Subalgebras and Ideals of BCK/BCI-Algebras in the Framework of the Hesitant Intersection

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doi:10.5666/KMJ.2016.56.2.371

Article

Kyungpook Mathematical Journal 2016; 56(2): 371-386

Published online June 1, 2016

Subalgebras and Ideals of BCK/BCI-Algebras in the Framework of the Hesitant Intersection

Young Bae Jun

Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea

Received: September 7, 2014; Accepted: December 10, 2015

Abstract

Go to

Abstract
1. Introduction
2. Preliminaries
3. Subalgebras and Ideals of BCK/BCI-Algebras Based on the Hesitant Intersection
Acknowledgements
References

Using the hesitant intersection (⋒), the notions of ⋒-hesitant fuzzy subalgebras, ⋒-hesitant fuzzy ideals and ⋒-hesitant fuzzy p-ideals are introduced, and their relations and related properties are investigated. Conditions for a ⋒-hesitant fuzzy ideal to be a ⋒-hesitant fuzzy p-ideal are provided. The extension property for ⋒-hesitant fuzzy p-ideals is established.

Keywords: Hesitant intersection, Hesitant fuzzy subalgebra, Hesitant fuzzy ideal, , Hesitant fuzzy $p$-ideal.

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Subalgebras and Ideals of BCK/BCI-Algebras Based on the Hesitant Intersection
Acknowledgements
References

The notions of Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets and fuzzy multisets etc. are a generalization of fuzzy sets. The concept of hesitant fuzzy sets, which is introduced by Torra [6, 7], is another generalization of fuzzy sets. The hesitant fuzzy set is very useful to express peoples hesitancy in daily life, and it is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. Xu and Xia [11] proposed a variety of distance measures for hesitant fuzzy sets, based on which the corresponding similarity measures can be obtained. They investigated the connections of the aforementioned distance measures and further develop a number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures. Xu and Xia [12] defined the distance and correlation measures for hesitant fuzzy information and then discussed their properties in detail. Also, hesitant fuzzy set theory is used in decision making problem etc.(see [5, 8, 9, 10, 12]), and is applied to residuated lattices and MTL-algebras (see [2, 4]).

In this paper, we introduce the notions of hesitant fuzzy subalgebras, hesitant fuzzy ideals and hesitant fuzzy p-ideals based on the hesitant intersection (⋒), briefly, ⋒-hesitant fuzzy subalgebras, ⋒-hesitant fuzzy ideals and ⋒-hesitant fuzzy p-ideals, in BCK/BCI-algebras. We investigate their relations and related properties. We provide conditions for a ⋒-hesitant fuzzy ideal to be a ⋒-hesitant fuzzy p-ideal. We finally establish the extension property for ⋒-hesitant fuzzy p-ideals.

2. Preliminaries

Go to

Abstract
1. Introduction
2. Preliminaries
3. Subalgebras and Ideals of BCK/BCI-Algebras Based on the Hesitant Intersection
Acknowledgements
References

An algebra (L; *, 0) of type (2, 0) is calleda BCI-algebra if it satisfies the following conditions:

(I) (∀x, y, z ∈ L) (((x * y) * (x * z)) * (z * y) = 0),
(II) (∀x, y ∈ L) ((x * (x * y)) * y = 0),
(III) (∀x ∈ L) (x * x = 0),
(IV) (∀x, y ∈ L) (x * y = 0, y * x = 0 ⇒ x = y).

If a BCI-algebra L satisfies the following identity:

(V) (∀x ∈ L) (0 * x = 0),

then L is called a BCK-algebra.

Any BCK/BCI-algebra L satisfies the following conditions:

(2.1)(∀x∈L) (x*0=x),

(2.2)(∀x,y,z∈L) (x≤y⇒x*z≤y*z, z*y≤z*x),

(2.3)(∀x,y,z∈L) ((x*y)*z=(x*z)*y),

(2.4)(∀x,y,z∈L) ((x*z)*(y*z)≤x*y)

where x ≤ y if and only if x * y = 0.

Any BCI-algebra X satisfies the following conditions:

(2.5)(∀x,y,z∈X) (0*(0*((x*z)*(y*z)))=(0*y)*(0*x)),

(2.6)(∀x,y∈X) (0*(0*(x*y))=(0*y)*(0*x)),

(2.7)(∀x∈X) (0*(0*(0*x))=0*x).

A BCI-algebra L is said to be p-semisimple(see [1]) if 0 * (0 * x) = x for all x ∈ L.

Every p-semisimple BCI-algebra L satisfies:

(2.8)(∀x,y,z∈L) ((x*z)*(y*z)=x*y).

A nonempty subset S of a BCK/BCI-algebra L is called a subalgebra of L if x * y ∈ S for all x, y ∈ S. A subset A of a BCK/BCI-algebra L is called an ideal of L if it satisfies:

(2.9)0∈A,

(2.10)(∀x∈L) (x*y∈A, y∈A⇒x∈A).

A subset A of a BCI-algebra L is called a p-ideal of L (see [13]) if it satisfies (2.9) and

(2.11)(∀x,y,z∈L) ((x*z)*(y*z)∈A, y∈A⇒x∈A).

Note that an ideal A of a BCI-algebra L is a p-ideal of L if and only if the following assertion is valid:

(2.12)(∀x,y,z∈L) ((x*z)*(y*z)∈A⇒x*y∈A).

We refer the reader to the books [1, 3] for further information regarding BCK/BCI-algebras.

3. Subalgebras and Ideals of BCK/BCI-Algebras Based on the Hesitant Intersection

Go to

Abstract
1. Introduction
2. Preliminaries
3. Subalgebras and Ideals of BCK/BCI-Algebras Based on the Hesitant Intersection
Acknowledgements
References

Let L be a set. A hesitant fuzzy set on L (see [6]) is defined in terms of a function ℋ that when applied to L returns a subset of [0, 1], that is, ℋ : L → ℘ ([0, 1]).

Given a hesitant fuzzy set ℋ on L, we define Infℋ and Supℋ, respectively, as follows:

(3.1)InfH(x)={minimum of H(x)if H(x) is finite,infimum of H(x)otherwise,

and

(3.2)SupH(x)={maximum of H(x)if H(x) is finite,supremum of H(x)otherwise

for all x ∈ L. It is obvious that Infℋ and Supℋ are fuzzy sets in L.

For a hesitant fuzzy set ℋ on L and x, y ∈ L, we define

(3.3)H(x)⋓H(y):={t∈H(x)∪H(y)∣t≥max{InfH(x),InfH(y)}}

and

(3.4)H(x)⋒H(y):={t∈H(x)∪H(y)∣t≤min{SupH(x),SupH(y)}}.

We say that ℋ(x) ⋓ ℋ(y) (resp., ℋ(x) ⋒ ℋ(y)) is the hesitant union (resp., hesitant intersection) of ℋ(x) and ℋ(y).

Proposition 3.1

For any hesitant fuzzy set ℋ on L, we have

(∀x ∈ L) (ℋ(x) ⋓ ℋ(x) = ℋ(x)).
(∀x ∈ L) (ℋ(x) ⋒ ℋ(x) = ℋ(x)).
(∀a, b, x, y ∈ L) (ℋ(a) ⊆ ℋ(x), ℋ(b) ⊆ ℋ(y) ⇒ ℋ(a) ⋒ ℋ(b) ⊆ ℋ(x) ⋒ ℋ(y)).
(∀a, b, x, y ∈ L) (ℋ(a) ⊆ ℋ(x), ℋ(b) ⊆ ℋ(y) ⇒ ℋ(a) ⋓ ℋ(b) ⊆ ℋ(x) ⋓ ℋ(y)).

Proof

(1) and (2) are straightforward.
(3) Let a, b, x, y ∈ L be such that ℋ(a) ⊆ ℋ(x) and ℋ(b) ⊆ ℋ(y). Then Supℋ(a) ≤ Supℋ(x) and Supℋ(b) ≤ Supℋ(y). If t ∈ ℋ(a) ⋒ ℋ(b), then t∈H(a)∪H(b)⊆H(x)∪H(y)
and t ≤ min{Supℋ(a), Supℋ(b)} ≤ min{Supℋ(x), Supℋ(y)}.
Hence t ∈ ℋ(x) ⋒ ℋ(y), and so ℋ(a) ⋒ ℋ(b) ⊆ ℋ(x) ⋒ ℋ(y).
(4) Let a, b, x, y ∈ L be such that ℋ(a) ⊆ ℋ(x) and ℋ(b) ⊆ ℋ(y). Then Infℋ(a) ≥ Infℋ(x) and Infℋ(b) ≥ Infℋ(y). If t ∈ ℋ(a) ⋓ ℋ(b), then t∈H(a)∪H(b)⊆H(x)∪H(y)
and t ≥ max{Infℋ(a), Infℋ(b)} ≥ max{Infℋ(x), Infℋ(y)}.
Hence t ∈ ℋ(x) ⋓ ℋ(y), and so ℋ(a) ⋓ ℋ(b) ⊆ ℋ(x) ⋓ ℋ(y).

Definition 3.2

A hesitant fuzzy set on a BCK/BCI-algebra L is called a hesitant fuzzy subalgebra of L based on the intersection (∩) (briefly, ∩-hesitant fuzzy subalgebra of L) if it satisfies:

(3.5)(∀x,y∈L) (H(x*y)⊇H(x)∩H(y)).

Definition 3.3

A hesitant fuzzy set on a BCK/BCI-algebra L is called a hesitant fuzzy subalgebra of L based on the hesitant intersection (⋒) (briefly, ⋒-hesitant fuzzy subalgebra of L) if it satisfies:

(3.6)(∀x,y∈L) (H(x*y)⊇H(x)⋒H(y)).

Example 3.4

Let L = {0, 1, 2, 3} be a BCK-algebra (see [3]) with the following Cayley table:

*012300000110102220033210

(1) Define a hesitant fuzzy set ℋ on L as follows: H:L→P([0,1]), x↦{[0.3,0.8]if x=0,[0.3,0.7]if x=1,[0.3,0.5]if x∈{2,3}.
It is easy to check that ℋ is a ⋒-hesitant fuzzy subalgebra of L.
(2) Define a hesitant fuzzy set on L as follows: G:L→P([0,1]), x↦{[0.2,0.8]if x=0,[0.2,0.7]if x=1,[0.2,0.4]if x=2,[0.2,0.6]if x=3.
Then is not a ⋒-hesitant fuzzy subalgebra of L since G(3)⋒G(1)={t∈G(3)∪g(1)∣t≤min{SupG(3),SupG(1)}={t∈[0.2,0.7]∣t≤min{0.6,0.7}}=[0.2,0.6]⊈[0.2,0.4]=G(2)=G(3*1).

It is clear that every ⋒-hesitant fuzzy subalgebra is a ∩-hesitant fuzzy subalgebra, but the converse is not true in general as seen in the following example.

Example 3.5

Let L = {0, a, b, c, d} be a BCI-algebra (see [1]) with the following Cayley table:

*0abcd000bcdaa0bcdbbb0dccccd0bdddcb0

Define a hesitant fuzzy set ℋ on L as follows:

H:L→P([0,1]), x↦{[0,0.9]if x=0,[0.2,0.7]if x=a,[0.2,0.3]if x=b,[0.4,0.5,0.6]if x=c,[0.6,0.7]if x=d.

It is routine to check that ℋ is a ∩-hesitant fuzzy subalgebra of L. Note that

H(b)⋒H(d)={x∈H(b)∪H(d)∣x≤min{SupH(b),SupH(d)}={x∈(0.2,0.3]∪[0.6,0.7]∣x≤min{0.3,0.7}}=[0.2,0.3],

and so ℋ(b * d) = ℋ(c) = {0.4, 0.5, 0.6} ⊉ (0.2, 0.3] = ℋ(b) ⋒ ℋ(d). Therefore ℋ is not a ⋒-hesitant fuzzy subalgebra of L.

For any hesitant fuzzy set ℋ on a BCK/BCI-algebra L and ɛ ∈ ℘ ([0, 1]), we consider the set

Hɛ:={x∈L∣ɛ⊆H(x)}

which is called the hesitant ɛ-level set on L.

Theorem 3.6

If ℋ is a ⋒-hesitant fuzzy subalgebra of a BCK/BCI-algebra L, then the hesitant ɛ-level set ℋ_ɛon L is a subalgebra of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀.

Proof

Assume that ℋ is a ⋒-hesitant fuzzy subalgebra of a BCK/BCI-algebra L and let ɛ ∈ ℘ ([0, 1]) be such that ℋ_ɛ ≠ ∅︀. If x, y ∈ ℋ_ɛ, then ɛ ⊆ ℋ(x) and ɛ ⊆ ℋ(y). It follows from (3.6) and Proposition 3.1(3) that

(3.7)H(x*y)⊇H(x)⋒H(y)⊇ɛ

and that x * y ∈ ℋ_ɛ. Therefore ℋ^ɛ is a subalgebra of L.

The converse of Theorem 3.6 is not true in general as seen in the following example.

Example 3.7

Let L = {0, 1, 2, a, b} be a BCI-algebra (see [1]) with the following Cayley table:

*012ab0000aa1101ba2220aaaaaa00bbab10

Define a hesitant fuzzy set ℋ on L as follows:

H:L→P([0,1]), x↦{[0.3,0.8)if x=0,(0.3,0.5]if x=1,[0.4,0.7]if x=2,(0.4,0.6)if x=a,(0.4,0.5]if x=b.

Then we have

Hɛ={{0}if ɛ⊆[0.3,0.8),ɛ⊈(0.3,0.5) and ɛ⊈[0.4,0.7],{0,2}if ɛ⊆[0.4,0.7] and ɛ⊈(0.4,0.6),{0,2,a}if ɛ⊆(0.4,0.6) and ɛ⊈(0.4,0.5],{0,1}if ɛ⊆(0.3,0.5) ɛ⊈(0.4,0.5],Lif ɛ⊆(0.4,0.5],∅otherwise,

and so ℋ_ɛ is a subalgebra of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀. Since

H(2)⋒H(b)={t∈H(2)∪H(b)∣t≤min{SupH(2),SupH(b)}={t∈[0.4,0.7]∣t≤min{0.7,0.5}}=[0.4,0.5]⊈(0.4,0.6)=H(a)=H(2*b),

ℋ is not a ⋒-hesitant fuzzy subalgebra of L.

Theorem 3.8

Let ℋ be a hesitant fuzzy set on a BCK/BCI-algebra L such that

(3.8)(∀x,y∈L) (H(x)⋒H(y)=H(y)∩H(y)).

If the hesitant ɛ-level set ℋ_ɛon L is a subalgebra of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀, then ℋ is a ⋒-hesitant fuzzy subalgebra of L.

Proof

Assume that the set ℋ_ɛ := {x ∈ L | ɛ ⊆ ℋ(x)} is a subalgebra of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀. For any x, y ∈ L, let ℋ(x) = ɛ_x and ℋ(y) = ɛ_y. Take ɛ = ɛ_x ∩ ɛ_y. Then x, y ∈ ℋ_ɛ, and so x * y ∈ ℋ_ɛ. It follows from (3.8) that

H(x*y)⊇ɛ=ɛx∩ɛy=ɛx⋒ɛy= H(x)⋒H(y).

Therefore ℋ is a ⋒-hesitant fuzzy subalgebra of L.

Definition 3.9

A hesitant fuzzy set on a BCK/BCI-algebra L is called a hesitant fuzzy ideal of L based on the intersection (∩) (briefly, ∩-hesitant fuzzy ideal of L) if it satisfies:

(3.9)(∀x∈L) (H(x)⊆H(0)),

(3.10)(∀x,y∈L) (H(x*y)∩H(y)⊆H(x)).

Definition 3.10

A hesitant fuzzy set on a BCK/BCI-algebra L is called a hesitant fuzzy ideal of L based on the hesitant intersection (⋒) (briefly, ⋒-hesitant fuzzy ideal of L) if it satisfies the condition (3.9) and

(3.11)(∀x,y∈L) (H(x*y)⋒H(y)⊆H(x)).

Example 3.11

Let (Z,+, 0) be an additive group of integers. Note that (Z,−, 0) is the adjoint BCI-algebra of (Z,+, 0). For any BCI-algebra (Y, *, 0), let L := Y ×Z. Then (L, ⊗, (0, 0)) is a BCI-algebra (see [1]) in which the operation ⊗ is given by

(∀(x,m),(y,n)∈L) ((x,m)⊗(y,n)=(x*y,m-n)).

For a subset A := Y × N₀ of L where N₀ is the set of nonnegative integers, let ℋ be a hesitant fuzzy set on L defined by

H:L→P([0,1]), x↦{[0.3,0.9)if x∈A,[0.3,0.6]otherwise.

It is routine to verify that ℋ is a ⋒-hesitant fuzzy ideal of L.

It is clear that every ⋒-hesitant fuzzy ideal is a ∩-hesitant fuzzy ideal, but the converse is not true in general as seen in the following example.

Example 3.12

Let L = {0, e, a, b, c} be a BCI-algebra (see [1]) with the following Cayley table:

*0eabc000abcee0abcaaa0cbbbbc0acccba0

Define a hesitant fuzzy set ℋ on L as follows:

H:L→P([0,1]), x↦{[0,1)if x=0,[0.2,0.7]if x=e,(0.2,0.3]if x=a,{0.4,0.5}if x=b,[0.6,0.7)if x=c.

Then ℋ is a ∩-hesitant fuzzy subalgebra of L. Note that

H(a*c)⋒H(c)=H(b)⋒H(c)={t∈[0.4,0.5]∪[0.6,0.7]∣t≤min{0.5,0.7}}=[0.4,0.5]⊈(0.2,0.3)=H(a).

Hence ℋ is not a ⋒-hesitant fuzzy ideal of L.

Proposition 3.13

Every ⋒-hesitant fuzzy ideal ℋ of a BCI-algebra L satisfies the following assertion:

(3.12)(∀x∈L) (H(x)⊆H(0*(0*x))).

Proof

For every x ∈ L, we have

H(x)=H(x)⋒H(x)⊆H(0)⋒H(x)=H((0*(0*x))*x)⋒H(x)⊆H(0*(0*x))

by Proposition 3.1, (III), (2.3) and (3.11).

Theorem 3.14

If ℋ is a ⋒-hesitant fuzzy ideal of a BCK/BCI-algebra L, then the hesitant ɛ-level set ℋ_ɛon L is an ideal of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀. Proof. Suppose that ℋ is a ⋒-hesitant fuzzy ideal of a BCK/BCI-algebra L. Let x, y ∈ L and ɛ ∈ ℘ ([0, 1]) be such that x * y ∈ ℋ_ɛ and y ∈ ℋ_ɛ. Then ɛ ⊆ ℋ(x * y) and ɛ ⊆ ℋ(y). It follows from (3.9), (3.11) and Proposition 3.1(3) that

H(0)⊇H(x)⊇H(x*y)⋒H(y)⊇ɛ.

Hence 0 ∈ ℋ_ɛ and x ∈ ℋ_ɛ. Therefore ℋ_ɛ is an ideal of L.

The following example shows that the converse of Theorem 3.14 is not true in general.

Example 3.15

Consider the BCI-algebra L in Example 3.11. For a subset A := Y × N₀ of L where N₀ is the set of nonnegative integers, let ℋ be a hesitant fuzzy set on L defined by

H:L→P([0,1]), x↦{[0.3,0.9)if x∈A,[0.4,0.6]otherwise.

Then ℋ_ɛ is an ideal of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀. For any a ∈ Y, we have

H((a,-3)⊗(a,3))⋒H(a,3)=H(0,-6)⋒H(a,3)={t∈H(0,-6)∪H(a,3)∣t≤min{SupH(0,-6),SupH(a,3)}}={t∈[0.3,0.9]∣t≤min{0.6,0.9}}=[0.3,0.6]⊈(0.4,0.6)=H(a,-3).

Hence ℋ is not a ⋒-hesitant fuzzy ideal of L.

We provide a condition for the converse of Theorem 3.14 to be true.

Theorem 3.16

Let ℋ be a hesitant fuzzy set on a BCK/BCI-algebra L satisfying the condition (3.8). If the hesitant ɛ-level set ℋ_ɛon L is an ideal of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀, then ℋ is a ⋒-hesitant fuzzy ideal of L.

Proof

For any x ∈ L, let ℋ(x) = ɛ_x. Then x ∈ ℋ_{ɛ_x}, and so ℋ_{ɛ_x} is an ideal of L by assumption. Thus 0 ∈ ℋ_{ɛ_x}, and hence ℋ(0) = ɛ_x = ℋ(x). For any x, y ∈ L, let ℋ(x * y) = ɛ_x_*_y and ℋ(y) = ɛ_y. Taking ɛ = ɛ_x_*_y ∩ ɛ_y implies that x * y ∈ ℋ_ɛ and y ∈ ℋ_ɛ. Hence x ∈ ℋ_ɛ, and it follows from the condition (3.8) that

H(x)⊇ɛ=ɛx*y∩ɛy=ɛx*y⋒ɛy=H(x*y)⋒H(y).

Therefore ℋ is a ⋒-hesitant fuzzy ideal of L.

Theorem 3.17

Let ɛ₁and ɛ₂be subintervals of [0, 1] such that

ɛ₂ ⊊ ɛ₁, Infɛ₁ = Infɛ₂and Supɛ₂ ∈ ɛ₂,
Infɛ₁ ∈ ɛ₁and Infɛ₂ ∈ ɛ₂(or, Infɛ₁ ∉ ɛ₁and Infɛ₂ ∉ ɛ₂).

Define a hesitant fuzzy set ℋ on a BCK/BCI-algebra L as follows:

H:L→P([0,1]), x↦{ɛ1if x∈A,ɛ2otherwise,

where A is a nonempty proper subset of L. Then ℋ is a ⋒-hesitant fuzzy ideal (resp., subalgebra) of L if and only if A is an ideal (resp., subalgebra) of L.

Proof

Note that

Hɛ={Aif ɛ⊆ɛ1 and ɛ⊈ɛ2,Lif ɛ⊆ɛ2,∅otherwise.

If ℋ is a ⋒-hesitant fuzzy ideal of L, then ℋ_ɛ is an ideal of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀ by Theorem 3.14. Hence A is an ideal of L.

Conversely, suppose that A is an ideal of L. Then ℋ_ɛ is an ideal of L for all ɛ ∈ ℘ ([0, 1]) with ℋ_ɛ ≠ ∅︀. Let x, y ∈ L. If x, y ∈ A, then

H(x)⋒H(y)={t∈H(x)∪H(y)∣t≤min{SupH(x),SupH(y)}}=ɛ1=H(x)∩H(y).

If x, y ∈ L A, then

H(x)⋒H(y)={t∈H(x)∪H(y)∣t≤min{SupH(x),SupH(y)}}=ɛ2=H(x)∩H(y).

If x ∈ A and y ∈ L A, then

H(x)⋒H(y)={t∈H(x)∪H(y)∣t≤min{SupH(x),SupH(y)}}={t∈ɛ1∣t≤min{Supɛ1,Supɛ2}}={t∈ɛ1∣t≤Supɛ2}=ɛ2=H(x)∩H(y).

Similarly, if x ∈ LA and y ∈ A, then ℋ(x) ⋒ ℋ(y) = ℋ(x)∩ℋ(y). Thus ℋ satisfies the condition (3.8), and therefore ℋ is a ⋒-hesitant fuzzy ideal of L by Theorem 3.16. By the similar way, we can prove that ℋ is a ⋒-hesitant fuzzy subalgebra of L if and only if A is a subalgebra of L.

Proposition 3.18

For every ⋒-hesitant fuzzy ideal ℋ of a BCK/BCI-algebra L, the following assertions are valid.

(∀x, y ∈ L) (x ≤ y ⇒ ℋ(x) ⊇ ℋ(y)),
(∀x, y, z ∈ L) (x * y ≤ z ⇒ ℋ(x) ⊇ ℋ(y) ⋒ ℋ(z)),

Proof

Assume that x ≤ y for all x, y ∈ L. Then x * y = 0, which implies from (3.9), Proposition 3.1 and (3.11) that H(y)=H(y)⋒H(y)⊆H(0)⋒H(y)=H(x*y)⋒H(y)⊆H(x).
Let x, y, z ∈ L be such that x * y ≤ z. Then (x * y) * z = 0, and so H(z)=H(z)⋒H(z)⊆H(0)⋒H(z)=H((x*y)*z)⋒H(z)⊆H(x*y)
by (3.9), Proposition 3.1 and (3.11). It follows from Proposition 3.1 and (3.11) that H(y)⋒H(z)⊆H(x*y)⋒H(y)⊆H(x).

Proposition 3.19

For every ⋒-hesitant fuzzy ideal ℋ of a BCK/BCI-algebra L, the following assertions are equivalent.

(1) (∀x, y ∈ L) (ℋ((x * y) * y) ⊆ ℋ(x * y)),
(2)(∀x, y, z ∈ L) (ℋ((x * y) * z) ⊆ ℋ((x * z) * (y * z))).

Proof

Suppose that (1) is true and let x, y, z ∈ L. Note that

((x*(y*z))*z)*z=((x*z)*(y*z))*z≤(x*y)*z

by (2.3),(2.4) and (2.2). It follows from Proposition 3.18(1), (1) and (2.3) that

H((x*y)*z)⊆H(((x*(y*z))*z)*z)⊆H((x*(y*z))*z)=H((x*z)*(y*z)),

which shows that (2) is valid.

Now, assume that (2) holds and take z := y in (2). Then

H((x*y)*y)⊆H((x*y)*(y*y))=H((x*y)*0)=H(x*y)

by using (III) and (2.1). Thus (1) is valid.

We consider relations between a ⋒-hesitant fuzzy subalgebra and a ⋒-hesitant fuzzy ideal.

Theorem 3.20

In a BCK-algebra, every ⋒-hesitant fuzzy ideal is a ⋒-hesitant fuzzy subalgebra.

Proof

Let ℋ be a ⋒-hesitant fuzzy ideal of a BCK-algebra L. Using (3.11), (2.3), (III), (V), (3.9) and Proposition 3.1, we have

H(x*y)⊇H((x*y)*x)⋒H(x)=H((x*x)*y)⋒H(x)=H(0*y)⋒H(x)=H(0)⋒H(x)⊇H(x)⋒H(y)

for all x, y ∈ L. Hence ℋ is a ⋒-hesitant fuzzy subalgebra of L.

The converse of Theorem 3.20 is not true in general. In fact, consider a BCK-algebra L = {0, 1, 2} with the following Cayley table:

*012000011002220

Let ℋ be a hesitant fuzzy set on L defined by

H:L→P([0,1]), x↦{[0.3,0.8)if x=0,[0.3,0.6]if x=1,[0.3,0.7]if x=2.

Then ℋ is a ⋒-hesitant fuzzy subalgebra of L, but it is not a ⋒-hesitant fuzzy ideal of L since

H(1*2)⋒H(2)=H(0)⋒H(2)={t∈H(0)∪H(2)∣t≤min{SupH(0),SupH(2)}={t∈[0.3,0.8)∣t≤min{0.8,0.7}=[0.3,0.7]⊈[0.3,0.6]=H(1).

In a BCI-algebra, any ⋒-hesitant fuzzy ideal may not be a ⋒-hesitant fuzzy subalgebra. In fact, the ⋒-hesitant fuzzy ideal ℋ of L in Example 3.11 is not a ⋒-hesitant fuzzy subalgebra of L since

H(a,0)⋒H(a,2)={t∈H(a,0)∪H(a,2)∣t≤{SupH(a,0),SupH(a,2)}}={t∈[0.3,0.9)∣t≤0.9}=[0.3,0.9]⊈[0.3,0.6]=H((a,0)⊗(a,2))

for all a ∈ Y.

Definition 3.21

A hesitant fuzzy set ℋ on a BCI-algebra L is called a hesitant fuzzy p-ideal of L based on the hesitant intersection (⋒) (briefly, ⋒-hesitant fuzzy p-ideal of L) if it satisfies (3.9) and

(3.13)(∀x,y,z∈L) (H((x*z)*(y*z))⋒H(y)⊆H(x)).

Example 3.22

Let L = {0, a, b, c} be a BCI-algebra (see [1]) with the following Cayley table.

*0abc00abcaa0cbbbc0accba0

Define a hesitant fuzzy set ℋ on L as follows:

H:L→P([0,1]), x↦{(0.4,0.7)if x∈{0,b}(0.4,0.5]otherwise,

It is routine to verify that ℋ is a ⋒-hesitant fuzzy p-ideal of L.

Theorem 3.23

Let L be a BCI-algebra. Then every ⋒-hesitant fuzzy p-ideal of L is a ⋒-hesitant fuzzy ideal of L.

Proof

Let ℋ be a ⋒-hesitant fuzzy p-ideal of L. Since x * 0 = x for all x ∈ X, it follows from taking z := 0 in (3.13) that

H(x)⊇H((x*0)*(y*0))⋒H(y)=H(x*y)⋒H(y)

for all x, y ∈ L. Therefore ℋ is a ⋒-hesitant fuzzy ideal of L.

The following example shows that the converse of Theorem 3.23 is not true in general.

Example 3.24

Consider a BCI-algebra L = {0, 1, a, b, c} with the following Cayley table (see [1]).

*01abc000cba110cbaaaa0cbbbba0ccccba0

Define a hesitant fuzzy set ℋ on L as follows:

H:L→P([0,1]), x↦{(0.2,0.9)if x=0,(0.2,0.7]if x=1,(0.2,0.5]otherwise,

Then ℋ is a ⋒-hesitant fuzzy ideal of L. But it is not a ⋒-hesitant fuzzy p-ideal of L since

H((1*a)*(0*a))⋒H(0)=H(c*c)⋒H(0)=H(0)=(0,2,0.9)⊈(0.2,0.7]=H(1).

Proposition 3.25

Every ⋒-hesitant fuzzy p-ideal ℋ of a BCI-algebra L satisfies the following assertion:

(3.14)(∀x∈L) (H(0*(0*x))⊆H(x)).

Proof

Let ℋ be a ⋒-hesitant fuzzy p-ideal of L. If we put z := x and y := 0 in (3.13), then

H(x)⊇H((x*x)*(0*x))⋒ H(0)=H(0*(0*x))⋒H(0)⊇ H(0*(0*x))

for all x ∈ L by (III), (3.9) and Proposition 3.1.

Proposition 3.26

Every ⋒-hesitant fuzzy p-ideal ℋ of a BCI-algebra L satisfies:

(3.15)(∀x,y,z∈L) (H(x*y)⊆H((x*z)*(y*z))).

Proof

Let ℋ be a ⋒-hesitant fuzzy p-ideal of L. Then it is a ⋒-hesitant fuzzy ideal of L by Theorem 3.23. Using (3.11), (2.4) and Proposition 3.1, we have

H((x*z)*(y*z))⊇H(((x*z)*(y*z))*(x*y)⋒ H(x*y)=H(0)⋒H(x*y)⊇H(x*y)

for all x, y, z ∈ L.

We provide conditions for a ⋒-hesitant fuzzy ideal to be a ⋒-hesitant fuzzy p-ideal.

Theorem 3.27

Let ℋ be a ⋒-hesitant fuzzy ideal of L such that

(3.16)(∀x,y,z∈L) (H(x*y)⊇H((x*z)*(y*z))).

Then ℋ is a ⋒-hesitant fuzzy p-ideal of L.

Proof

If the condition (3.16) is valid, then

H(x)⊇H(x*y)⋒H(y)⊇H((x*z)*(y*z))⋒H(y)

for all x, y, z ∈ L by (3.11) and Proposition 3.1. Therefore ℋ is a ⋒-hesitant fuzzy p-ideal of L.

Theorem 3.28

If a ⋒-hesitant fuzzy ideal ℋ of L satisfies the condition (3.14), then it is a ⋒-hesitant fuzzy p-ideal of L.

Proof

Let x, y, z ∈ L. Using Proposition 3.13, (2.5), (2.6) and (3.14), we have

H((x*z)*(y*z))⊆H(0*(0*((x*z)*(y*z))))=H((0*y)*(0*x))=H(0*(0*(x*y)))⊆H(x*y).

It follows from Theorem 3.27 that ℋ is a ⋒-hesitant fuzzy p-ideal of L.

Theorem 3.29

In a p-semisimple BCI-algebra, every ⋒-hesitant fuzzy ideal is a ⋒-hesitant fuzzy p-ideal.

Proof

Let ℋ be a ⋒-hesitant fuzzy ideal of a p-semisimple BCI-algebra L. Using (3.11) and (2.8), we have

H(x)⊇H(x*y)⋒H(y)=H((x*z)*(y*z))⋒H(y)

for all x, y, z ∈ L. Therefore ℋ is a ⋒-hesitant fuzzy p-ideal of L.

Theorem 3.30

(Extension property for ⋒-hesitant fuzzy p-ideals) Let ℋ andbe ⋒-hesitant fuzzy ideals of a BCI-algebra L such that ℋ(0) = (0) and ℋ(x) ⊆ (x) for all x ∈ L. If ℋ is a ⋒-hesitant fuzzy p-ideal of L, then so is.

Proof

Assume that ℋ is a ⋒-hesitant fuzzy p-ideal of X. Using (2.6), (2.7) and (III), we have 0 * (0 * (x * (0 * (0 * x)))) = 0 for all x ∈ X. It follows from hypothesis and (3.14) that

G(x*(0*(0*x)))⊇H(x*(0*(0*x)))⊇H(0*(0*(x*(0*(0*x)))))=H(0)=G(0),

and that

G(x)⊇G(x*(0*(0*x)))⋒G(0*(0*x))⊇G(0)⋒G(0*(0*x))⊇G(0*(0*x))⋒G(0*(0*x))=G(0*(0*x))

by (3.11), (3.9) and Proposition 3.1. Therefore is a ⋒-hesitant fuzzy p-ideal of X by Theorem 3.28.

Acknowledgements

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Abstract
1. Introduction
2. Preliminaries
3. Subalgebras and Ideals of BCK/BCI-Algebras Based on the Hesitant Intersection
Acknowledgements
References

The author wishs to thank the anonymous reviewers for their valuable suggestions.

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. Subalgebras and Ideals of BCK/BCI-Algebras Based on the Hesitant Intersection
Acknowledgements
References

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