Soft<i> BCH</i><i>-</i>Algebras of the Power Sets

Shuker Mahmood Khalil; Mayadah Abd Ulrazaq

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American Journal of Mathematics and Statistics

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

2018; 8(1): 1-7

doi:10.5923/j.ajms.20180801.01

Soft BCH-Algebras of the Power Sets

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Shuker Mahmood Khalil, Mayadah Abd Ulrazaq

Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq

Correspondence to: Shuker Mahmood Khalil, Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq.

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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, in the first we introduce the concept of algebra of the power set and new notions connected to it are investigated and discussed like subalgebra of the power set, soft algebra of the power set and soft subalgebra of the power set. Then some binary operations between two soft algebras of the power set are studied. Further, we state the relations between soft algebra of the power set and soft algebra of the power set. Moreover, several examples are given to illustrate the notations introduced in this work.

Keywords: Soft sets theory, Proper algebra, algebras

Cite this paper: Shuker Mahmood Khalil, Mayadah Abd Ulrazaq, Soft BCH-Algebras of the Power Sets, American Journal of Mathematics and Statistics, Vol. 8 No. 1, 2018, pp. 1-7. doi: 10.5923/j.ajms.20180801.01.

Article Outline

1. Introduction

2. Preliminaries

3. Soft BCH-algebra of the Power Sets

1. Introduction

algebras two classes of abstract algebras are introduced by Imai and Iseki ([22], [23]). The class of BCK-algebras is a proper subclass of the class of

algebras. Also, Hu and Li introduced a wider class of abstract algebras, it is said to be BCH-algebras ([5], [6]). Next, the concept of

algebras, which is another useful generalization of BCK –algebras are introduced (see [2], [3], [20]). After then, the concept of

algebra is introduced and studied [9]. The basic notions of soft sets theory are introduced by Molodtsov ([18]) to deal with uncertainties when solving problems in practice as in engineering, social science, environment, and economics. This notion is convenient and easy to apply as it is free from the difficulties that appear when using other mathematical tools as theory of theory of fuzzy sets, rough sets and theory of vague sets etc. Moreover, many researches on soft sets theory and some of their applications are studied (see [10]-[15]). On other word, many authors applied the notion of soft set on several classes of algebras like soft

algebras [7] and soft

algebras [16]. In recent years, for any

(finite set

), the notations of

algebra of the power set,

algebra of the power set, soft

algebra of the power set, soft edge

algebra of the power set are introduced (see [16], [17]). The aim of this paper is to introduce new branch of the pure algebra it's called

algebra of the power set. Then some binary operations between two soft

algebras of the power set are stated. Further, we study the relations between soft

algebra of the power set and soft

algebra of the power set. Also, several examples are given to illustrate the notations introduced in this work.

2. Preliminaries

In this section we recall the basic background needed in our present work.

Definition 2.1: ([20]) A

algebra is a non-empty set

with a constant 0 and a binary operation* satisfying the following axioms:

(i)-

(ii)-

(iii)-

and

imply that

for all x, y in X.

Definition 2.2: ([19]) A

algebra

is said to be

algebra if

satisfies the following additional axioms:

(1).

(2).

for all

Definition 2.3 ([8]) A

algebra

is a non-empty set

with a constant

and a binary operation* satisfying the following axioms:

(i)-

(ii)-

(iii)-

imply that

(iv)- For all

imply that

Definition 2.4: ([21]) A

algebra is a non-empty set

with a constant 0 and a binary operation* satisfying the following axioms:

(i)-

(ii)-

and

imply that

(iii)-

for all x, y ,z in X.

Definition 2.4: ([5]) A

algebra is a non-empty set

with a constant 0 and a binary operation* satisfying the following axioms:

(i)-

(ii)-

and

imply that

(iii)-

for all x, y ,z in X.

Definition 2.5: ([18]) Let

be an initial universe set and let

be a set of parameters. The power set of

is denoted by

. Let

be a subset of

. A pair

is said to be a soft set over

is a set-valued functions of

into the set of all subsets of the set

Definition 2.6: ([11]) Let

and

be two soft sets over

, then their union is the soft set

where

and for all

We write

Further, [4] for any two soft sets

and

over

their intersection is the soft set

over

, and we write

where

and

for all

Definition 2.7: ([8]) Let

be a soft set over

. Then

is said to be a soft

algebra over

is a

algebra for all

Definition 2.8: ([7]) Let

be a soft set over

. Then

is said to be a soft

algebra over

is a

algebra for all

Definition 2.9: ([7]) Let

be a soft set over

. Then

is said to be a soft

algebra over

is a

algebra for all

Definition 2.10: ([16]) Let

be a soft set over

. Then

is said to be a soft

algebra over

is a

algebra for all

Definition 2.10: ([24]) Let

be a soft set over

. Then

is said to be a soft

algebra over

is a

algebra for all

Definition 2.11: ([16]) Let

be non-empty set and

be a power set of

. Then

with a constant

and a binary operation* is said to be

algebra of the power set of

satisfying the following axioms:

(i)-

(ii)-

(iii)-

and

imply that

for all

Definition 2.12: ([16]) Let

be a

algebra of the power set of

. Then

is said to be

algebra of the power set of

if it satisfies the following additional axioms:

(1).

(2).

Definition: 2.13: ([16]) Let

be a

algebra of the power set of

. Then

is said to be a

algebra of the power set of

if it satisfies the identity

for all

Definition 2.14: ([16]) Let

be a

algebra of the power set of

. Then

is said to be a

algebra of the power set of

if it satisfies the identity

for all

Definition 3.1: ([17]) Let

be non-empty set and

be a power set of

. Then

with a constant

and a binary operation

is said to be

algebra of the power set of

satisfying the following axioms:

(i)-

(ii)-

and

imply that

for all

(iii)-

for all

Definition 2.15: ([16], [17]) Let

be a

algebra

algebra,

algebra) of the power set of

and let

be a collection of some random subsets of

. Then

is said to be

subalgebra (resp.

subalgebra,

subalgebra) of the power set of

, if

for any

Definition 2.16: ([16], [17]) Let

be a

algebra (resp.

algebra,

algebra) of the power set of

and let

be a set valued function , where

is a collection of some random subsets of

defined by

for all

where

is an arbitrary binary operation from

. Then the pair

is a soft set over

. Further,

is said to be a soft

algebra (resp. soft ρ-algebra, soft

algebra, soft

algebra) of the power set of

, if

is a

subalgebra (resp.

subalgebra,

subalgebra) of the power set of

for all

3. Soft BCH-algebra of the Power Sets

In this section we introduce the notion of soft

algebra of the power set and soft

subalgebra of the power set. We will illustrate the definitions with examples.

Definition 3.1 Let

be non-empty set and

be a power set of

. Then

with a constant

and a binary operation

is said to be

algebra of the power set of

satisfying the following axioms:

(i)-

(ii)-

and

imply that

for all

(iii)-

for all

Definition 3.2 A

algebra of power set is said to be proper

algebra of power set if it satisfies

, for some

Example 3.3 Let

and let

be a binary operation defined by the following table:

Table (1)

Then

is a proper

algebra of the power set of

, since there are

such that

Definition 3.4 Let

be a

algebra of the power set of

and let

be a collection of some random subsets of

. Then

is said to be

subalgebra of the power set of

, if

, for any

Example 3.5 Let

algebra of the power set of

in example (3.3). Then

are

subalgebra of the power set of

Remark 3.6 we will show that not necessary every

algebra of the power set is

algebra of the power set.

Example 3.7 Let

and let

be a binary operation defined by

for all

. Then

is a

algebra of the power set of

with the following table (2).

In other word,

, for some

. Then

is not

algebra. Also, for example,

and

are

subalgebra of the power set of

, but not

subalgebra.

Example 3.8: Let

and let

be a binary operation defined by

for all

. Then

is a

algebra of the power set of

with the following table (3).

Table (2)

Table (3)

Remarks 3.9:

(1) It is not necessary every

algebra of the power set of

algebra of the power set. In example (3.8), let

. Then

is not

algebra of the power set of

, since

Also, let

Then

is not

of the power set of

, since

Also,

is a

algebra of the power set of

. On the other hand,

and

are

subalgebras. However,

and

are not

subalgebras. Further, see example (3.7)

is not ρ-algebra.

(2) Further, if

is a

algebra of the power set of

satisfying

for any

Then

is a d-algebra of the power set of X.

Definition 3.10 Let

be a

algebra of the power set of

and let

be a set valued function , where

is a collection of some random subsets of

defined by

for all

where

is an arbitrary binary operation from

. Then the pair

is a soft set over

. Further,

is said to be a soft

algebra of the power set of

, if

is a

subalgebra of the power set of

for all

Example: 3.11 Let

be a proper

algebra of the power set of

with the following table:

Table (4)

Let

be a soft set over

where

and

is a set valued function defined by

for all

. Then

which are soft

subalgebras of the power set of

. Hence

is a soft

algebra of the power set of

. The next example shows that there exist set-valued functions

where

the soft set is not a soft

algebra of the power set of

Example 3.12: Consider the

algebra in example (3.11) with a set valued function defined by

for all

We have

is not a soft

algebra of the power set of

, since there exists

, but

is not soft

subalgebras of the power set of

Definition: 3.13 Let

be a soft

algebra of the power set of

, and let

, where

is defined by

for all

. Then

is defined by

for all

Lemma 3.14: If

is a soft

algebra of the power set of

, then

is a soft

algebra of the power set of

, for any

Proof: Let

be a

algebra of the power set of

and let

be a soft

algebra of the power set of

, then

is a

subalgebra of the power set of

, for all

. Moreover, for all

we have

, but

(since

). Hence

is a

subalgebra of the power set of

, for all

. Then

is a soft

algebra of the power set of

Definition 3.15: Let

be a soft

algebra of the power set of

. Then

is said to be a null soft

algebra of the power set of

for all

. Also,

is said to be an absolutely soft

algebra of the power set of

for all

Example 3.16 Let

be the

algebra of the power set of

in example (3.11) and let

where

are defined by

and

. Thus,

and hence

is an absolutely soft

algebra of the power set

. Moreover,

and hence

is a null soft

algebra of the power set

Lemma: 3.17 Let

be a

algebra of the power set of

. Then

is a

algebra of the power set of

, if

is a

algebra of the power set of

Proof: Since

is a

algebra of the power set of

. Then

satisfying the following axioms:

(i)-

(ii)-

(iii)-

and

imply that

for all

Also, let

be a

algebra of the power set of

, then

for all. Hence, from (i) we consider that

Further, from (ii) we have

and this implies that

Therefore, the following are hold:

(1)-

(2)-

and

imply that

(3)-

for all

. The

is a

algebra of the power set of

Corollary 3.18 If

is a

algebra of the power set of

. Then

is a

algebra of the power set of

, if

and

for all

Proof: Since

is a

algebra of the power set of

. Then

satisfying the following axioms:

(i)-

(ii)-

(iii)-

and

imply that

, for all

(iv)-

for all

Therefore, we consider only the following are hold:

1)-

2)-

and

imply that

Then, to prove that

is a

algebra of the power set of

, we need also to prove that

for any

Thus we have to show that

and hence we have

For any

we have the all cases that are cover all probabilities as following:

(1) If

(2) If

(3) If

, but

(4) If

, then

Hence, for all (1),(2),(3), and (4) we consider that

is a

algebra of the power set of

, since

for any

(5) If

thus since

and

is a

algebra of the power set of

. Then from (iv) we have

and this implies that

and hence

Then

is a

algebra of the power set of

and by[Lemma (3.17)] we have

is a

algebra of the power set of

Remark 3.19 From [Lemma (3.17) and Corollary (3.18)] we consider that, if

is a

algebra of the power set of

and

for all

Then

is a

algebra of the power set of

Theorem 3.20: Let

and

be two soft

algebras over

. If

then the union

is a soft

algebra of the power set of

Proof: Since

and by definition (2.6), we have for all

then

is a

subalgebra of the power set of

. Similarly, if

then

is a

subalgebra of the power set of

. Hence

is a soft

algebra of the power set of

. Thus the union of two soft

algebras of the power set of

is a soft

algebra of the power set of

Example 3.21: In example (3.4), let

and

be two soft sets over

where

and

. Define

for all

and

for all

. Note that

Thus, we have

and

Then

and

which are

subalgebras of the power set of

. Hence,

is a soft

algebra of the power set of

Remark 3.22: The condition

is important as if

then the theorem does not apply. In above example, if

and

. Then

which is not a

subalgebra of the power set of

. Therefore, is not a soft

algebra of the power set of

Theorem 3.23: Let

and

be two soft

algebras over

. If

, then the union

is a soft

algebra of the power set of

Proof: Since

, where

and

for all

[by definition (2.6)], Note that

is a mapping and so

is a soft set over

. We have,

is a

subalgebra of the power set of

. Hence,

is a soft

algebra of the power set of

. Therefore, the intersection of two soft

algebras is a soft

algebra.

Example 3.24: Consider the algebra in example (3.21) with

and

. Then

Note that both are

subalgebras of the power set of

. Hence,

is a soft

algebras of the power set of

Theorem 3.25: Let

be a

algebra of the power set of

with the condition

for any

. If

is a soft

algebra of the power set of

, then

is a soft

algebra of the power set of

Proof: Straightforward from Definitions [(3.10), (2.7)] and remark [(2)-(3.9)].

Theorem 3.26: Let

be a

algebra of the power set of

and

is a soft

algebra of the power set of

. Then

is a soft

algebra of the power set of

, if

algebra of the power set.

Proof: Straightforward from Definitions [(3.10), (2.7)] and [Lemma (3.17)].

The author would like to thank from the anonymous reviewers for carefully reading of the manuscript and giving useful comments, which will help to improve the paper.

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Paper Information

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Soft BCH-Algebras of the Power Sets

Article Outline

1. Introduction

2. Preliminaries

3. Soft BCH-algebra of the Power Sets

References