GENERALIZED INT-SOFT SUBALGEBRAS OF BE-ALGEBRAS

Abstract

The notion of θ-generalized int-soft subalgebras of BE-algebras is introduced, and related properties are investigated. Relations between int-soft subalgebras and θ-generalized int-soft subalgebras are discussed, and characterizations of θ-generalized int-soft subalgebras are considered.

1. Introduction

In 1966, Imai and Iséki [4] and Iséki [5] introduced two classes of abstract algebras. BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. As a generalization of a BCK-algebra, Kim and Kim [7] introduced the notion of a BE-algebra, and investigated several properties. In [1], Ahn and So introduced the notion of ideals in BE-algebras. They gave several descriptions of ideals in BE-algebras.

Various problems in system identification involve characteristics which are essentially non-probabilistic in nature [12]. In response to this situation Zadeh [13] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [14].

Uncertainties can’t be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [10]. Maji et al. [9] and Molodtsov [10] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [10] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [9] described the application of soft set theory to a decision making problem. Maji et al. [8] also studied several operations on the theory of soft sets. Chen et al. [3] presented a new definition of soft set parametrization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors.

Jun and Ahn [6] introduced the notion of int-soft subalgebras of a BE-algebra, and investigated their properties. They considered characterization of an int-soft subalgebra, and solved the problem of classifying int-soft subalgebras by their inclusive subalgebras.

In this paper, we consider a generalization of the paper [6]. We introduce the notion of θ-generalized int-soft subalgebras of BE-algebras, and investigate related properties. We discuss relations between int-soft subalgebras and θ-generalized int-soft subalgebras. We consider characterizations of θ-generalized int-soft subalgebras.

 

2. Preliminaries

Let K(τ) be the class of all algebras of type τ = (2, 0). By a BE-algebra we mean a system (X; ∗, 1) ∈ K(τ) in which the following axioms hold (see [7]):

A relation “≤” on a BE-algebra X is defined by

A nonempty subset S of a BE-algebra X is called a subalgebra of X if x∗y ∈ S for all x, y ∈ S.

Molodtsov [10] defined the soft set in the following way: Let U be an initial universe set and E be a set of parameters. Let denotes the power set of U and A ⊂ E.

A pair is called a soft set (see [10]) over U, where is a mapping given by

In other words, a soft set over U is a parameterized family of subsets of the universe U. For ε ∈ A, may be considered as the set of ε-approximate elements of the soft set . Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [10].

For a soft set over U and a subset γ of U, the γ-inclusive set of , denoted by , is defined to be the set

 

3. θ-generalized int-soft subalgebras

In what follows, we take a BE-algebra X, as a set of parameters, and let unless otherwise specified.

Definition 3.1 ([6]). A soft set over U is called an int-soft subalgebra of X if it satisfies:

Definition 3.2. If a soft set over U satisfies the following assertion:

then we say that is a θ-generalized int-soft subalgebra of X.

Obviously, every int-soft subalgebra is a θ-generalized int-soft subalgebra for all . Also, if θ = U then every θ-generalized int-soft subalgebra is an int-soft subalgebra. For every soft set over U, it is cleat that if is a θ-generalized int-soft subalgebra of X.

For a soft set over U, we know that there exists nonempty subset θ of U such that is a θ-generalized int-soft subalgebra, but not an int-soft subalgebra as seen in the following example.

Example 3.3. Let X = {1, a, b} be a BE-algebra with the following Cayley table:

Let be a soft set over U = ℤ(; the set of integers) defined as follows:

Then is a θ-generalized int-soft subalgebra of X with θ = 10ℤ, but it is not an int-soft subalgebra of X since . Also, is not a θ-generalized int-soft subalgebra of X with θ = 4ℤ or θ = 8ℤ.

Example 3.4. Let X = {1, a, b, c, d, 0} be a BE-algebra ([1]) with the following Cayley table:

Let be a soft set over U = ℤ(; the set of integers) defined as follows:

Then is a θ-generalized int-soft subalgebra of X with θ = 12ℤ, but it is not an int-soft subalgebra of X since

Theorem 3.5. A soft set over U is a θ-generalized int-soft subalgebra of X if and only if is a subalgebra of X for all with γ ⊆ θ.

Proof. Assume that is a θ-generalized int-soft subalgebra of X. Let with γ ⊆ θ. Then . It follows from (3.2) that

and so that is a subalgebra of X for all with γ ⊆ θ.

Conversely, suppose that is a subalgebra of X for all with γ ⊆ θ. Let x, y ∈ X be such that . Take γ = θ ∩ γx ∩ γy. Then . Thus

which shows that is a θ-generalized int-soft subalgebra of X. □

Lemma 3.6. Every θ-generalized int-soft subalgebra over U satisfies the following inclusion:

Proof. Using (2.1) and (3.2), we have

for all x ∈ X. □

Proposition 3.7. For any θ-generalized int-soft subalgebra over U, if a fixed element x ∈ X satisfies , then

Proof. Assume that a fixed element x ∈ X satisfies . Then

for all y ∈ X. □

Proposition 3.8. Let be a θ-generalized int-soft subalgebra over U. If a fixed element x ∈ X satisfies the following condition:

then

Proof. Taking y = 1 in (3.5) implies that by (2.3). It follows from Lemma 3.6 that □

Theorem 3.9. For every with ϑ ⊆ θ, every θ-generalized int-soft subalgebra is a ϑ-generalized int-soft subalgebra.

Proof. Let be a θ-generalized int-soft subalgebra over U and let with ϑ ⊆ θ. For any x, y ∈ X, we have

Therefore is a ϑ-generalized int-soft subalgebra over U for all with ϑ ⊆ θ. □

The following example shows that the converse of Theorem 3.9 is not true in general.

Example 3.10. Consider the soft set which is given in Example 3.4. Note that it is a ϑ-generalized int-soft subalgebra of X with ϑ = 12ℤ. If we take θ = 6ℤ, then ϑ ⊆ θ and is a θ-generalized int-soft subalgebra of X. But if we take θ = 4ℤ, then ϑ ⊆ θ and

Hence is not a θ-generalized int-soft subalgebra of X with θ = 4ℤ.

Theorem 3.11. If is a θ-generalized int-soft subalgebra over U, then the set

is a subalgebra of X.

Proof. Let . It follows from (3.2) that

and so that is a subalgebra of X. □

Theorem 3.12. For a subset S of X, define a soft set over U as follows:

where with τ ⊊ γ ∩ θ. Then is a θ-generalized int-soft subalgebra over U if and only if S is a subalgebra of X. Moreover,

Proof. Assume that is a θ-generalized int-soft subalgebra over U. Let x, y ∈ S. Then and so x ∗ y ∈ S. Thus S is a subalgebra of X.

Conversely, suppose that S is a subalgebra of X. Let x, y ∈ X. If x, y ∈ S, then x ∗ y ∈ S. Hence . If x ∉ S or y ∉ S, then . Therefore is a θ-generalized int-soft subalgebra over U. Moreover, we have

This completes the proof. □

For any BE-algebras X and Y, let μ : X → Y be a function and be soft sets over U.

(1) The soft set

where , is called the soft pre-image of under μ (see [6]).

(2) The soft set

where

is called the soft image of under μ (see [6]).

Theorem 3.13. Let μ : X → Y be a homomorphism of BE-algebras and a soft set over U. If is a θ-generalized int-soft subalgebra over U, then the soft pre-image under μ is also a θ-generalized int-soft subalgebra over U.

Proof. For any x1, x2 ∈ X, we have

Hence is also a θ-generalized int-soft subalgebra over U. □

Theorem 3.14. Let μ : X → Y be a homomorphism of BE-algebras and a soft set over U. If is a θ-generalized int-soft subalgebra over U and μ is injective, then the soft image μ under μ is also a θ-generalized int-soft subalgebra over U.

Proof. Let y1, y2 ∈ Y. If at least one of μ−1(y1) and μ−1(y1) is empty, then the inclusion

is clear. Assume that μ−1(y1) ≠ ∅ and μ−1(y2) ≠ ∅. Then

Therefore is a θ-generalized int-soft subalgebra over U. □

For any soft set over U and be a soft set over U where

Theorem 3.15. If is an int-soft subalgebra over U, then so is

Proof. For any x, y ∈ X and , we have

Hence is an int-soft subalgebra over U for all □

We pose a question as follows.

Question. Let be a soft set over U such that is an int-soft subalgebra over U for some an int-soft subalgebra over U?

The answer to the question above is false. In fact, let be a soft set over U which is not an int-soft subalgebra over U. If we take is an int-soft subalgebra over U.

Let be a subclass of such that

Theorem 3.16. Let be a soft set over U such that is an int-soft subalgebra over U for some , then is an int-soft subalgebra over U.

Proof. For any x, y ∈ X, we have

and so by (3.6). Therefore is an int-soft subalgebra over U. □

The following example shows that if is a θ-generalized int-soft subalgebra over U, then Theorem 3.15 is false.

Example 3.17. Consider the θ(= 12ℤ)-generalized int-soft subalgebra over X which is given in Example 3.4. For δ = ℕ, the soft set over U(= ℤ) is described as follows:

Since , we know that is not an int-soft subalgebra over U.

Theorem 3.18. If is a θ-generalized int-soft subalgebra over U, then is a (θ ∩ δ)-generalized int-soft subalgebra over U for all

Proof. Let x, y ∈ X. Then

and thus is a (θ ∩ δ)-generalized int-soft subalgebra over U for all □