Bipolar Fuzzy α-ideal of BP-algebra

Osama Rashad El-Gendy

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American Journal of Mathematics and Statistics

p-ISSN: 2162-948X e-ISSN: 2162-8475

2020; 10(2): 33-37

doi:10.5923/j.ajms.20201002.01

Bipolar Fuzzy α-ideal of BP-algebra

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Osama Rashad El-Gendy

Batterjee Medical College for Sciences & Technology, Jeddah, Saudi Arabia

Correspondence to: Osama Rashad El-Gendy, Batterjee Medical College for Sciences & Technology, Jeddah, Saudi Arabia.

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This work is licensed under the Creative Commons Attribution International License (CC BY).
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Abstract

In this paper, the concept of bipolar fuzzy α-ideal of BP-algebra is introduced. We introduced α-ideal and fuzzy α-ideal. Several theorems are presented in this regard. The homomorphic image and inverse image of the bipolar fuzzy α-ideal are studied.

Keywords: BP-algebra, α-ideal, fuzzy α-ideal, Bipolar fuzzy -ideal

Cite this paper: Osama Rashad El-Gendy, Bipolar Fuzzy α-ideal of BP-algebra, American Journal of Mathematics and Statistics, Vol. 10 No. 2, 2020, pp. 33-37. doi: 10.5923/j.ajms.20201002.01.

Article Outline

1. Introduction

2. Preliminaries

3. α-ideal & Fuzzy α-ideal of BP-algebra

4. Bipolar Fuzzy α-ideal of BP-algebra

5. Conclusions and Future Research

1. Introduction

Y. Imai and K. Iséki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras [7,8]. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. In [4,5], Q. P. Hu and X. Li introduced a wide class of abstract: BCH-algebras. They had shown that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. In [27], J. Neggers, S. S. Ahn and H. S. Kim introduced Q-algebras which is a generalization of BCK / BCI-algebras and obtained several results. In 2002, Neggers and Kim [19,26,28] introduced a new notion, called a B-algebra, and obtained several results. In 2007, Walendziak [30] introduced a new notion, called a BF-algebra, which is a generalization of B-algebra. In 2013, Ahn and Han. [2] introduced a new notion, called BP-algebra which is related to several classes of algebra. In 1965, the concept of fuzzy sets, a remarkable idea in mathematics, was proposed by Zadeh [31]. In this traditional concept of fuzzy set, the membership degree expresses belongingness of an element to a fuzzy set. The membership degree of an element ranges over the interval [0, 1]. When the membership degree of an element is 1, then the element completely belongs to its corresponding fuzzy set, and the membership degree of an element is 0 means an element does not belong to the fuzzy set. Based on this tool, different fuzzy algebraic structures have been developed by many researchers, The fuzzy structures of BCK/BCI-algebras worked out by many researchers such as Jun [16,17,18,25], Liu [24], Bej and Pal [3], Jana et al. and others [10-15] have done much investigations on BCK/BCI/G/B-algebras related to these algebras. In 1994, the notion of bipolar fuzzy sets was proposed by Zhang [32,33] as a generalization of fuzzy sets [31]. Bipolar-valued fuzzy sets [22,23] are seen as an extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to [−1, 1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [−1, 0) of an element indicates that the element somewhat satisfies the implicit counter-property. Bipolar fuzzy sets have various applications in fuzzy algebras. For example, bipolar fuzzy ideals [1] in LA-semigroups, bipolar fuzzy sub-algebras and ideals [21] of BCK/BCI-algebras, bipolar fuzzy a-ideals in BCK/BCI-algebras [20] and bipolar valued fuzzy BCK/BCI-algebras [29] are some of them. The aim of this paper is to apply the notion of the bipolar fuzzy set to α-ideal of BP-algebra. The notions of α-ideal, fuzzy α-ideal and bipolar fuzzy α-ideal are defined, and a lot of properties are investigated. The homomorphic image and the inverse image of the bipolar fuzzy α-ideal are studied. Several theorems and basic properties that are related to the bipolar fuzzy α-ideal of BP-algebra are investigated. In section 5, we conclude and present some topics for future research.

2. Preliminaries

In this section, some elementary aspects necessary for this paper are included.

Definition 2.1 [2]. An algebra

is called BP-algebra if it satisfies the following axioms:

In X, we can define a binary relation “≤” by x ≤ y if and only if

Example 2.2 [2]. Let

. Define

as the following table:

Then

is a BP-algebra.

Theorem 2.3 [2]. If

is a BP-algebra, then following conditions hold: for any

Example 2.4 [2]. Let

Define

as the following table:

Then

is a BP-algebra

3. α-ideal & Fuzzy α-ideal of BP-algebra

In this section,

and fuzzy

of BP-algebra are defined and some important properties are presented.

Definition 3.1. A nonempty subset S of a BP-algebra X is called a subalgebra of X if

for all

Definition 3.2. A non-empty subset

of a BP-algebra

is called a

if for all

Definition 3.3. Let

and

be BP-algebra. A mapping

is said to be a homomorphism if

, for all

Definition 3.4. Let

be a BP-algebra. A fuzzy set

is called a fuzzy

if it satisfies:

Example 3.5. Consider a BP-algebra X = {0, a, b, c} in which the

operation is given by example 2.4. Let

be such that

. Define the mapping

and

Then routine calculations give that

is a fuzzy

of X.

Proposition 3.6. If

is a fuzzy α-ideal of BP-algebra X, then

implies

, for all

Proof. Let

be a Fuzzy

of BP-algebra X.

By definition 2.1, if

then

and

given that

for all

Then

4. Bipolar Fuzzy α-ideal of BP-algebra

In this section, we defined bipolar fuzzy

of BP-algebra and examined some related properties.

Definition 4.1 [22]. A bipolar fuzzy set

is defined as

where

and

are mappings. The positive membership degree

denotes the satisfaction degree of an element

to the property corresponding to a bipolar fuzzy set

and the negative membership degree

denotes the satisfaction degree of an element

to some implicit counter property of

and

this case is regarded as having only a positive satisfaction degree for

and

does not satisfy the property of

but somewhat satisfies the counter property of

In some cases, it is possible for an element

to be

and

when the membership function of the property overlaps that of the counter property of its portion of the domain (Lee [26]). We shall use the symbol

for the bipolar fuzzy set

Definition 4.2 [32]. For every two bipolar fuzzy set

and

we define

Proposition 4.3 [21]. A bipolar fuzzy set

is called a bipolar fuzzy subalgebra of

if it satisfies

and

for all

Definition 4.4 [21]. A bipolar fuzzy set

is called a bipolar fuzzy ideal of

if it satisfies the following conditions

Definition 4.5. A bipolar fuzzy set

of a BP-algebra

is called a bipolar fuzzy

if it satisfies the following conditions

Example 4.6. Consider a BP-algebra X = {0, a, b, c} in which the

operation is given by example 2.4. Define a bipolar fuzzy set

by,

and

routine calculation gives that

is a bipolar fuzzy

Theorem 4.7. The intersection of any set of bipolar fuzzy

in BP-algebra

is also a bipolar fuzzy

Proof. Let

be a family of bipolar fuzzy

in BP-algebra

. Then for any

And

Proposition 4.8. Let

be a bipolar fuzzy

of BP-algebra. If

holds in

, then

and

Proof. Let

holds in

. Then

Since

and

Then

And

Proposition 4.9. Let

be a bipolar fuzzy

of BP-algebra

. If

holds in

, then

and

Proof. Let

holds in

. Then

Since

Then

And

Then

Definition 4.10. Let

be a bipolar fuzzy

of BP-algebra

and

We define

is called upper

cut of

and lower

cut of

of the bipolar fuzzy

Theorem 4.11. Let

be a bipolar fuzzy

of BP-algebra

. Then for every

of BP-algebra.

Proof. Assume that

is a bipolar fuzzy

. For

and

where

Now we get that

and

this implies that

Next, let

This means

and

also,

and

Then

and

This implies that

Hence

is a

of BP-algebra.

Definition 4.12. Let

and

be BP-algebras, and let

be a mapping from the set

and the set

. If

and

are bipolar fuzzy sets of

and

respectively. Then

And

For all

is called the image of

under

. Similarly, the inverse image

defined as,

and

for all

Theorem 4.13. An into homomorphic inverse image of a bipolar fuzzy

of BP-algebra is also bipolar fuzzy

Proof. Let

be an into homomorphism of BP-algebras. Assume that

is a bipolar fuzzy

, and

is a bipolar fuzzy

. Then for all

and

Now, let

Then

And,

Hence the inverse image of a bipolar fuzzy

of BP-algebra is also bipolar fuzzy

Definition 4.14. A bipolar fuzzy subset

has

and

properties if for any subset

there exist

such that

and

Theorem 4.15. An onto homomorphic image of a bipolar fuzzy

of BP-algebra with

and

properties is a bipolar fuzzy

Proof. Let

be an onto homomorphism of BP-algebras and

is a bipolar fuzzy

. Let

is a bipolar fuzzy

with

and

properties. Then for all

we get

and

Since

is a bipolar fuzzy

We have

and

Note that

such that and are the zero elements of

and

respectively.

Thus

and

This implies that

and

for all

For any

let

be such that

and

Then

Similarly, we have

and

Then

Hence the onto homomorphic image of a bipolar fuzzy

of BP-algebra is also bipolar fuzzy

5. Conclusions and Future Research

To investigate the structure of an algebraic system, it is clear that

with special properties plays an important role. In the present paper, we have applied the notion of the bipolar fuzzy set theory to

of BP-algebra and investigated some of their useful properties. In the future, these definitions and fundamental results can be applied to some different algebraic structures. There are more topics that could take advantage of

. Like for example cubic intuitionistic

of BP-algebra, cubic fuzzy

of BP-algebra, and fuzzy soft

in BP-algebra. There are many other aspects which should be explored and studied in the area of BP-algebra such as anti-fuzzy

of BP-algebra, interval-valued fuzzy

of BP-algebra, intuitionistic fuzzy

of BP-algebra, doubt intuitionistic fuzzy

of BP-algebra, fuzzy derivations

of BP-algebra, and interval-valued intuitionistic fuzzy

of BP-algebra. It is our hope that this work would other foundations for further study of the theory of BP-algebra.

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Paper Information

Journal Information

Bipolar Fuzzy α-ideal of BP-algebra

Article Outline

1. Introduction

2. Preliminaries

3. α-ideal & Fuzzy α-ideal of BP-algebra

4. Bipolar Fuzzy α-ideal of BP-algebra

5. Conclusions and Future Research

References