vP(a◇b), ∀a,b∈A and l=1,2,…,m.

Proposition 28.

An m-polar PFI P=(μP,ηP,vP) is an m-polar PFII iff P is both m-polar PFCI and m-polar PFPII.

Proof.

Suppose that P is m-polar PFII. Then by Proposition 25 (i) and Proposition 4,

pl∘μP((a◇b)◇c)∧pl∘μP(b◇c) ⩽pl∘μP((a◇c)◇c) =pl∘μP((a◇c)◇(a◇(a◇c))) [by Proposition 25 (ii)] =pl∘μP(a◇c) [by Proposition 20 (iii)], pl∘ηP((a◇b)◇c)∧pl∘ηP(b◇c) ⩽pl∘ηP((a◇c)◇c) =pl∘ηP((a◇c)◇(a◇(a◇c))) [by Proposition 25 (ii)] =pl∘ηP(a◇c) [by Proposition 20 (iii)] and pl∘vP((a◇b)◇c)∧pl∘vP(b◇c) ⩾pl∘vP((a◇c)◇c) =pl∘vP((a◇c)◇(a◇(a◇c))) [by Proposition 25 (ii)] =pl∘vP(a◇c) [by Proposition 20 (iii)]

Therefore, P is an m-polar PFPII.

By Proposition 25 (iii) and Proposition 3 we get,

pl∘μP(a◇b) ⩽pl∘μP(a◇(b◇(b◇a)))◇(b◇(a◇(b◇(b◇a))))) =pl∘μP((a◇(b◇(b◇a))) [by Proposition 20 (iii)], pl∘ηP(a◇b) ⩽pl∘ηP(a◇(b◇(b◇a)))◇(b◇(a◇(b◇(b◇a))))) =pl∘ηP((a◇(b◇(b◇a))) [by Proposition 20 (iii)] and pl∘vP(a◇b) ⩾pl∘vP(a◇(b◇(b◇a)))◇(b◇(a◇(b◇(b◇a))))) =pl∘vP((a◇(b◇(b◇a))) [by Proposition 20 (iii)].

Therefore, P is an m-polar PFCI of A.

Conversely, let P be both m-polar PFPII and m-polar PFCI of A. Since (b◇(b◇a))◇(b◇a)⩽a◇(b◇a), by Proposition 3,

pl∘μP(a◇(b◇a)) ⩽pl∘μP((b◇(b◇a))◇(b◇a)), pl∘ηP(a◇(b◇a)) ⩽pl∘ηP((b◇(b◇a))◇(b◇a)) and pl∘vP(a◇(b◇a)) ⩾pl∘vP((b◇(b◇a))◇(b◇a))

By Proposition 22,

pl∘μP((b◇(b◇a))◇(b◇a))=pl∘μP(b◇(b◇a)), pl∘ηP((b◇(b◇a))◇(b◇a))=pl∘ηP(b◇(b◇a)) and pl∘vP((b◇(b◇a))◇(b◇a))=pl∘vP(b◇(b◇a))

therefore it is obtained that

pl∘μP(a◇(b◇a))⩽pl∘μP(b◇(b◇a)),(1)

pl∘ηP(a◇(b◇a))⩽pl∘ηP(b◇(b◇a))(2)

and pl∘vP(a◇(b◇a))⩾pl∘vP(b◇(b◇a)) (3)

Also, a◇b⩽a◇(b◇a). Therefore, by Proposition 3,

pl∘μP(a◇(b◇a))⩽pl∘μP(a◇b), pl∘ηP(a◇(b◇a))⩽pl∘ηP(a◇b) and pl∘vP(a◇(b◇a))⩾pl∘vP(a◇b)

Since P is an m-polar PFCI therefore by Proposition 27,

pl∘μP(a◇b)=pl∘μP(a◇(b◇(b◇a))), pl∘ηP(a◇b)=pl∘ηP(a◇(b◇(b◇a))) and pl∘vP(a◇b)=pl∘vP(a◇(b◇(b◇a)))

Hence it is obtained that

pl∘μP(a◇(b◇a))⩽pl∘μP(a◇(b◇(b◇a))),(4)

pl∘ηP(a◇(b◇a))⩽pl∘ηP(a◇(b◇(b◇a)))(5)

and pl∘vP(a◇(b◇a))⩾pl∘vP(a◇(b◇(b◇a))) (6)

Combining (1) and (4), (2) and (5), (3) and (6) it is obtained that

pl∘μP(a◇(b◇a)) ⩽pl∘μP(a◇(b◇(b◇a)))∧pl∘μP(b◇(b◇a)) ⩽pl∘μP(a), pl∘ηP(a◇(b◇a)) ⩽pl∘ηP(a◇(b◇(b◇a)))∧pl∘ηP(b◇(b◇a)) ⩽pl∘ηP(a) and pl∘vP(a◇(b◇a)) ⩾pl∘vP(a◇(b◇(b◇a)))∨pl∘vP(b◇(b◇a)) ⩾pl∘vP(a).

So, by Proposition 20 (ii), P is an m-polar PFII of A.

8. CONCLUSION

In this paper, we have initiated the notion of m-polar PFI and m-polar PFII of BCK algebra. We have studied some basic results related to them. We have established a relationship between m-polar PFI and m-polar PFII of a BCK algebra. We have also investigated a relationship between m-polar PFI and m-polar PFCI. We have studied some properties of m-polar PFI under homomorphism of BCK algebra. It is our hope that our works will help the researchers to study some other types of algebraic structures in context of m-polar PFS.

CONFLICT OF INTEREST

Authors declare that they have no conflict of interest.

AUTHORS' CONTRIBUTIONS

S. Dogra Writing, reviewing and editing. M. Pal reviewing, editing and supervision.

Funding statement

There is no funding source for this work.

ACKNOWLEDGMENTS

Authors are thankful to the reviewers for their valuable suggestions towards the improvement of the paper.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 13 - 1
Pages: 409 - 420
Publication Date: 2020/04/14
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.200330.001 How to use a DOI?
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/).

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ris enw bib

TY  - JOUR
AU  - Shovan Dogra
AU  - Madhumangal Pal
PY  - 2020
DA  - 2020/04/14
TI  - m-Polar Picture Fuzzy Ideal of a BCK Algebra
JO  - International Journal of Computational Intelligence Systems
SP  - 409
EP  - 420
VL  - 13
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.200330.001
DO  - 10.2991/ijcis.d.200330.001
ID  - Dogra2020
ER  -

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