Roughness in n -ary semigroups based on fuzzy ideals

Abstract

A novel congruence relation U (μ, t) on an n-ary semigroup S is established. We show that U (μ, t) is a congruence relation on an n-ary semigroup S if μ is a fuzzy ideal of S. Based on this idea, we construct the lower and upper approximations in an n-ary semigroup. Furthermore, we introduce the notions of rough ideals and rough prime ideals by means of fuzzy ideals of an n-ary semigroup. In particular, the concepts of rough n-ary semigroups, rough homomorphisms are introduced and some relative properties are also investigated.

Keywords

Rough set Upper (lower) approximation Approximation space n-ary semigroup (Prime) ideal Rough (prime) ideal Rough homomorphism

1 Introduction

The concept of rough sets was originally proposed by Pawlak [22] as a formal tool for modeling and processing incomplete information in information systems. Since then the subject has been investigated by many researchers. The theory of rough sets 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. A key notion in the Pawlak rough set model is the equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all equivalence classes which are subsets of the set, and the upper approximation is the union of all equivalence classes which have a non-empty intersection with the set. However, the restrictive for many practical applications. For this reason, some more general models have been put forward, such as [26 –28].

It is a natural question to ask what happens if we substitute the universe set with an algebraic system. Some researchers have studied the algebraic properties of rough sets. Biswas and Nanda [3] introduced the notion of rough subgroups. Kuroki and Wang [19] gave some properties of the lower and upper approximations with respect to the normal subgroups. Later, Kuroki [18] introduced the notion of a rough ideal in a semigroup and gave some properties of such notion. Jun [21] studied the rough set theory in BCK-algebras. Davvaz [6] investigated the concept of rough ideals with respects to an ideal of rings. In particular, Davvaz [7] considered the relationships between rough sets, fuzzy sets and ring theory, and presented a definition of the lower and upper approximations of subset of a ring by means of a fuzzy ideal. Also, rough modules have been investigated by Davvaz and Mahdavipour [10]. In [24], Xiao introduced the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup, and gave some properties of such ideals. In 2008, Davvaz [8] applied the concepts of approximation spaces and rough sets in n-ary algebraic systems. The other results can be found in [2 , 23].

The generalization of algebraic structures was in active research for a long time, it was first initiated by Kasner [16] in 1904. In the following decades and nowadays, a number of different n-ary systems have been studied in depth in different contexts. In [11 –13], Dudek studied some properties of n-ary semigroups, proved some results and presented many examples of n-ary groups. Later Crombez et al. [4, 5] gave the generalized rings and named it as (m, n)-rings and introduced their quotient structure. In addition, Alan et al. [1] proposed a new class of mathematical structures called (m, n)-semirings (which generalize the usual semirings) and described their basic properties. Up till now, the theory of n-ary systems has many applications, for example, lattices and binary relations [14], fuzzy sets, soft sets and rough sets (see [8 , 25]) and so on.

In this paper, we consider the relationship between rough sets, fuzzy sets and n-ary semigroups. The remaining part of this paper is organized as follows: In Section 2, we recall some concepts and results on n-ary semigroups and rough sets. In Section 3, the approximations based on fuzzy ideals are studied. In Section 4, we introduce the notions of rough ideals and rough prime ideals in an n-ary semigroup, and give some properties of such ideals. Finally, we give the concepts of rough n-ary semigroups, rough homomorphisms of an n-ary semigroup, and investigate some related properties.

2 Preliminaries

A non-empty set S together with one n-ary operation f : Sⁿ → S, where n ≥ 2, is called an n-ary groupoid and is denoted by (S, f). According to the general convention used in the theory of n-ary groupoids, the sequence of elements x_i, x_i+1, …, x_j is denoted by $x_{i}^{j}$ . In the case j < i, it is the empty symbol. If x_i+1 = x_i+2 = … = x_i+t = x, then we write $\overset{(t)}{x}$ instead of $x_{i + 1}^{i + t}$ . In this convention, $f (x_{1}, x_{2}, \dots, x_{n}) = f (x_{1}^{n}),$ and $\begin{matrix} f (x_{1}, \dots, x_{i}, \underset{t}{\underset{︸}{x, \dots, x}}, x_{i + t + 1}, \dots, x_{n}) \\ = f (x_{1}^{i}, \overset{(t)}{x}, x_{i + t + 1}^{n}) . \end{matrix}$

An n-ary groupoid (S, f) is called (i, j)-associativeif $\begin{matrix} f (x_{1}^{i - 1}, f (x_{i}^{n + i - 1}), x_{n + i}^{2 n - 1}) \\ = f (x_{1}^{j - 1}, f (x_{j}^{n + j - 1}), x_{n + j}^{2 n - 1}) \end{matrix}$

hold for all x₁, x₂, …, x_2n-1 ∈ S. If this identity holds for all 1 ≤ i ≤ j ≤ n, then we say that the operation f is associative, and (S, f) is called an n-ary semigroup. An n-ary semigroup (S, f) is called idempotent if f (x, ⋯ , x) = x for all x ∈ S.

A non-empty subset A of an n-ary semigroup (S, f) is an n-ary subsemigroup if (A, f) is an n-ary subsemigroup, i.e., if it is closed under the operation f. Throughout this paper, unless otherwise mentioned, S will denote an n-ary semigroup with zero element 0, and (S₁, f) , (S₂, g) are shorthand for S₁, S₂.

Definition 2.1. [12] An n-ary semigroup (S, f) has zero element 0 if it satisfies: $f (x_{1}^{i - 1}, 0, x_{i + 1}^{n}) = 0$ for all x₁, x₂, …, x_n ∈ S and 1 ≤ i ≤ n.

Definition 2.2. [13] A non-empty subset I of (S, f) is called an i-ideal of S if for every x₁, ⋯ , x_i-1, x_i+1, …, x_n ∈ S with a ∈ I, then $f (x_{1}^{i - 1}, a, x_{i + 1}^{n}) \in I$ . I is called an ideal of S if I is an i-ideal for every 1 ≤ i ≤ n.

Definition 2.3. [14] An ideal P of an n-ary semigroup S is a prime ideal of S such that $f (x_{1}^{n}) \in P$ for every x₁, x₂, …, x_n ∈ S implies x_i ∈ P for some 1 ≤ i ≤ n.

Definition 2.4. [1] Let R be an equivalence relation of (S, f) . R is called a congruence relation of (S, f) if it satisfies: (x_i, y_i) ∈ R implies $(f (x_{1}^{n}), f (y_{1}^{n})) \in R$

for all 1 ≤ i ≤ n and x₁, x₂, …, x_n, y₁, y₂, …, y_n∈S .

Definition 2.5. [7] For an approximation space(U, θ), by a rough approximation in (U, θ) we mean a mapping (U, θ, -) P (U) → P (U) × P (U) defined for every X ∈ P (U) by $(U, θ, X) = ((\underline{U}, θ, X), (\bar{U}, θ, X))$ , where $(\underline{U}, θ, X) = {x \in U | [x]_{θ} \subseteq X},$ and $(\bar{U}, θ, X) = {x \in U | [x]_{θ} \cap X \neq \emptyset} .$ $(\underline{U}, θ, X) ((\bar{U}, θ, X))$ is called lower (upper) rough approximation of X in (U, θ). $(U, θ, X) = ((\underline{U}, θ, X), (\bar{U}, θ, X))$ is called a rough set if $(\underline{U}, θ, X) \neq (\bar{U}, θ, X)$ .

Definition 2.6. [25] A fuzzy set μ on an n-ary semigroup (S, f) is called a fuzzy k-ideal if $μ (f (x_{1}^{n})) \geq μ (x_{k})$ holds for all x₁, x₂, …, x_n ∈ S. If μ is a fuzzy k-ideal for every k = 1, 2, …, n, then it is called a fuzzy ideal. Clearly, if μ satisfies $μ (f (x_{1}^{n})) \geq μ (x_{1}) \lor μ (x_{2}) \lor \dots \lor μ (x_{n})$ , then it is a fuzzy ideal of (S, f).

3 Approximations based on fuzzy ideals

In this section, we give the definition of a t-level relation of a fuzzy ideal μ, and then we investigate some relative properties.

Definition 3.1. Let μ be a fuzzy ideal of S. For each t ∈ [0, μ (0)], the set U (μ, t) = {(x, y) ∈ S × S| (μ (x) ∧ μ (y)) ∨ Id_S (x, y) ≥ t} is called a t-level relation of μ, where ${Id}_{S} (x, y) = {\begin{matrix} 1 & if x = y, \\ 0 & otherwise, \end{matrix}$ for all (x, y) ∈ S × S.

Lemma 3.2. Let μ be a fuzzy ideal of S and t ∈ [0, μ (0)]. Then U (μ, t) is a congruence relation on S.

Proof. For all x ∈ S, (μ (x) ∧ μ (x)) ∨ Id_S (x, x) =1 ≥ t and so (x, x) ∈ U (μ, t), hence U (μ, t) is reflexive. Clearly, U (μ, t) is symmetric. Let(x, y) ∈ U (μ, t) and (y, z) ∈ U (μ, t). Then we have(μ (x) ∧ μ (y)) ∨ Id_S (x, y) ≥ t and (μ (y) ∧ μ (z)) ∨ Id_S (y, z) ≥ t. If x = y = z, (x, z) ∈ U (μ, t) is straightforward. If x = y ≠ z, then μ (y) ∧ μ (z) ≥ t and (μ (x) ∧ μ (z)) ∨ Id_S (x, z) = μ (x) ∧ μ (z) = μ(y) ∧ μ (z) ≥ t, which implies (x, z) ∈ U (μ, t). Ifx ≠ y = z, then μ (x) ∧ μ (y) ≥ t and (μ (x) ∧ μ (z))∨Id_S (x, z) = μ (x) ∧ μ (z) = μ (x) ∧ μ (y) ≥ t, so (x, z) ∈ U (μ, t). If x ≠ y ≠ z, then μ (x) ∧ μ (y) ≥ t, μ (y) ∧ μ (z) ≥ t and (μ (x) ∧ μ (z)) ∨ Id_S (x, z) = μ (x) ∧ μ (z) ≥ μ (x) ∧ μ (y) ∧ μ (z) ≥ t, thus (x, z)∈U (μ, t) and U (μ, t) is transitive. Hence U (μ, t) is an equivalence relation on S.

Let (x_i, y_i) ∈ U (μ, t) (k = 1, 2, …, n). Then (μ (x_k) ∧ μ (y_k)) ∨ Id_S (x_k, y_k) ≥ t. If $f (x_{1}^{n}) = f (y_{1}^{n})$ , $(f (x_{1}^{n}), f (y_{1}^{n})) \in U (μ, t)$ is obvious. If $f (x_{1}^{n}) \neq f (y_{1}^{n})$ , then at most n - 1 equations hold as follows: $x_{1} = y_{1}, x_{2} = y_{2}, \dots, x_{n} = y_{n} .$

Since μ is a fuzzy ideal of (S, f), there exist i (0 ≤ i ≤ n - 1) equations hold in above, we have $\begin{matrix} (μ (f (x_{1}^{n})) \land μ (f (y_{1}^{n}))) \lor {Id}_{S} (f (x_{1}^{n}), f (y_{1}^{n})) \\ = μ (f (x_{1}^{n})) \land μ (f (y_{1}^{n})) \\ \geq (μ (x_{1}) \lor μ (x_{2}) \dots \lor μ (x_{n})) \land (μ (y_{1}) \\ \lor μ (y_{2}) \dots \lor μ (y_{n})) \\ \geq (μ (x_{1}) \land μ (y_{1})) \lor (μ (x_{2}) \land μ (y_{2})) \lor \dots \\ \lor (μ (x_{n}) \land μ (y_{n})) \\ = μ (x_{1}) \lor μ (x_{2}) \lor \dots \lor μ (x_{i}) \lor (μ (x_{i + 1}) \\ \land μ (y_{i + 1})) \lor \dots \lor (μ (x_{n}) \land μ (y_{n})) \\ \geq (μ (x_{i + 1}) \land μ (y_{i + 1})) \lor \dots \lor (μ (x_{n}) \land μ (y_{n})) \\ \geq t \lor \dots \lor t \\ = t . \end{matrix}$

Hence $(f (x_{1}^{n}), f (y_{1}^{n})) \in U (μ, t)$ . Therefore, U (μ, t) is a congruence relation on S. This completes the proof. □

In this case, we say x is congruent to y mod μ, written x ≡ _ty (modμ) if (μ (x) ∧ μ (y)) ∨ Id_S (x, y) ≥ t for all x, y ∈ S and t ∈ [0, μ (0)]. According to Definition 3 and Lemma 3, we can get many useful properties of these congruence relations.

We denote by [x] _(μ,t) the equivalence class of U (μ, t) containing x of S.

Lemma 3.3.Let μ be a fuzzy ideal of S and t ∈ [0, μ (0)]. If for all x₁, x₂, …, x_n ∈ S, then $f ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) \subseteq [f (x_{1}^{n})]_{(μ, t)} .$

Proof. For all a ∈ f ([x₁] _(μ,t), [x₂] _(μ,t), …, [x_n]_(μ,t)), then there exist b_i ∈ [x_i] _(μ,t) (i = 1, 2, …, n)such that a = f (b₁, b₂, …, b_n). Since (x_i, b_i)∈U (μ, t), we have (f (x₁, x₂, …, x_n) , f (b₁, b₂, …, b_n)) ∈ U (μ, t), this means $f (b_{1}, b_{2}, \dots, b_{n}) \in [f (x_{1}^{n})]_{(μ, t)}$ , which implies that $a \in [f (x_{1}^{n})]_{(μ, t)}$ .

Therefore $f ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) \subseteq [f (x_{1}^{n})]_{(μ, t)}$ .

Definition 3.2. Let μ be a fuzzy ideal of S and t ∈ [0, μ (0)]. Then U (μ, t) is called a complete congruence relation on S if $f ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) = [f (x_{1}^{n})]_{(μ, t)}$ for all x₁, x₂, …, x_n ∈ S.

Example 3.5. Let (Z₄, f) be a 4-ary semigroup derived from the multiplication semigroup Z₄. We define a fuzzy set by μ (0) =1 and μ (x) =0.5 for all x ≠ 0. Then μ is a fuzzy ideal of(Z₄, f). Let t = 0.6, then U (μ, t) = {(0, 0) , (0, 1) , (0, 2) , (0, 3) , (1, 1) , (2, 2) , (3, 3)}. Hence we have[0] _(μ,t) = {0, 1, 2, 3} , [1] _(μ,t) = {0, 1} , [2] _(μ,t) ={0, 2} , [3] _(μ,t) = {0, 3}, then f ({0, 1, 2, 3} , {0, 1} , {0, 2} , {0, 3}) = [f (0, 1, 2, 3)] _(μ,t) = [0] _(μ,t) = {0, 1, 2, 3}.

Let μ be a fuzzy ideal of an n-ary semigroup S and t ∈ [0, μ (0)], we know U (μ, t) is an equivalence (congruence) relation on S. Therefore, when U = S and θ is the above equivalence relation, then we use (S, μ, t) instead of approximation space (U, θ).

Let μ be a fuzzy ideal of an n-ary semigroup S and U (μ, t) be a t-level congruence relation of μ on S. Let X be a non-empty subset of S. Then the sets $\underline{U} (μ, t, X) = {x \in S | [x]_{(μ, t)} \subseteq X}$ and $\bar{U} (μ, t, X) = {x \in S | [x]_{(μ, t)} \cap X \neq \emptyset},$ are called, respectively, the lower and upper approximations of the set X with respect to U (μ, t).

Proposition 3.6. [7] Let μ and ν be two fuzzy ideals of S. If A and B are non-empty subsets of S, then

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

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

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

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

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

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

(7) μ ⊆ ν, then $\bar{U} (μ, t, A) \subseteq \bar{U} (ν, t, A)$ and $\underline{U} (μ, t, A) \supseteq \underline{U} (ν, t, A)$ ,

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

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

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

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

Theorem 3.7. Let μ be a fuzzy ideal of S. If A₁, A₂, …, A_n are non-empty subsets of S, then

$f (\bar{U} (μ, t, A_{1}), \bar{U} (μ, t, A_{2}), \dots, \bar{U} (μ, t, A_{n})) \subseteq \bar{U} (μ, t, f (A_{1}^{n})) .$

Proof. Suppose that $x \in f (\bar{U} (μ, t, A_{1}), \bar{U} (μ, t, A_{2}),$ $\dots, \bar{U} (μ, t, A_{n}))$ . Then there exists $a_{i} \in \bar{U} (μ, t, A_{i})$ for i = 1, 2, …, n such that x = f (a₁, a₂, …, a_n). Therefore there exist elements x_i ∈ S for i = 1, 2, …, n such that x_i ∈ [a_i] _(μ,t) ∩ A_i for i = 1, 2, …, n. By Lemma 3, U (μ, t) is a congruence relation on S, then we have $f (x_{1}, x_{2}, \dots, x_{n}) \in f ([a_{1}]_{(μ, t)}, [a_{2}]_{(μ, t)}, \dots, [a_{n}]_{(μ, t)}) \subseteq [f (a_{1}^{n})]_{(μ, t)}$ .

On the other hand, $f (x_{1}, x_{2}, \dots, x_{n}) \in f (A_{1}, A_{2}, \dots, A_{n}) = f (A_{1}^{n})$ , so $f (x_{1}, x_{2}, \dots, x_{n}) \in [f (a_{1}^{n})]_{(μ, t)} \cap f (A_{1}^{n})$ , which implies $x = f (a_{1}, a_{2}, \dots, a_{n}) = f (a_{1}^{n}) \in \bar{U} (μ, t, f (A_{1}^{n}))$ . Hence we have $f (\bar{U} (μ, t, A_{1}), \bar{U} (μ, t, A_{2}), \dots, \bar{U} (μ, t, A_{n})) \subseteq \bar{U} (μ, t, f (A_{1}^{n}))$ . □

The following example shows that converse of Theorem 3 is not true in general.

Example 3.8. Let S = {a, b, c, d, e} be a set and f a 3-ary operation defined on S by the formula f (xyz) = (x * y) * z for all x, y, z ∈ S. Where * is defined by the table:

*	a	b	c	d	e
a	b	b	d	d	d
b	b	b	d	d	d
c	d	d	c	d	c
d	d	d	d	d	d
e	d	d	c	d	c

Then S is a 3-ary semigroup. Let μ be a fuzzy ideal of S such that the congruence classes are the subsets [a] _(μ,t) = {a, b} , [c] _(μ,t) = {c, d} , [e] _(μ,t) = {e}. Consider A₁ = {a, b} , A₂ = {b, c} , A₃ = {d}. Then $\bar{U} (μ, t, A_{1}) = {a, b}, \bar{U} (μ, t, A_{2}) = {a, b, c, d},$ $\bar{U} (μ, t, A_{3}) = {c, d}$ . Thus $f (\bar{U} (μ, t, A_{1}), \bar{U} (μ,$ $t, A_{2}), \bar{U} (μ, t, A_{3})) = {d}$ and $\bar{U} (μ, t, f (A_{1}, A_{2},$ A₃)) = {x ∈ S| [x] _(μ,t) ∩ f (A₁, A₂, A₃)} = {c, d}.Therefore, $\bar{U} (μ, t, f (A_{1}, A_{2}, A_{3})) ⊈ f (\bar{U} (μ, t, A_{1}),$ $\bar{U} (μ, t, A_{2}), \bar{U} (μ, t, A_{3}))$ .

Theorem 3.9. Let U (μ, t) be a complete congruence relation on S. If A₁, A₂, …, A_n are non-empty subsets of S, then

$f (\underline{U} (μ, t, A_{1}), \underline{U} (μ, t, A_{2}), \dots, \underline{U} (μ, t, A_{n})) \subseteq \underline{U} (μ, t, f (A_{1}^{n})) .$

Proof. Suppose that $x \in f (\underline{U} (μ, t, A_{1}), \underline{U} (μ, t, A_{2}),$ $\dots, \underline{U} (μ, t, A_{n}))$ . Then there exist $a_{i} \in \underline{U} (μ, t, A_{i})$ for i = 1, 2, …, n such that x = f (a₁, a₂, …, a_n). So we have [a_i] _(μ,t) ⊆ A_i for i = 1, 2, …, n. Since U (μ, t) is a complete congruence relation on S, we have [f (a₁, a₂, …, a_n)] _(μ,t) = f ([a₁] _(μ,t), [a₂] _(μ,t), …, [a_n] _(μ,t)) ⊆ f (A₁, A₂, …, A_n), and so $x = f (a_{1}, a_{2}, \dots, a_{n}) \in \underline{U} (μ, t, f$ $(A_{1}^{n}))$ . Therefore $f (\underline{U} (μ, t, A_{1}), \underline{U} (μ, t, A_{2}), \dots, \underline{U} (μ, t, A_{n})) \subseteq \underline{U} (μ, t, f (A_{1}^{n}))$ . □

The following example shows that if U (μ, t) is not a complete congruence relation on S, then Theorem 3 does not hold.

Example 3.10. Consider the 3-ary semigroup in Example 3. Let μ be a fuzzy ideal of S such that the congruence classes are the subsets [a] _(μ,t) = {a, b} , [c] _(μ,t) = {c, d} , [e] _(μ,t) = {e}. Let A₁ = {a, b} , A₂ = {c, d} , A₃ = {e}, then we have $\underline{U} (μ, t, A_{1}) = {a, b}, \underline{U} (μ, t, A_{2}) = {c, d}, \underline{U} (μ, t,$ A₃) = {e}, but $f (\underline{U} (μ, t, A_{1}), \underline{U} (μ, t, A_{2}), \underline{U} (μ, t,$ $A_{n})) = {d} ⊈ \underline{U} (μ, t, f (A_{1}^{n})) = \emptyset$ .

4 Rough (prime) ideals in an n-ary semigroup

In this section, we introduce the notions of rough ideals and rough prime ideals in n-ary semigroups, and give some properties of such ideals.

Definition 4.1. Let μ be a fuzzy ideal of S and t ∈ [0, μ (0)]. Then a non-empty subset A of S is called an upper (lower) rough n-ary subsemigroup of S if $\bar{U} (μ, t, A)$ $(\underline{U} (μ, t, A))$ is an n-ary subsemigroup of S. And A is called an upper (lower) rough (prime) ideal of S if $\bar{U} (μ, t, A)$ $(\underline{U} (μ, t, A))$ is a (prime) ideal of S.

Theorem 4.2. Let μ be a fuzzy ideal of S and t ∈ [0, μ (0)]. Then

(1) If A is an n-ary subsemigroup of S, then A is an upper rough n-ary subsemigroup of S,

(2) If A is an ideal of S, then A is an upper rough ideal of S.

Proof. (1) Let A be an n-ary subsemigroup of S. Then f (A, A, …, A) ⊆ A and $\bar{U} (μ, t, f (A, A,$ $\dots, A)) \subseteq \bar{U} (μ, t, A)$ . By Theorem 3, we have $f (\bar{U} (μ, t, A), \dots, \bar{U} (μ, t, A)) \subseteq \bar{U} (μ, t, f (A, A,$ $\dots, A)) \subseteq \bar{U} (μ, t, A)$ . This means $\bar{U} (μ, t, A)$ is an n-ary subsemigroup of S. Hence A is an upper rough n-ary subsemigroup of S. (2) Let A be an ideal of S. Then $f (\overset{(i - 1)}{S}, A, \overset{(n - i)}{S}) \subseteq A$ for all i = 1, 2, …, n. By Theorem 3, we have $f (\overset{(i - 1)}{\bar{U} (μ, t, S)}, \bar{U} (μ, t, A), \overset{(n - i)}{\bar{U} (μ, t, S)}) \subseteq \bar{U} (μ, t, f$ $(\overset{(i - 1)}{S}, A, \overset{(n - i)}{S})) \subseteq \bar{U} (μ, t, A)$ . This means $\bar{U} (μ, t, A)$ is an ideal of S. Hence A is an upper rough ideal of S. □

Remark 4.3. The above theorem shows that the notion of an upper rough n-ary subsemigroup (ideal) is an extended notion of a usual n-ary subsemigroup (ideal) of an n-ary semigroup. The following example shows that the converse of Theorem 4 does not hold in general.

Example 4.4. Let S = {- i, 0, i} be a set with a 3-ary operation f as the usual multiplication of complex numbers. Then (S, f) is a 3-ary semigroup. Let μ be a fuzzy ideal of S such that the congruence classes are [i] _(μ,t) = [- i] _(μ,t) = {- i, i} , [0] _{ (μ, t)} = {0}. Then A = {i} is not a 3-ary subsemigroup of S, but $\bar{U} (μ, t, A) = {- i, i}$ is a 3-ary subsemigroup. Moreover, B = {0, i} is not an ideal of S, but $\bar{U} (μ, t, B) = {- i, 0, i}$ is an ideal.

Theorem 4.5. Let U (μ, t) be a complete congruence relation on S. Then

(1) If A is an n-ary subsemigroup of S, then A is an lower rough n-ary subsemigroup of S,

(2) If A is an ideal of S, then A is a lower rough ideal of S.

Proof. It is similar to the proof of Theorem 4.

The following example shows that if U (μ, t) is a complete congruence relation on an n-ary semigroup S, then $\underline{U} (μ, t, A)$ is an n-ary subsemigroup of S even if A is not an n-ary subsemigroup of S.

Example 4.6. Consider the n-ary semigroup and the congruence class in Example 4. Then A = {0, i} is not a 3-ary subsemigroup of S but $\underline{U} (μ, t, A) = {0}$ is a 3-ary subsemigroup of S.

Theorem 4.7. Let μ be an ideal of S and t ∈ [0, μ (0)]. If A₁, A₂, …, A_n are 1, 2, …, n-ideal of S, respectively, then

(1) $\bar{U} (μ, t, f (A_{1}^{n})) \subseteq \bar{U} (μ, t, A_{1}) \cap \bar{U} (μ, t, A_{2}) \dots \cap \bar{U} (μ, t, A_{n})$ ,

(2) $\underline{U} (μ, t, f (A_{1}^{n})) \subseteq \underline{U} (μ, t, A_{1}) \cap \underline{U} (μ, t, A_{2}) \dots \cap \underline{U} (μ, t, A_{n})$ .

Proof. (1) Since A₁ is a 1-ideal of S, we have f (A₁, A₂, …, A_n) ⊆ f (A₁, S, …, S) ⊆ A₁, f (A₁, A₂, …, A_n) ⊆ f (S, A₂, …, S) ⊆ A₂, …, f (A₁, A₂, …, A_n) ⊆ f (S, S, …, A_n) ⊆ A_n. Thus f (A₁, $A_{2}, \dots, A_{n}) = f (A_{1}^{n}) \subseteq A_{1} \cap A_{2} \cap \dots \cap A_{n}$ . Thenit follows from Proposition 3, $\bar{U} (μ, t, f (A_{1}^{n})) \subseteq \bar{U} (μ, t, A_{1} \cap A_{2}, \dots, A_{n}) \subseteq \bar{U} (μ, t, A_{1}) \cap \bar{U} (μ, t,$ $A_{2}) \dots \bar{U} (μ, t, A_{n})$ .

(2) It is similar to the proof of (1). □

Proposition 4.8. If S is an idempotent n-ary semigroup in Theorem 4.7, then $\bar{U} (μ, t, f (A_{1}^{n})) = \bar{U} (μ, t, A_{1}) \cap \bar{U} (μ, t, A_{2}) \dots \cap \bar{U} (μ, t, A_{n})$ .

Proof. For all $x \in \bar{U} (μ, t, A_{1}) \cap \bar{U} (μ, t, A_{2}) \dots \cap \bar{U} (μ, t, A_{n})$ , then $x \in \bar{U} (μ, t, A_{1})$ , $x \in \bar{U} (μ, t, A_{2})$ , …, $x \in \bar{U} (μ, t, A_{n})$ . This means [x] _(μ,t)∩ A₁ ≠ ∅, [x] _(μ,t)∩ A₂ ≠ ∅,…, [x] _(μ,t)∩ A_n ≠ ∅. So there exist a₁, a₂, …, a_n ∈ S such that a₁ ∈ [x] _(μ,t), a₁ ∈ A₁, a₂ ∈ [x] _(μ,t), a₂ ∈ A₂,…,a_n ∈ [x] _(μ,t), a_n ∈ A_n. Since S is an idempotent n-ary semigroup, $f (a_{1}^{n}) \in f ([x]_{(μ, t)}, [x]_{(μ, t)}, \dots, [x]_{(μ, t)}) \subseteq [f (x, x, \dots, x)]_{(μ, t)} = [x]_{(μ, t)}$ and $f (a_{1}^{n}) \in f (A_{1}^{n})$ . Therefore $f (a_{1}^{n}) \in [x]_{(μ, t)} \cap f (A_{1}^{n})$ . This implies $x \in \bar{U} (μ, t, f (A_{1}^{n}))$ . So $\bar{U} (μ, t, A_{1}) \cap \bar{U} (μ, t, A_{2}) \dots \cap \bar{U} (μ, t, A_{n}) \subseteq \bar{U} (μ, t, f (A_{1}^{n}))$ . Hence by Theorem 4.7, we have $\bar{U} (μ, t, f (A_{1}^{n})) = \bar{U} (μ, t, A_{1}) \cap \bar{U} (μ, t, A_{2}) \dots \cap \bar{U} (μ, t, A_{n})$ . □

Lemma 4.9. Let μ be a fuzzy ideal of an n-ary semigroup S and t ∈ [0, 1]. Then [0] _(μ,t) is an ideal of S.

Proof. For all x₁, x₂, …, x_n ∈ S and a ∈ [0] _(μ,t), then we have $f (x_{1}, x_{2}, \dots, a, \dots, x_{n}) \in f ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [0]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) \subseteq [f (x_{1}^{i - 1}, 0, x_{i + 1}^{n})]_{(μ, t)} = [0]_{(μ, t)}$ . This means $f (x_{1}^{i - 1}, a, x_{i + 1}^{n}) \in [0]_{(μ, t)}$ . Hence [0] _(μ,t) is an ideal of S. □

Proposition 4.10. Let μ be a fuzzy ideal of an n-ary semigroup S and t ∈ [0, μ (0)]. Then $\underline{U} (μ, t, [0]_{(μ, t)}) = [0]_{(μ, t)}$ .

Proof. By Proposition 3, we have $\underline{U} (μ, t, [0]_{(μ, t)}) \subseteq [0]_{(μ, t)}$ . Now, we show that $[0]_{(μ, t)} \subseteq \underline{U} (μ, t, [0]_{(μ, t)})$ . For every x ∈ [0] _(μ,t), then (0, x) ∈ U (μ, t). Let y ∈ [x] _(μ,t), then (x, y) ∈ U (μ, t). According to U (μ, t) is a congruence relation, we have (0, y) ∈ U (μ, t), this implies y ∈ [0] _(μ,t). Hence [x] _(μ,t) ⊆ [0] _(μ,t). This means $x \in \underline{U} (μ, t, [0]_{(μ, t)})$ . Therefore $\underline{U} (μ, t, [0]_{(μ, t)}) = [0]_{(μ, t)}$ . □

Corollary 4.11. Let μ be a fuzzy ideal of an n-ary semigroup S and t ∈ [0, μ (0)]. Then [0] _(μ,t) is a lower rough ideal of S.

Proposition 4.12. Let μ be a fuzzy ideal of an n-ary semigroup S and t ∈ [0, μ (0)]. Then μ_t = [0] _(μ,t).

Proof. For all x ∈ μ_t, then μ (x) ≥ t and μ (0) ≥ μ (x) ≥ t. So (μ (x) ∧ μ (0)) ∨ Id_S (x, 0) ≥ μ (x) ∧ μ (0) ≥ t, by Definition 3, we have (x, 0) ∈ U (μ, t). It implies x ∈ [0] _(μ,t). This means μ_t ⊆ [0] _(μ,t). On the other hand, for all x ∈ [0] _(μ,t), then (x, 0) ∈ U (μ, t) and (μ (x) ∧ μ (0)) ∨ Id_S (x, 0) ≥ t. If x = 0, μ (x) = μ (0) ≥ t is obvious, so x ∈ μ_t, it implies [0] _(μ,t) ⊆ μ_t. If x ≠ 0, then μ (x) ∧ μ (0) ≥ t, μ (x) ≥ t, this means x ∈ μ_t. Hence [0] _(μ,t) ⊆ μ_t. Therefore μ_t = [0] _(μ,t). □

Theorem 4.13. Let U (μ, t) be a complete congruence relation. If A is a prime ideal of S, then $\underline{U} (μ, t, A)$ is a prime ideal of an n-ary semigroup S if $\underline{U} (μ, t, A) \neq \emptyset$ .

Proof. Since A is an ideal of S, by Lemma 4 and Proposition 4, we know that $\underline{U} (μ, t, A)$ is an ideal of S. Let $f (x_{1}^{n}) \in \underline{U} (μ, t, A)$ for some x₁, x₂, …, x_n ∈ S, then we have $f ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) \subseteq [f (x_{1}^{n})]_{(μ, t)} \subseteq A$ . We suppose that $\underline{U} (μ, t, A)$ is not a prime ideal, then there exist x₁, x₂ … , x_n ∈ S such that $f (x_{1}^{n}) \in \underline{U} (μ, t, A)$ but $x_{1} \notin \underline{U} (μ, t, A), x_{2} \notin \underline{U} (μ, t, A), \dots, x_{n} \notin \underline{U} (μ, t, A)$ . Thus [x₁] _(μ,t) ⊈ A, [x₂] _(μ,t) ⊈ A, …, [x_n] _(μ,t) ⊈ A. Then exist $x_{1}^{'} \in [x_{1}]_{(μ, t)}, x_{1}^{'} \notin A, x_{2}^{'} \in [x_{2}]_{(μ, t)}, x_{2}^{'} \notin A, \dots, x_{n}^{'} \in [x_{n}]_{(μ, t)}, x_{n}^{'} \notin A$ . Thus $f (x_{1}^{'}, x_{2}^{'}, \dots, x_{n}^{'}) \in f ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) \subseteq A$ . Since A is a prime ideal of S, we have $x_{i}^{'} \in A$ for some 1 ≤ i ≤ n. It contradicts with the supposition. Hence $\underline{U} (μ, t, A) \neq \emptyset$ is a prime ideal of S. □

Theorem 4.14. Let U (μ, t) be a complete congruence relation. If A is a prime ideal of S, then A is an upper rough prime ideal of S.

Proof. Since A is a prime ideal of S, by Theorem 4, we know that $\bar{U} (μ, t, A)$ is an ideal of S. Let $f (x_{1}^{n}) \in \bar{U} (μ, t, A)$ for some x₁, x₂, …, x_n ∈ S. Then we have $[f (x_{1}^{n})]_{(μ, t)} \cap A = f ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) \cap A \neq \emptyset$ . So there exist $x_{1}^{'} \in [x_{1}]_{(μ, t)}, x_{2}^{'} \in [x_{2}]_{(μ, t)}, \dots, x_{n}^{'} \in [x_{n}]_{(μ, t)}$ such that $f (x_{1}^{'}, x_{2}^{'}, \dots, x_{n}^{'}) \in A$ . Since A is a prime ideal, we have $x_{i}^{'} \in A$ for some 1 ≤ i ≤ n. Thus [x_i] _(μ,t)∩ A ≠ ∅, and so $x_{i} \in \bar{U} (μ, t, A)$ is a prime ideal of S. □

5 Rough n-ary semigroups and homomorphisms

In this section, we introduce the concepts of rough n-ary semigroups, rough homomorphisms of an n-ary semigroup, and investigate some relative properties.

Definition 5.1. Let (S, μ, t) be an approximation space. A non-empty subset A of S is called a rough n-ary semigroup of S if it satisfies $f (x_{1}^{n}) \in \bar{U} (μ, t, A)$ for all x₁, x₂, …, x_n ∈ A.

Proposition 5.2. If A is a rough n-ary semigroup of S, then $\bar{U} (μ, t, A)$ is a rough n-ary semigroup of S.

Proof. For all $x_{1}, x_{2}, \dots, x_{n} \in \bar{U} (μ, t, A)$ , then [x₁] _(μ,t)∩ A = ∅ , [x₂] _(μ,t) ∩ A = ∅ , …, [x_n] _(μ,t) ∩ A = ∅, hence there exist y₁ ∈ [x₁] _(μ,t) ∩ A, y₂ ∈ [x₂] _(μ,t) ∩ A, …, y_n ∈ [x_n] _(μ,t) ∩ A, we have y₁ ∈ [x₁] _(μ,t), y₂ ∈ [x₂] _(μ,t), …, y_n ∈ [x_n] _(μ,t) and y₁, y₂, …, y_n ∈ A. Since A is a rough n-ary semigroup of S, $f (y_{1}^{n}) \in \bar{U} (μ, t, A)$ , this means $[f (y_{1}^{n})]_{(μ, t)} \cap A \neq \emptyset$ . Since U (μ, t) is a congruence relation and (x₁, y₁) ∈ U (μ, t) , (x₂, y₂) ∈ U (μ, t) , …, (x_n, y_n) ∈ U (μ, t), we have $(f (x_{1}^{n}), f (y_{1}^{n})) \in U (μ, t)$ , it implies $[f (x_{1}^{n})]_{(μ, t)} = [f (y_{1}^{n})]_{(μ, t)}$ and so $[f (y_{1}^{n})]_{(μ, t)} \cap A \neq \emptyset$ . This means $f (x_{1}^{n}) \in \bar{U} (μ, t, A) = \bar{U} (μ, t, \bar{U} (μ, t, A))$ . Therefore $\bar{U} (μ, t, A)$ is a rough n-ary semigroup of S. □

Definition 5.3. Let A be a rough n-ary semigroup of S. A non-empty subset B of S is called a rough n-ary subsemigroup of A if it satisfies:

(1) B ⊆ A,

(2) B is a rough n-ary semigroup of S.

Definition 5.4. Let (S₁, μ₁, t) and (S₂, μ₂, t) be two approximation spaces and A₁, A₂ be rough n-ary semigroups of S₁ and S₂, respectively. If a mapping $φ : \bar{U} (μ_{1}, t, A_{1}) \to \bar{U} (μ_{2}, t, A_{2})$ satisfies: $φ (f (x_{1}^{n})) = g (φ (x_{1}), φ (x_{2}), \dots, φ (x_{n}))$ for all $x_{1}, x_{2}, \dots, x_{n} \in \bar{U} (μ_{1}, t, A_{1})$ . Then φ is called a rough homomorphism from A₁ to A₂.

If such a mapping is surjective, injective or bijective, then it is called a rough epimorphism, a rough monomorphism or a rough isomorphism, respectively.

Definition 5.5. Let (S₁, μ₁, t) and (S₂, μ₂, t) be two approximation spaces and A₁, A₂ be a rough n-ary semigroup of S₁ and S₂, respectively. If φ is a rough homomorphism from A₁ to A₂, then $\ker φ = {(x, y) | φ (x) = φ (y), x, y \in \bar{U} (μ_{1}, t, A_{1})}$ is called a rough homomorphism kernel of A₁ and $φ (\bar{U} (μ_{1}, t, A_{1})) = {φ (x) | x \in \bar{U} (μ_{1}, t, A_{1})}$ is called a rough homomorphism image of φ.

Proposition 5.6. Let (S₁, μ₁, t) and (S₂, μ₂, t) be two approximation spaces and A₁, A₂ be a rough n-ary semigroup of S₁ and S₂, respectively. If φ is a rough homomorphism from A₁ to A₂, then

(1) φ is rough monomorphism if and only if $\ker φ = {(x, x) | x \in \bar{U} (μ_{1}, t, A_{1})}$ ,

(2) φ is rough epimorphism if and only if $φ (\bar{U} (μ_{1}, t, A_{1})) = (\bar{U} (μ_{2}, t, A_{2}))$ .

Proof. It is straightforward. □

Proposition 5.7. Let (S₁, μ₁, t) and (S₂, μ₂, t) be two approximation spaces and A₁, A₂ are rough n-ary semigroups of S₁ and S₂, respectively. If φ is a rough homomorphism from A₁ to A₂, then

(1) kerφ is a congruence relation over A₁,

(2) $φ (\bar{U} (μ_{1}, t, A_{1}))$ is a rough n-ary subsemigroup of $\bar{U} (μ_{2}, t, A_{2})$ .

Proof. (1) Obviously. (2) By Proposition 5, we know $\bar{U} (μ_{1}, t, A_{1})$ and $\bar{U} (μ_{2}, t, A_{2})$ are a rough n-ary semigroup of S₁ and S₂ respectively. For all y₁, y₂, …, $y_{n} \in φ (\bar{U} (μ_{1}, t, A_{1}))$ , according to Definition 5, then there exist $x_{1}, x_{2}, \dots, x_{n} \in \bar{U} (μ_{1}, t, A_{1})$ such that φ (x₁) = y₁, φ (x₂) = y₂, …, φ (x_n) = y_n. So we have $g (y_{1}^{n}) = g (φ (x_{1}), φ (x_{2}), \dots, φ (x_{n})) = φ (f (x_{1}^{n})) \in φ (\bar{U} (μ_{1}, t, \bar{U} (μ_{1}, t, A_{1}))) = φ (\bar{U} (μ_{1}, t,$ $A_{1})) \subseteq \bar{U} (μ_{1}, t, φ (\bar{U} (μ_{1}, t, A_{1})))$ . This means $φ (\bar{U} (μ_{1}, t, A_{1}))$ is a rough n-ary subsemigroup of S₂. By Definition 5, $φ (\bar{U} (μ_{1}, t, A_{1}))$ is a rough n-ary subsemigroup of $\bar{U} (μ_{2}, t, A_{2})$ . □

Now, we introduce the notion of rough quotient n-ary semigroups and investigate some relative properties.

Definition 5.8. Let (S, μ, t) be an approximation space and H a rough n-ary semigroup of S. Then the set $\begin{matrix} \bar{U} (μ, t, H) / U (μ, t) = {[x]_{(μ, t)} \in H / U (μ, t) \\ | [x]_{(μ, t)} \cap H \neq \emptyset} \end{matrix}$ is called the upper approximation of H/U (μ, t) with respect to U (μ, t), where H/U (μ, t) = {[x] _(μ,t)|x∈H}.

Theorem 5.9.If H is a rough n-ary semigroup of S, then (H/U (μ, t) , F) is a rough n-ary semigroup under the n-ary operation defined by $F ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) = [f (x_{1}^{n}]_{(μ, t)}$ for all x₁, x₂, …, x_n ∈ H

Proof. We shall first show that F is well defined. Let x₁, x₂, …, x_n, y₁, y₂, ⋯ , y_n ∈ H be such that [x₁] _(μ,t) = [y₁] _(μ,t), [x₂] _(μ,t) = [y₂] _(μ,t), …, [x_n] _(μ,t) = [y_n] _(μ,t) It follows that (x₁, y₁) ∈ U (μ, t) , (x₂, y₂) ∈ U (μ, t) , …, (x_n, y_n) ∈ U (μ, t). Since U (μ, t) is a congruence relation over S, we have $(f (x_{1}^{n}), f (y_{1}^{n})) \in U (μ, t)$ , this means $[f (x_{1}^{n}]_{(μ, t)} = [f (y_{1}^{n}]_{(μ, t)}$ . Hence F is well defined. For all x₁, x₂, …, x_n ∈ H, then [x₁] _(μ,t), [x₂] _(μ,t), …, [x_n] _(μ,t) ∈ H/U (μ, t). Since H is a rough n-ary semigroup, $f (x_{1}^{n}) \in \bar{U} (μ, t, H)$ , this implies $[f (x_{1}^{n})]_{(μ, t)} \cap H \neq \emptyset$ . There exists $a \in [f (x_{1}^{n})]_{(μ, t)}$ and a ∈ H. For all a ∈ H, a ∈ [a] _(μ,t) ⊆ H/U (μ, t), hence H ⊆ H/U (μ, t). It implies $[f (x_{1}^{n})]_{(μ, t)} \cap H / U (μ, t) \neq \emptyset$ , this means $F ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)}) = [f (x_{1}^{n}]_{(μ, t)} \in \bar{U} (μ, t, H / U (μ, t))$ . Therefore (H/U (μ, t) , F) is a rough n-ary semigroup under F. □

Theorem 5.10. Let (S, μ, t) be an approximation space and H a rough n-ary semigroup of S. Then there exists a rough epimorphism between H and H/U (μ, t).

Proof. We define a mapping $θ : \bar{U} (μ, t, H) \to \bar{U} (μ, t, H) / U (μ, t)$ by θ (x) = [x] _(μ,t) for all $x \in \bar{U} (μ, t, H)$ , θ is well defined and a surjective is obvious. Moreover, for every $x_{1}, x_{2}, \dots, x_{n} \in \bar{U} (μ, t, H)$ , $θ (f (x_{1}^{n})) = [f (x_{1}^{n})]_{(μ, t)} = F ([x_{1}]_{(μ, t)}, [x_{2}]_{(μ, t)}, \dots, [x_{n}]_{(μ, t)})$ . Therefore θ is a rough epimorphism. □

Theorem 5.11. Let (S₁, μ₁, t) and (S₂, μ₂, t) be two approximation spaces and A₁, A₂ be a rough n-ary semigroup of S₁ and S₂, respectively. If φ is a rough homomorphism from A₁ to A₂, then there exists a rough ismorphism between A₁/kerφ and φ (A₁).

Proof. By Proposition 5 and Theorem 5, we know A₁/kerφ is a rough n-ary semigroup of S₁. For all y₁, y₂, …, y_n ∈ φ (A₁), then there exist x₁, x₂, …, x_n ∈ A₁ such that φ (x₁) = y₁, φ (x₂) = y₂, …, φ (x_n) = y_n, hence $g (y_{1}^{n}) = g (φ (x_{1}), φ (x_{2}), \dots, φ (x_{n})) = φ (f (x_{1}^{n})) \in φ (\bar{U} (μ, t, A_{1})) \subseteq \bar{U} (μ, t, φ (A_{1}))$ . This means φ (A₂) is a rough n-ary semigroup of S₂. We define a mapping $δ : \bar{U} (μ_{1}, t, A_{1} / \ker φ) \to \bar{U} (μ_{2}, t, φ (A_{1}))$ by δ ([x] _kerφ) = φ (x) for all x ∈ A₁.

(1) δ is well defined:

If [x] _kerφ = [y] _kerφ, since kerφ is a congruence relation over A₁, (x, y) ∈ kerφ, φ (x) = φ (y). Hence δ is well defined.

(2) δ is a homomorphism:

For all x₁, x₂, …, x_n ∈ A₁, then

$[x_{1}]_{\ker φ}, [x_{2}]_{\ker φ}, \dots, [x_{n}]_{\ker φ} \in A_{1} / \ker φ \subseteq \bar{U} (μ_{1}, t, A_{1} / \ker φ)$ , $δ (F ([x_{1}]_{\ker φ}, [x_{2}]_{\ker φ}, \dots, [x_{n}]_{\ker φ})) = δ ([f (x_{1}^{n})]_{\ker φ}) = φ (f (x_{1}^{n})) = g (φ (x_{1}), φ (x_{2}), \dots, φ (x_{n})) = g ([x_{1}]_{\ker φ}, [x_{2}]_{\ker φ}, \dots, [x_{1}]_{\ker φ})$ . This means δ is a homomorphism.

(3) δ is a monomorphism:

Suppose that δ ([x] _kerφ) = δ ([y] _kerφ), we have φ (x) = φ (y), and so (x, y) ∈ kerφ. This implies [x] _kerφ = [y] _kerφ. So δ is a monomorphism.

(4) δ is an epimorphism:

For all $y \in φ (A_{1}) \subseteq \bar{U} (μ_{2}, t, φ (A_{1}))$ , there exists $x \in A_{1} \subseteq \bar{U} (μ, t, A_{1})$ such that φ (x) = y and δ ([x] _kerφ) = φ (x) = y. This means δ is an epimorphism. This completes the proof. □

Corollary 5.12. Let (S₁, μ₁, t) and (S₂, μ₂, t) be two approximation spaces and A₁, A₂ be a rough n-ary semigroup of S₁ and S₂, respectively. If φ is a rough epimorphism from A₁ to A₂, then there exists a rough ismorphism between A₁/kerφ and A₂.

6 Conclusions

It is well known that ideals (fuzzy ideals) and congruence relations always play important roles in the study of algebraic structures. In this paper, we concern a relationship between rough sets, fuzzy set and n-ary semigroups. We give a definition of the lower and upper approximations of a subset of n-ary semigroup with respect to a fuzzy ideal. Furthermore, we introduce the notions of rough ideals and rough prime ideals by means of fuzzy ideals of an n-ary semigroup. In particular, we investigate the concepts of rough n-ary semigroups, rough homomorphisms about an n-ary semigroup, and give some relative properties.

In the future study of n-ary semigroups, we can apply the congruence relations in Definition 3 of other uncertain theory of n-ary semigroups, such as, soft rough n-ary semigroups, rough soft n-ary semigroups, and soft rough fuzzy n-ary semigroups and so on. We hope this theory can be served as a foundation of some applied fields, such as decision making, data analysis, and forecasting.

Acknowledgments The authors are very thankful for the reviewers to give some valuable comments to improve this paper.

This research is partially supported by a grant of National Natural Science Foundation of China (11561023).

References

Alam

S.E.

, Rao

and Davvaz

, (m, n)-semirings and a generalized fault-Tolerance algebra of systems, J Appl Math (2013), 10, Art. ID 482391.

Aslam

, Abdullah

, Davvaz

and Yaqoob

, Rough M-hypersystems and fuzzy M-hypersystems in Γ-semihypergroups, Neural Comput & Applic21 (2012), 281–287.

Biswas

and Nanda

, Rough groups and rough subgroups, Bull Polish Acad Sci Math42 (1994), 251–254.

Crombez

, On (m; n)-rings, Abh Math Semin Univ Hambg37 (1972), 180–199.

Crombez

and Timm

, On (m; n)-quotient rings, Abh Math Semin Univ Hambg37 (1972), 200–203.

Davvaz

, Roughness in rings, Inform Sci164 (2004), 147–163.

Davvaz

, Roughness based on fuzzy ideals, Inform Sci176 (2006), 2417–2437.

Davvaz

, Approximation in n-ary algebraic systems, Soft Comput12 (2008), 409–418.

Davvaz

and Dudek

W.A.

, Fuzzy n-ary groups as a generalization of Rosenfeld’s fuzzy groups, J Mult-Valued Logic Soft Comput15 (2009), 451–469.

10.

Davvaz

and Mahdavipour

, Roughness in modules, Inform Sci176 (2006), 3658–3674.

11.

Dudek

W.A.

, Remarks on n-groups, Demonstratio Math13 (1980), 165–181.

12.

Dudek

W.A.

, On (i; j)-associative n-groupoids with the non-empty center, Ricer Mat (Napoli)35 (1986), 105–111.

13.

Dudek

W.A.

, Idempotents in n-ary semigroups, Southeast Asian Bull Math25 (2001), 97–104.

14.

Heidari

and Davvaz

, n-ary ΓGreen-semigroups, operators and ’s relations, Afr Mat25 (2014), 841–856.

15.

Hila

, Vougiouklis

and Naka

, On the structure of soft m-ary semihypergroups, Politehn Univ Bucharest Sci Bull Ser A Appl Math Phys76 (2014), 91–106.

16.

Kasner

, An extension of the group concepts, Bull Amer Math Soc10 (1904), 290–291.

17.

Kazanci

and Davvaz

, On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings, Inform Sci178(5) (2008), 1343–1354.

18.

Kuroki

, Rough ideals in semigroups, Inform Sci100 (1997), 139–163.

19.

Kuroki

and Wang

P.P.

, The lower and upper approximations in a fuzzy group, Inform Sci90 (1996), 203–220.

20.

Marichal

and Mathonet

, A description of n-ary semigroups polynomial-derived from integral domains, Semigroup Forum83 (2011), 241–249.

21.

Jun

Y.B.

, Roughness of ideals in BCK-algebras, Sci Math Jpn57 (2003), 165–169.

22.

Pawlak

, Rough sets, Int J Comput Inform Sci11 (1982), 341–356.

23.

Rasouli

and Davvaz

, Roughness in MV-algebras, Inform Sci180 (2010), 737–747.

24.

Xiao

and Zhang

, Rough prime ideals and rough fuzzy prime ideals in semigroups, Inform Sci176 (2006), 725–733.

25.

Zhou

, Xiang

and Zhan

, A novel study on fuzzy congruences on n-ary semigroups, Politehn Univ Bucharest Sci Bull Ser A Appl Math Phys (2015), in press.

26.

Zhu

, Relationships among basic concepts in covering-based rough sets, Inform Sci179 (2009), 2479–2486.

27.

Zhu

, Relationships between generalized rough sets based on binary relation and covering, Inform Sci179(3) (2009), 210–225.

28.

Zhu

, Generalized rough sets based on relations, Inform Sci177(22) (2007), 4997–5011.