Rough sets induced by fuzzy ideals in BCK -Algebras

Abstract

In this paper by considering of congruence relations induced by fuzzy ideals, we study rough sets in BCK-algebras. To this purpose we clarify a lower and upper approximations for any subset of X in this algebras. Finally, lower and upper rough ideals with respect to fuzzy ideal μ of X are discussed.

Keywords

Fuzzy ideal lower and upper approximation regular congruence relation rough ideal rough set

1 Introduction

The concept of rough set was originally proposed by Pawlak [27, 28] as a formal tool for modeling and processing in complete information in information systems. It seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences, especially in research areas such as machine learning, intelligent systems, inductive reasoning, pattern recognition, knowledge discovery, decision analysis and expert systems. Rough set theory, a new mathematical approach to deal with inexact, uncertain or vague knowledge, has recently received wide attention on the research areas in both of the real-life applications and the theory itself. Rough set theory is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. There are at least two methods for the development of this theory, the constructive and axiomatic approaches. In constructive methods, lower and upper approximations are constructed from the primitive notions such as equivalence relations on a universe [28, 30] and neighborhood systems [32, 35]. In Pawlak rough sets, the equivalence classes are the building blocks for the construction of the lower and upper approximations. Some authors, for example, Bonikowaski [1], Iwinski [13], J. Pomykala and J.A. Pomykala [29] showed that the set of rough sets forms a stone algebra. Comer [3] presented an interesting discussion of rough sets and various algebras related to the study of algebraic logic, such as stone algebras and relation algebras. It is a natural question to ask what does happen if we substitute an algebraic system instead of the universe set. Biswas and Nanda [2] introduced the notion of rough subgroups. Kuroki in [17] introduced the notion of a rough ideal in a semigroup. Kuroki and Wang [18] gave some properties of the lower and upper approximations with respect to the normal subgroups. Also, Kuroki and Mordeson in [19] studied the structure of rough sets and rough groups. Jun [15] discussed the roughness of Γ-subsemigroups and ideals in Γ-semigroups. In [16], as a generalization of ideals in BCK-algebras, the notion of rough ideals is discussed. In [4 –7] Davvas applied the concept of approximation spaces in the theory of algebraic hyper-structures. Mordeson [26] used covers of the universal set to define approximation operators on the power set of the given set. In [8] Davvaz concerned a relationship between rough sets and ring theory and considered a ring as a universal set and introduced the notion of rough ideals and rough subrings with respect to an ideal of a ring. Also, rough modules have been investigated by Davvaz and Mahdavipour [9]. In [33] the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup introduced. Based on constructive method, extensive research has also been carried out to compare the theory of rough sets with other theories of uncertainty such as fuzzy sets and conditional events. M. Irfan Ali, B. Davvaz, M. Shabir and S. Tanveer [10, 11] discussed on Some properties of generalized rough sets and characterized Roughness in Semirings. The theory of fuzzy sets was introduced by Zadeh [36] in 1965. This theory has provided a useful mathematical tool for describing the behavior of systems that are too complex or ill-defined to admit precise mathematical analysis by classical methods and tools. Extensive applications of the fuzzy set theory have been found in various fields. Since Rosenfeld [31] applied the notion of fuzzy sets to algebra and introduced the notion of fuzzy subgroups, the literature of various fuzzy algebraic concepts has been growing very rapidly. Liu [21] for example introduced and examined the notion of fuzzy ideal of a ring, subsequently, among others, Liu himself [22] and Mukherjee and Sen [25] fuzzified certain standard concerns a concept and result on rings and ideals. The purpose of this paper is the application of rough sets to finding lower and upper approximation for subsets of a given BCK-algebra by using fuzzy ideals on X. We characterize approximation of subsets of X and study upper and lower rough ideals and obtain theirs basic results.

2 Preliminaries

In this section we give some definitions and results of rough sets, BCK-algebras and fuzzy subsets which we need to extending our work. Let U be a universal set. For an equivalence relation θ on U, the set of the elements of U that are related to x ∈ U is called the equivalence class of x and is denoted by [x] _θ. Moreover, let U/θ denote the family of all equivalence classes induced on U by θ. For any X⊆U, we write X^c to denote the complement of X in U, that is the set U ∖ X. A pair (U, θ) where U ≠ φ and θ is an equivalence relation on U, is called an approximation space. The interpretation in rough set theory is that our knowledge of the objects in U extends only up to membership in the class of θ and our knowledge about a subset X of U is limited to the class of θ and their unions.

Definition 2.1. [28] For an approximation space (U, θ), by a rough approximation in (U, θ) we mean a mapping $(U, θ, -) : P (U) \to P (U) \times P (U)$ defined for every X ∈ P (U) by $(U, θ, X) = ((\underline{U}, θ, X), (\bar{U}, θ, X))$ , where $\begin{matrix} (\underline{U}, θ, X) & = & {x \in U | [x]_{θ} \subseteq X}, (\bar{U}, θ, X) \\ = & {x \in U | [x]_{θ} \cap X \neq φ} \end{matrix}$ $(\underline{U}, θ, X)$ is called a lower rough approximation of X in (U, θ), where $(\bar{U}, θ, X)$ is called a upper rough approximation of X in (U, θ).

Given an approximation space (U, θ), a pair (A, B) ∈ P (U) × P (U) is called a rough set in (U, θ) if and only if (A, B) = (U, θ, X) for some X ∈ P (U). Let (U, θ) be an approximation space and X be a non-empty subset of U, then the following hold:

If $(\underline{U}, θ, X) = (\bar{U}, θ, X)$ , then X is called definable,

If $(\underline{U}, θ, X) = φ$ , then X is called empty interior,

If $(\bar{U}, θ, X) = U$ , then X is called empty exterior.

(for more see [28]. The lower approximation of X in (U, θ) is the greatest definable set in U contained in X. The upper approximation of X in (U, θ) is the least definable set in U containing X. Therefore, we have:

\begin{matrix} (\underline{U}, θ, X) & = & \cup {S | S \subseteq X, S is definable}, \\ (\bar{U}, θ, X) & = & \cap {S | X \subseteq S, S is definable} . \end{matrix}

Definition 2.2. [14] A BCK-algebra is a set X with a binary operation * and a constant 0 satisfying the following axioms, for any x, y, z ∈ X:

((x * y) * (x * z)) * (z * y) =0,

(x * (x * y)) * y = 0,

x * x = 0,

x * y = 0 and y * x = 0 imply x = y,

0 * x = 0.

In a BCK-algebra, we can define a partial ordering ≤ by putting x ≤ y if and only if x * y = 0. In a BCK-algebra the following hold:

(x * z) * (y * z) ≤ x * y,

(x * (x * (x * y)) = x * y,

x * 0 = x,

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

x ≤ y imply x * z ≤ y * z and z * y ≤ z * x.

A BCK-algebra X is called commutative if x ∧ y = y ∧ x for all x, y ∈ X, where x ∧ y = y * (y * x). A BCK-algebra X is called implicative if x = x * (y * x) for all x, y ∈ X. It is well known that an implicative BCK-algebra is commutative. A BCK-algebra is called a BCK-algebra with condition (S), if for any a, b ∈ X, the set A (a, b) = {x ∈ X|x * a ≤ b} has a greatest element in X. This greatest element is denoted by a ∘ b, (see [12 , 24]. Recall that a non empty subset I of a BCK-algebra X is an ideal of X if 0 ∈ I and x * y ∈ I, y ∈ I imply that x ∈ I for all x, y, z ∈ X. A BCK-algebra X is called associative if x * (y * z) = (x * y) * z for any x, y, z ∈ X. If A, B be two non empty subset of a BCK-algebra X, then A * B is defined as follow: $A * B = {a * b | a \in A, b \in B} .$

Definition 2.3. [36] Let μ and λ be two fuzzy subsets of a set X. The inclusion λ⊆μ is defined by λ (x) ≤ μ (x) for all x ∈ X, and μ∩ λ is defined by: $(μ \cap λ) (x) = min {μ (x), λ (x)} for all x \in X .$

Definition 2.4. [34] A fuzzy subset μ of a BCK-algebra X is said to be a fuzzy ideal of X if it satisfies: (F₁) μ (0) ≥ μ (x) forall x ∈ X, (F₂) μ (x) ≥ min {μ (x * y) , μ (y)} forall x, y ∈ X .

Lemma 2.5. [34] Let μ be a fuzzy subset of a BCK-algebra X. Then μ is a fuzzy ideal of X if and only if $\begin{matrix} (x * y) * z & = & 0 implies that μ (x) \geq μ (y) \land μ (z) \\ for all x, y, z \in X, \end{matrix}$ where μ (y) ∧ μ (z) = min {μ (y) , μ (z)} .

Lemma 2.6. [34] If x ≤ y, then μ (x) ≥ μ (y).

Definition 2.7. [14] Let (X, * , θ) be a BCK-algebra and θ be an equivalence relation on X. Then

θ is called a right congruence relation on X if (x, y) ∈ θ implies (x * u, y * u) ∈ θ for all u ∈ X .

θ is called a left congruence relation on X if (x, y) ∈ θ implies (u * x, u * y) ∈ θ for all u ∈ X .

θ is called a congruence relation on X if it be a left and right congruence relation or for any (x, y) ∈ θ and (u, v) ∈ θ, we have (x * u, y * v) ∈ θ.

3 Congruence relation with respect to fuzzy ideals of BCK-algebra

In this section by X we mean a BCK-algebra, then we introduce and study congruence relation on BCK-algebra induced by fuzzy ideals. For a nonempty set by F (X) we mean the set of all fuzzy subsets of X. Define operation * on F (X) by: $\begin{matrix} (μ * λ) (x) & = & {Sup}_{a * b \leq x} {min {μ (a), λ (b)}} \\ for all x \in X . \end{matrix}$

Recall that for μ ∈ F (X) and t ∈ [0, 1], the set U (μ, t) = {(x, y) ∈ X × X : μ (x * y) ≥ t, μ (y * x) ≥ t} is a t-level relation of μ.

Lemma 3.1. Let μ be a fuzzy ideal of BCK-algebra X and t ∈ [0, 1]. Then every nonempty U (μ, t) is a congruence relation on X.

Proof. For any element x of X, we have μ (x * x) = μ (0) ≥ t and so (x, x) ∈ U (μ, t). Now, if (x, y) ∈ U (μ, t), then (y, x) ∈ U (μ, t). Let (x, y) ∈ U (μ, t) and (y, z) ∈ U (μ, t). Since μ is a fuzzy ideal of X, then by Lemma 2.5, μ (x * z) ≥ μ (y * z) ∧ μ (x * y) ≥ t and μ (z * x) ≥ μ (z * y) ∧ μ (y * x) ≥ t and so we have (x, z) ∈ U (μ, t). Therefore U (μ, t) is an equivalence relation on X. Now, let (x, y) ∈ U (μ, t) and a ∈ X. Since (x * a) * (y * a) ≤ x * y, by Lemma 2.6 we have: $μ ((x * a) * (y * a)) \geq μ (x * y) \geq t .$

By similar way, we show that μ ((y * a) * (x * a)) ≥ t and so (x * a, y * a) ∈ U (μ, t). By similar manner, we have (a * x, a * y) ∈ U (μ, t) for all a ∈ X and (x, y) ∈ U (μ, t). Therefore U (μ, t) is a congruence relation on X. □

Lemma 3.2. Let μ and λ be fuzzy ideals of a BCK-algebra X and t ∈ [0, 1]. Then U (μ ∩ λ , t) = U (μ, t) ∩ U (λ , t).

Proof. $\begin{matrix} (x, y) \in U (μ \cap λ, t) \\ \Leftrightarrow (μ \cap λ) (x * y) \geq t and (μ \cap λ) (y * x) \geq t, \\ \Leftrightarrow min {μ (x * y), λ (x * y)} \geq t and \\ min {μ (y * x), λ (y * x)} \geq t, \\ \Leftrightarrow μ (x * y) \geq t, λ (x * y) \geq t, \\ and μ (y * x) \geq t, λ (y * x) \geq t, \\ \Leftrightarrow (x, y) \in U (μ, t) and (x, y) \in U (λ, t), \\ \Leftrightarrow (x, y) \in U (μ, t) \cap U (λ, t) . \end{matrix}$

Note that we denote the equivalence class of U (μ, t) containing x by [x] _(μ,t). □

Lemma 3.3. Let μ and λ be two fuzzy ideals of a BCK-algebra X such that λ⊆μ and t ∈ [0, 1]. Then [x] _(λ,t)⊆ [x] _(μ,t) for all x ∈ X.

Proof. $\begin{matrix} (x, y) \in U (λ, t) & \Rightarrow & λ (x * y) \geq t, λ (y * x) \geq t, \\ \Rightarrow & μ (x * y) \geq t, μ (y * x) \geq t, \\ \Rightarrow & (x, y) \in U (μ, t) . □ \end{matrix}$

Definition 3.4. Let μ and λ be two fuzzy ideals of a BCK-algebra X. The composition of congruence relation U (μ, t) and U (λ , t) is defined as follow: $\begin{matrix} U (μ, t) \circ U (λ, t) = {(a, b) \in X \times X : \exists c \in X \\ such that (a, c) \in U (μ, t), (c, b) \in U (λ, t)} . \end{matrix}$

It is easy to verify that U (μ, t) ∘ U (λ , t) is a congruence relation and it is denoted by U (μ ∘ λ , t).

Theorem 3.5. Let μ and λ be two fuzzy ideals of a BCK-algebra X and t ∈ [0, 1]. Then U (μ ∘ λ , t) ⊆U (μ * λ , t).

Proof. Assume that (a, b) ∈ U (μ ∘ λ , t), then there exist c ∈ X such that (a, c) ∈ U (μ, t) , (c, b) ∈ U (λ , t). Hence, μ (a * c) ≥ t and λ (b * c) ≥ t. Thus, we have: $\begin{matrix} (μ * λ) (a * b) & = & {Sup}_{u * v \leq a * b} {min {μ (u), λ (v)}} \\ \geq & min {μ (a * c), λ (b * c)} \geq t . \end{matrix}$

By a similar way we get (μ * λ) (b * a) ≥ t. Therefore, (a, b) ∈ U (μ * λ , t). □

4 Rough sets in BCK-algebras

Let μ be a fuzzy ideal of a BCK-algebra X and U (μ, t) be a t-level congruence relation of μ on X. For a non-empty subset A of X, the sets $\begin{matrix} \underline{U} (μ, t, A) & = & {x \in X | [x]_{(μ, t)} \subseteq A}, \\ \bar{U} (μ, t, A) & = & {x \in X | [x]_{(μ, t)} \cap A \neq φ} . \end{matrix}$ are called lower and upper approximations of the set A with respect to U (μ, t), respectively.

Proposition 4.1. Let μ be a fuzzy ideal of a BCK-algebra X. Then for every approximation space (X, μ, t) and every subset A and B of X, we have:

$\underline{U} (μ, t, φ) = φ = \bar{U} (μ, t, φ)$ ,

$\underline{U} (μ, t, X) = X = \bar{U} (μ, t, X)$ ,

$\underline{U} (μ, t, A) \subseteq A \subseteq \bar{U} (μ, t, A)$ ,

If A⊆B, then $\underline{U} (μ, t, A) \subseteq \underline{U} (μ, t, B)$ and $\bar{U} (μ, t, A) \subseteq \bar{U} (μ, t, B)$ ,

$\underline{U} (μ, t, \underline{U} (μ, t, A)) = \underline{U} (μ, t, A)$ ,

$\bar{U} (μ, t, \bar{U} (μ, t, A)) = \bar{U} (μ, t, A)$ ,

$\bar{U} (μ, t, \underline{U} (μ, t, A)) = \underline{U} (μ, t, A)$ ,

$\underline{U} (μ, t, \bar{U} (μ, t, A)) = \bar{U} (μ, t, A)$ ,

$\underline{U} (μ, t, A) = (\bar{U} (μ, t, A^{c}))^{c}$ ,

$\bar{U} (μ, t, A) = (\underline{U} (μ, t, A^{c}))^{c}$ ,

$\underline{U} (μ, t, A \cap B) = \underline{U} (μ, t, A) \cap \underline{U} (μ, t, B)$ ,

$\bar{U} (μ, t, A \cap B) \subseteq \bar{U} (μ, t, A) \cap \bar{U} (μ, t, B)$ ,

$\underline{U} (μ, t, A \cup B) \supseteq \underline{U} (μ, t, A) \cup \underline{U} (μ, t, B)$ ,

$\bar{U} (μ, t, A \cup B) = \bar{U} (μ, t, A) \cup \bar{U} (μ, t, B)$ ,

$\underline{U} (μ, t, [x]_{(μ, t)}) = \bar{U} (μ, t, [x]_{(μ, t)})$ ,

for all x ∈ X.

Proof. Proof is similar to the proof of Theorem 2.1 of [17] and Proposition 4.1 of [4]. □

The following examples show that the converse of (12) and (13) in Proposition 4.1. is not true.

Example 4.2. Let X = {0, 1, 2, 3} be a BCK-algebra with Cayley table as follow:

*	0	1	2	3
0	0	0	0	0
1	1	0	0	1
2	2	1	0	2
3	3	3	3	0

Define μ : Xlo [0, 1] by μ (0) = μ (1) = μ (2) =0.7 and μ (3) =0.5. By a routine calculation μ is a fuzzy ideal of X and so we have: $\begin{matrix} U (μ, 0.7) \\ = {(x, y) | μ (x * y) \geq 0.7, μ (y * x) \geq 0.7} \\ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 1), (0, 2), \\ (1, 0), (1, 2), (2, 1), (2, 0)}, \\ U (μ, 0.5) = X \times X . \end{matrix}$

Now, let A = {0, 1} and B = {0, 2, 3}, then $\begin{matrix} \underline{U} (μ, 0.7, A) & = & {x \in X | [x]_{(μ, 0.7)} \subseteq A} = {}, \\ \underline{U} (μ, 0.7, B) & = & {x \in X | [x]_{(μ, 0.7)} \subseteq B} = {3}, \\ \underline{U} (μ, 0.7, A \cup B) & = & {x \in X | [x]_{(μ, 0.7)} \subseteq A \cup B} \\ = & {0, 1, 2, 3} . \end{matrix}$

Therefore, we see that $\underline{U}$ (μ, 0.7, A ∪ B) ⊈ $\underline{U}$ (μ, 0.7, A) ∪ $\underline{U}$ (μ, 0.7, B).

Example 4.3. In Example 4.2 if we assume that C = {1, 3} , D = {0, 2, 3}, then we have: $\begin{matrix} \bar{U} (μ, 0.7, C) \\ = {x \in X | [x]_{(μ, 0.7)} \cap C \neq φ} = {0, 1, 2, 3}, \\ \bar{U} (μ, 0.7, D) \\ = {x \in X | [x]_{(μ, 0.7)} \cap D \neq φ} = {0, 1, 2, 3}, \\ \bar{U} (μ, 0.7, C \cap D) \\ = {x \in X | [x]_{(μ, 0.7)} \cap (C \cap D) \neq φ} = {3} . \end{matrix}$

Therefore, $\bar{U} (μ, 0.7, C) \cap \bar{U} (μ, 0.7, D) ⊈ \bar{U} (μ, 0.7, C \cap D)$ . □

Theorem 4.4. Let μ and λ be fuzzy ideals of a BCK-algebra X and t ∈ [0, 1]. Then for a non-empty subset A of X, $\bar{U} (μ \cap λ, t, A) \subseteq \bar{U} (μ, t, A) \cap \bar{U} (λ, t, A) .$

Proof. We have: $\begin{matrix} x \in \bar{U} (μ \cap λ, t, A) \\ \Leftrightarrow [x]_{(μ \cap λ, t)} \cap A \neq φ \\ \Leftrightarrow \exists a \in [x]_{(μ \cap λ, t)} \cap A \\ \Leftrightarrow (a, x) \in U (μ \cap λ, t) and a \in A \\ \Leftrightarrow (μ \cap λ) (a * x) \geq t and (μ \cap λ) (x * a) \\ \geq t and a \in A \\ \Leftrightarrow min {μ (a * x), λ (a * x)} \geq t and \\ min {μ (x * a), λ (x * a)} \geq t and a \in A \\ \Leftrightarrow μ (a * x) \geq t, μ (x * a) \geq t and \\ λ (a * x) \geq t, λ (x * a) \geq t and a \in A \\ \Leftrightarrow (a, x) \in U (μ, t) and (a, x) \in U (λ, t) and a \in A \\ \Leftrightarrow a \in [x]_{(μ, t)} and a \in [x]_{(λ, t)} and a \in A \\ \Leftrightarrow [x]_{(μ, t)} \cap A \neq φ and [x]_{(λ, t)} \cap A \neq φ \\ \Leftrightarrow x \in \bar{U} (μ, t, A) and x \in \bar{U} (λ, t, A) \\ \Leftrightarrow x \in \bar{U} (μ, t, A) \cap \bar{U} (λ, t, A) . □ \end{matrix}$

In the following example we show that the equality in Theorem 4.4 does not hold in general.

Example 4.5. Let X = {0, 1, 2, 3} be a BCK-algebra with Cayley table given by:

*	0	1	2	3
0	0	0	0	0
1	1	0	1	1
2	2	2	0	2
3	3	3	3	0

Define μ : Xlo [0, 1] by μ (0) = μ (1) = μ (3) =0.7 and μ (2) =0.5. By routine calculations give that μ is a fuzzy ideal of X. Define λ : Xlo [0, 1] by λ(0) = λ (2) =0.7 and λ(1) = λ (3) =0.5. It is easy to verify that λ is a fuzzy ideal of X, so we have $\begin{matrix} U (μ, 0.7) \\ = {(x, y) | μ (x * y) \geq 0.7, μ (y * x) \geq 0.7} \\ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 1), (0, 3), \\ (1, 0), (1, 3), (3, 1), (3, 0)}, \\ U (λ, 0.7) \\ = {(x, y) | λ (x * y) \geq 0.7, λ (y * x) \geq 0.7} \\ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 2), (2, 0)}, \\ U (μ \cap λ, 0.7) = {(x, y) | (μ \cap λ) (x * y) \\ \geq 0.7, (μ \cap λ) (y * x) \geq 0.7} \\ = {(0, 0), (1, 1), (2, 2), (3, 3)} . \end{matrix}$

Now, let A = {1, 2}. Also, we have: $\begin{matrix} \bar{U} (μ, 0.7, A) & = & {x \in X | [x]_{(μ, 0.7)} \cap A \neq φ} \\ = & {0, 1, 2, 3}, \\ \bar{U} (λ, 0.7, A) & = & {x \in X | [x]_{(λ, 0.7)} \cap A \neq φ} \\ = & {0, 1, 2}, \\ \bar{U} (μ \cap λ, 0.7, A) & = & {x \in X | [x]_{(μ \cap λ, 0.7)} \cap A \neq φ} \\ = & {1, 2} . \end{matrix}$

Therefore, we see that $\bar{U} (μ \cap λ, 0.7, A) \neq \bar{U} (μ, 0.7, A) \cap \bar{U} (λ, 0.7, A) .$

Theorem 4.6. Let μ and λ be fuzzy ideals of a BCK-algebra X and t ∈ [0, 1]. Then for a non-empty subset A of X, $\underline{U}$ (μ, t, A) ∩ $\underline{U}$ (λ , t, A) ⊆ $\underline{U}$ (μ ∩ λ , t, A).

Proof. We have: $\begin{matrix} x \in \underline{U} (μ, t, A) \cap \underline{U} (λ, t, A) \\ \Leftrightarrow x \in \underline{U} (μ, t, A) and x \in \underline{U} (λ, t, A) \\ \Leftrightarrow [x]_{(μ, t)} \subseteq A and [x]_{(λ, t)} \subseteq A \\ \Leftrightarrow [x]_{(μ \cap λ, t)} \subseteq A (by Lemma 3.3) \\ \Leftrightarrow x \in \underline{U} (μ \cap λ, t, A) . \end{matrix}$

Now, in Example 4.5 we have: $\begin{matrix} \underline{U} (μ, 0.7, A) & = & {x \in X | [x]_{(μ, 0.7)} \subseteq A} = {2}, \\ \underline{U} (λ, 0.7, A) & = & {x \in X | [x]_{(λ, 0.7)} \subseteq A} = {1}, \\ \underline{U} (μ \cap λ, 0.7, A) & = & {x \in X | [x]_{(μ \cap λ, 0.7)} \subseteq A} \\ = & {1, 2} . \end{matrix}$

Thus, we have: $\underline{U} (μ \cap λ, 0.7, A) \neq \underline{U} (μ, 0.7, A) \cap \underline{U} (λ, 0.7, A) .$

Therefore the equality of Theorem 4.6 does not hold in general. □

Definition 4.7. A non-empty subset A of a BCK-algebra X is called a upper(resp. lower) rough ideal of X respect to μ, if $\bar{U} (μ, t, A)$ ( $\underline{U}$ (μ, t, A)) be an ideal of X. Note that a non-empty subset A of a BCK-algebra X is called rough ideal respect to μ, if A is an upper and lower rough ideal.

Theorem 4.8. Let μ be a fuzzy ideal of an associative BCK-algebra X. Then any ideal of X is an upper rough ideal of X.

Proof. We show that $\bar{U} (μ, t, A)$ is an ideal of X. Since A is an ideal of X, we have $0 \in A \subseteq \bar{U} (μ, t, A)$ . Now, let $x * y \in \bar{U} (μ, t, A)$ and $y \in \bar{U} (μ, t, A)$ , then [x * y] _(μ,t) ∩ A ≠ φ and [y] _(μ,t) ∩ A ≠ φ. Hence, there exist a ∈ [x * y] _(μ,t) ∩ A and b ∈ [y] _(μ,t) ∩ A. Since A is an ideal of X, we have a * b ∈ A. Also, we have: $\begin{matrix} μ (x * (a * b)) \\ \geq min {μ ((x * (a * b)) * (y * b)), μ (y * b)} \\ = min {μ ((x * (y * b)) * (a * b)), μ (y * b)} \\ = min {μ (((x * y) * b) * (a * b)), μ (y * b)} \\ \geq min {μ ((x * y) * a), μ (y * b)} \geq t . \end{matrix}$

On the other hand, we have: $\begin{matrix} μ ((a * b) * x) \\ \geq min {μ (((a * b) * x) * (y * b)), μ (y * b)} \\ = min {μ (((a * x) * b) * (y * b)), μ (y * b)} \\ \geq min {μ ((a * x) * y), μ (y * b)} \\ = min {μ (a * (x * y)), μ (y * b)} \geq t, \end{matrix}$

Hence, a * b ∈ [x] _(μ,t) ∩ A which implies $x \in \bar{U} (μ, t, A)$ . Therefore, A is an upper rough ideal of X. □

Example 4.9. Let X = {0, 1, 2} be a BCK-algebra with Cayley table given by:

*	0	1	2
0	0	0	0
1	1	0	0
2	2	2	0

Define μ : Xlo [0, 1] by μ (0) = μ (1) =0.5 and μ (2) =0.3. By routine calculations give that μ is a fuzzy ideal of X. Also, we have: $U (μ, 0.5) = {(0, 0), (1, 1), (2, 2), (0, 1), (1, 0)} .$

Now, let A = {0, 1}. Thus, we have: $\bar{U} (μ, 0.5, A) = {0, 1} = \underline{U} (μ, 0.5, A) .$

Therefore, A is a rough ideal of X. In the next example we show that the converse of Theorem 4.8 is not true and every ideal is not necessarily a lower rough BCK-ideal.

Example 4.10. Let (X, * , 0) be a BCK-algebra and μ be a fuzzy ideal as in Example 4.5. Then E = {1} is not an ideal of X but $\bar{U} (μ, 0.7, E) = {0, 1, 3}$ is an ideal of X. On the other hand I = {0, 2} is an ideal of X but $\underline{U} (μ, 0.7, I) = {2}$ is not an ideal of X.

Proposition 4.11.Let μ and λ be two fuzzy ideals of a BCK-algebra X such that λ⊆μ. Then for a non-empty subset A of X and t ∈ [0, 1], we have:

$\bar{U} (λ, t, A) \subseteq \bar{U} (μ, t, A)$ ,

$\underline{U}$ (μ, t, A) ⊆ $\underline{U}$ (λ , t, A).

Proof. (i) Let x be an arbitrary element of $\bar{U} (λ, t, A)$ . Then [x] _(λ,t) ∩ A ≠ φ. By lemma 3.3 we have [x] _(λ,t)⊆ [x] _(μ,t). Hence [x] _(μ,t) ∩ A ≠ φ which implies that $x \in \bar{U} (μ, t, A)$ .

(ii) Assume that x ∈ $\underline{U}$ (μ, t, A), then [x] _(μ,t)⊆A.

Now, by lemma 3.3 we obtain [x] _(λ,t)⊆A which implies that x ∈ $\underline{U}$ (λ , t, A). The next Lemma immediately follows from Proposition 4.11. □

Lemma 4.12. Suppose that μ and λ be fuzzy ideals of a BCK-algebra X and t ∈ [0, 1]. Let A be a non-empty subset of X, if U (λ , t) ⊆U (μ, t), then

$\bar{U} (λ, t, A) \subseteq \bar{U} (μ, t, A)$ ,

$\underline{U}$ (μ, t, A) ⊆ $\underline{U}$ (λ , t, A).

The next Theorem immediately follows from Theorem 3.5 and Lemma 4.12.

Theorem 4.13. Let μ and λ be fuzzy ideals of a BCK-algebra X and t ∈ [0, 1]. Then for a non-empty subset A of X, we have

$\bar{U} (μ \circ λ, t, A) \subseteq \bar{U} (μ * λ, t, A)$ ,

$\underline{U}$ (μ * λ , t, A) ⊆ $\underline{U}$ (μ ∘ λ , t, A).

Theorem 4.14. Let μ and λ be fuzzy ideals of a BCK-algebra X and t ∈ [0, 1]. Then If A be an ideal of X, then $\bar{U} (μ, t, A) * \bar{U} (λ, t, A) \subseteq \bar{U} (μ \circ λ, t, A)$ .

Proof. Suppose that z be any element of $\bar{U} (μ, t, A) * \bar{U} (λ, t, A)$ , then z = a * b for some $a \in \bar{U} (μ, t, A)$ and $b \in \bar{U} (λ, t, A)$ . Thus [a] _(μ,t) ∩ A ≠ φ and [b] _(λ,t) ∩ A ≠ φ and so there exist elements x and y in X such that x ∈ [a] _(μ,t) ∩ A and y ∈ [b] _(λ,t) ∩ A. Since A is an ideal of X, then x * y ∈ A. Also, we have: $\begin{matrix} λ ((x * b) * (x * y)) \geq λ (y * b) \geq t, \\ λ ((x * y) * (x * b)) \geq λ (b * y) \geq t, \\ μ ((x * b) * (a * b)) \geq μ (x * a) \geq t, \\ μ ((a * b) * (x * b)) \geq μ (a * x) \geq t . \end{matrix}$

Thus, we have (x * y, x * b) ∈ U (λ , t) and (x * b, a * b) ∈ U (μ, t). Hence (x * y, a * b) ∈ U (μ ∘ λ , t) and so x * y ∈ [a * b] _(μ∘λ,t) ∩ A which implies [a * b] _(μ∘λ,t) ∩ A ≠ φ. Therefore, $z = a * b \in \bar{U} (μ \circ λ, t, A)$ . □

5 Conclusions

This paper is intend to built up connection between rough sets, fuzzy sets and BCK-algebras. We have presented a definition of the lower and upper approximation of a subset of a BCK-algebra with respect to a fuzzy ideal. This definition and main results are easily extended to other algebraic structure such as BL-algebra, lattices, etc.

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