Generalizations of rough sets induced by binary relations approach in semigroups

Abstract

The Mareay’s rough set has been regarded as an approximation processing model in an approximation space induced by an arbitrary binary relation on the single universe. This is an effective mathematical tool for dealing with uncertain knowledge and vagueness in data for the single universe. Based on this induced notion, we firstly establish and verify two new classes, called a successor class and a core of a successor class induced by an arbitrary binary relation between two universes. Then we propose a generalization of the Mareay’s rough set in an approximation space based on cores of successor classes induced by an arbitrary binary relation between two universes, with a corresponding example. Some interesting algebraic properties of the new approximation processing model are investigated. We develop the use of the novel rough set in a semigroup under a preorder and compatible relation. We establish the notions of rough semigroups, rough ideals and rough completely prime ideals. Then we provide sufficient conditions of rough semigroups, rough ideals and rough completely prime ideals. Under homomorphism problems in semigroups, the relationship between the rough semigroup (resp. rough ideal and rough completely prime ideal) and the homomorphic image of the rough semigroup (resp. rough ideal and rough completely prime ideal) is demonstrated.

Keywords

Rough sets approximation spaces semigroups rough semigroups rough ideals rough completely prime ideals binary relations preorder relations compatible relations.

1. Introduction

In order to solve problems in information science and intelligent information processing, methods in fundamental mathematics are not always successfully used because incompleteness, granularity and uncertainty of information and knowledge are typical for these problems. Pawlak’s rough set theory, proposed by Pawlak [1], offers an alternative toolset to deal with imprecise, inconsistent, incomplete information and knowledge. Such a mathematical tool has been being used in information system, fuzzy system, decision support system and so on (see [2 –7]). One of the main research problems of Pawlak’s rough set theory is the approximation processing of non-empty sets based on equivalence relations. In terms of equivalence classes induced by an equivalence relation, a non-empty subset of the universe may be approximated by three subsets. The Pawlak’s upper approximation is the union of equivalent classes which have a nonempty intersection with the given set, and the Pawlak’s lower approximation is the union of equivalent classes which are subsets of the given set. The Pawlak’s rough set was defined by meaning of a pair of upper and lower approximations, where the difference of upper and lower approximations (The Pawlak’s boundary region) is a non-empty set. Because of novel thinking, Pawlak’s rough set theory has been becoming an attractive intelligent information processing tool in the field of artificial intelligence.

The classical approximation processing model of Pawlak’s rough set theory has possible uses in many systems, including groups (see [8 –10]), semigroups (see [11 –14]), BCK-algebras [15], rings (see [16, 17]), modules [18], hemirings [19], quantales [20], LA-semigroups [21] and pseudo-BCI algebras [22]. Most of these utilizations have already been proved in the research of mathematical reasoning systems. One of the interesting aspects is the development of Pawlak’s rough sets in semigroups. Because the semigroup is a non-complex algebraic structure and it has been being employed in computer science and data technology, especially in algebraic automata theory and algebraic engineering theory [23], it attracts researchers much more. Patently, work on hybrid notions of Pawlak’s rough set theory and semigroup theory has been progressing continuously. For example, Kuroki [11] introduced the notions of upper and lower approximation semigroups (resp. ideals) in semigroups induced by congruence relations, and provided sufficient conditions of upper and lower approximation semigroups (resp. ideals). Xiao and Zhang [12] proposed the notions of upper and lower approximation completely prime ideals in semigroups induced by congruence relations, and provided sufficient conditions of upper and lower approximation completely prime ideals. Also, they studied the relationship between upper and lower approximation completely prime ideals (resp. ideals) and the homomorphic image of upper and lower approximation completely prime ideals (resp. ideals) under homomorphism problems. Wang and Zhan [14] introduced upper and lower approximation semigroups (resp. ideals and completely prime ideals) induced by special congruence relations, and also provided sufficient conditions of upper and lower approximation semigroups (resp. ideals and completely prime ideals).

The approximation processing model based on an arbitrary binary relation on the single universe is one of the most important extensions of the approximation processing model of Pawlak’s rough set theory. This is an effective mathematical tool for dealing with uncertain knowledge and vagueness in data for the single universe (see [24 –29]). For instance, Yao [26] introduced and studied an approximation processing model by using successor neighborhoods induced by binary relations. Mareay [29] defined the notion of Mareay’s rough sets based on cores of the successor neighborhoods induced by binary relations on the single universe. This work was studied in topological spaces based on partitions induced by binary relations. These partitions were perfectly defined by applying of successor neighborhoods.

In this paper, after providing some fundamentals of semigroup theory and Mareay’s rough set theory in Section 2. Then, in Section 3, we firstly establish and verify two new classes, called a successor class and a core of a successor class induced by an arbitrary binary relation between two universes. Then we propose a generalization of the Mareay’s rough set in an approximation space based on cores of successor classes induced by an arbitrary binary relation between two universes, with a corresponding example. Some interesting algebraic properties of the new approximation processing model are investigated. In Section 4, we develop the use of the novel rough set in a semigroup under a preorder and compatible relation. We establish the notions of rough semigroups, rough ideals and rough completely prime ideals. Then we provide sufficient conditions of rough semigroups, rough ideals and rough completely prime ideals. In Section 5, the relationship between the rough semigroup (resp. rough ideal and rough completely prime ideal) and the homomorphic image of the rough semigroup (resp. rough ideal and rough completely prime ideal) is demonstrated under homomorphism problems in semigroups. In Section 6, we discuss to the important relationships of this work and approximation processing models in [1 , 29]. In the end, we give a conclusion of the research in Section 7.

2. Preliminaries

In this section we recall important definitions which will be used in subsequent sections. Recall that a semigroup [30] (S, ⊙) is defined as an algebraic system, where S is a non-empty set and ⊙ is an associative binary operation on S. Throughout this paper, S denotes a semigroup. A non-empty subset X of S is called a subsemigroup [31] of S if XX ⊆ X. A non-empty subset X of S is called a left (right) ideal [31] of S if SX ⊆ X (XS ⊆ X). A non-empty subset X of S is called an ideal [31] of S if it is both a left ideal and a right ideal of S. An ideal X of S is called a completely prime ideal [31] of S if for all s₁, s₂ ∈ S, s₁s₂ ∈ X implies s₁ ∈ X or s₂ ∈ X. Throughout this paper we assume that U and V are both non-empty universal sets.

Definition 2.1. [32] Let $P (U \times V)$ be a family of all subsets of U × V. An element in $P (U \times V)$ is referred to as a binary relation from U to V. An element in $P (U \times V)$ is called a binary relation on U if U and V are identical.

Definition 2.2. [32] Let Θ be a binary relation from U to V. Θ is called serial if for all u ∈ U, there exists v ∈ V such that (u, v) ∈ Θ.

Definition 2.3. [32] Let Θ be a binary relation on U.

Θ is called reflexive if for all u ∈ U, (u, u) ∈ Θ.

Θ is called transitive if for all u₁, u₂, u₃ ∈ U, (u₁, u₂) ∈ Θ and (u₂, u₃) ∈ Θ implies (u₁, u₃) ∈ Θ.

Θ is called symmetric if for all u₁, u₂ ∈ U, (u₁, u₂) ∈ Θ implies (u₂, u₁) ∈ Θ.

Θ is called anti-symmetric if for all u₁, u₂ ∈ U, (u₁, u₂) ∈ Θ and (u₂, u₁) ∈ Θ implies u₁ = u₂.

Θ is referred to as a preorder if it is reflexive and transitive.

Θ is referred to as an equivalence relation if it is reflexive, transitive and symmetric.

Definition 2.4. [30] Let Θ be a binary relation on S. Θ is called compatible if for all s₁, s₂, s₃ ∈ S, (s₁, s₂) ∈ Θ implies (s₁s₃, s₂s₃) ∈ Θ and (s₃s₁, s₃s₂) ∈ Θ. Θ is called a congruence relation if it is an equivalence relation and compatible.

For the remainder of this section we review the approximation processing of Mareay’s rough set theory [29]. Let Θ be a binary relation on U and u ∈ U. The successor neighborhood of u induced by Θ is referred to as S_Θ (u) : = {u′ ∈ U : (u, u′) ∈ Θ}. The core of the successor neighborhood of u induced by Θ is referred to as CS_Θ (u) : = {u′ ∈ U : S_Θ (u) = S_Θ (u′)}. Given a non-empty subset X of U, $\bar{Θ} (X) : = ⋃_{u \in U} {{CS}_{Θ} (u) : {CS}_{Θ} (u) \cap X \neq \emptyset}$ denotes a Mareay’s upper approximation of X, $\underline{Θ} (X) : = ⋃_{u \in U} {{CS}_{Θ} (u) : {CS}_{Θ} (u) \subseteq X}$ denotes a Mareay’s lower approximation of X, $Θ_{bnd} (X) : = \bar{Θ} (X) - \underline{Θ} (X)$ denotes a Mareay’s boundary region of X. Recall that $Θ (X) : = (\bar{Θ} (X), \underline{Θ} (X))$ is said to be a Mareay’s rough set of X if Θ_bnd (X)≠ ∅. Otherwise X is said to be a Mareay’s definable set.

Observe that every Pawlak’s rough set in [1] is a Mareay’s rough set, but the converse is not true in general. Hence the Mareay’s rough set is considered as a generalization of the Pawlak’s rough set whenever the equivalence property of a relation is contained in the approximation processing model of Mareay’s rough set theory.

3. Generalizations of rough sets induced by binary relations

In this section we introduce a generalization of the Mareay’s rough set, with a corresponding example. Some interesting algebraic properties are investigated. The starting point of this section we give two new classes as Definitions 3.1 and 3.3 below.

Definition 3.1. Let Θ be a binary relation from U to V. Given an element u ∈ U, the set S_Θ (u) : = {v ∈ V : (u, v) ∈ Θ} is called a successor class of u induced by Θ.

Remark 3.2. If Θ is a serial relation from U to V, then S_Θ (u)≠ ∅ for all u ∈ U.

Definition 3.3. Let Θ be a binary relation from U to V. For an element u₁ ∈ U, the set $C S_{Θ} (u_{1}) : = {u_{2} \in U : S_{Θ} (u_{1}) = S_{Θ} (u_{2})}$ is called a core of the successor class of u₁ induced by Θ. We denote by $C S_{Θ} (U)$ a family of CS_Θ (u) for all u ∈ U.

Directly from Definition 3.3, we can obtain Proposition 3.4 below.

Proposition 3.4. Let Θ be a binary relation from U to V. Then the following statements hold.

For all u ∈ U, u ∈ CS_Θ (u).

For all u₁, u₂ ∈ U, u₁ ∈ CS_Θ (u₂) if and only if CS_Θ (u₁) = CS_Θ (u₂).

The following remark is an immediate consequence of (2) in Proposition 3.4.

Remark 3.5.If Θ is a binary relation from U to V, then $C S_{Θ} (U)$ is the partition of U.

Proposition 3.6. Let Θ be a binary relation on U. Then we have the following statements.

If Θ is reflexive, then CS_Θ (u) ⊆ S_Θ (u) for all u ∈ U.

If Θ is an equivalence relation, then S_Θ (u) and CS_Θ (u) are identical classes for all u ∈ U.

Proof. The proof is straightforward, so we omit it. □

Proposition 3.7. Let Θ be a binary relation on U and u₁, u₂ ∈ U. If Θ is reflexive and anti-symmetric, then the following statements are equivalent.

u₁ ∈ CS_Θ (u₂).

CS_Θ (u₁) = CS_Θ (u₂).

u₁ = u₂.

Proof. By Proposition 3.4 and properties of Θ, we can prove that (1), (2) and (3) are equivalent. □

Definition 3.8. Let Θ be a binary relation from Uto V. The triple $(U, V, C S_{Θ} (U))$ is referred to as an approximation space based on $C S_{Θ} (U)$ (briefly, $C S_{Θ} (U)$ -approximation space). If U = V, then $(U, V, C S_{Θ} (U))$ is substituted by a pair (U, $C S_{Θ} (U))$ .

Definition 3.9. Let $(U, V, C S_{Θ} (U))$ be an $C S_{Θ} (U)$ -approximation space. For a given non-empty subset X of U, we define three sets as $\bar{Θ} (X) : = ⋃_{u \in U} {{CS}_{Θ} (u) : {CS}_{Θ} (u) \cap X \neq \emptyset}$ , $\underline{Θ} (X) : = ⋃_{u \in U} {{CS}_{Θ} (u) : {CS}_{Θ} (u) \subseteq X}$ and _{Θ
bnd} (X) : = $\bar{Θ} (X) - \underline{Θ} (X)$ .

$\bar{Θ} (X)$ is called an upper approximation of X in $(U, V, C S_{Θ} (U))$ (briefly, $C S_{Θ} (U)$ -upper approximation of X).

$\underline{Θ} (X)$ is called a lower approximation of X in $(U, V, C S_{Θ} (U))$ (briefly, $C S_{Θ} (U)$ -lower approximation of X).

_Θ
bnd (X) is called a boundary region of X in $(U, V, C S_{Θ} (U))$ (briefly, $C S_{Θ} (U)$ -boundary region of X).

$Θ (X) : = (\bar{Θ} (X), \underline{Θ} (X))$ is said to be a rough set of X in $(U, V, C S_{Θ} (U))$ (briefly, $C S_{Θ} (U)$ -rough set of X) if _Θ
bnd (X)≠ ∅.

X is said to be a definable set in $(U, V, C S_{Θ} (U))$ (briefly, $C S_{Θ} (U)$ -definable set) if _Θ
bnd (X) =∅.

According to Definition 3.9, it is easily verified that

\bar{Θ} (X) : = {u \in U : {CS}_{Θ} (u) \cap X \neq \emptyset}

and that

\underline{Θ} (X) : = {u \in U : {CS}_{Θ} (u) \subseteq X}

In view of Definition 3.9, we consider Example 3.10 as the following.

Example 3.10. Let U : = {u₁, u₂, u₃, u₄, u₅} be a set of electrical discharge machines (EDM) in a boat industry of a leading company and let V : = {v₁, v₂, v₃, v₄} be a set of the components with respect to elements in U.

Define an information of the damage values of all electrical discharge machines in U with respect to components in V as Table 1.

Table 1
The information table of damage values

	v ₁	v ₂	v ₃	v ₄
u ₁	Bad	Medium	Bad	Medium
u ₂	Bad	Medium	Bad	Medium
u ₃	Medium	Bad	Bad	Bad
u ₄	Medium	Medium	Medium	Medium
u ₅	Bad	Bad	Bad	Medium

Given a binary relation $Θ \in P (U \times V)$ and elements u ∈ U, v ∈ V, a pair (u, v) ∈ Θ is defined as a bad damage value of the electrical discharge machine u with respect to the component v under Θ. Then Θ : = {(u₁, v₁), (u₁, v₃), (u₂, v₁), (u₂, v₃), (u₃, v₂), (u₃, v₃), (u₃, v₄), (u₅, v₁), (u₅, v₂), (u₅, v₃)}. Suppose that a measurement expert committee assign X : = {u₂, u₃, u₅} which is a non-empty set of electrical discharge machines for the discharge under a global evaluation. Then the assessment of X in approximation space $(U, V, C S_{Θ} (U))$ is derived by a process as the following. According to Definition 3.1, it follows that

S_Θ (u₁) : = {v₁, v₃},

S_Θ (u₂) : = {v₁, v₃},

S_Θ (u₃) : = {v₂, v₃, v₄},

S_Θ (u₄) : =∅ and

S_Θ (u₅) : = {v₁, v₂, v₃}.

According to Definition 3.3, it follows that CS_Θ (u₁) : = {u₁, u₂},

CS_Θ (u₂) : = {u₁, u₂},

CS_Θ (u₃) : = {u₃},

CS_Θ (u₄) : = {u₄} and

CS_Θ (u₅) : = {u₅}.

According to Definition 3.9, it follows that

$\bar{Θ} (X) : = {u_{1}, u_{2}, u_{3}, u_{5}}$ ,

$\underline{Θ} (X) : = {u_{3}, u_{5}}$ and

$\underline{Θ} (X) : = {u_{1}, u_{2}}$ .

Therefore Θ (X) : = ({u₁, u₂, u₃, u₅}, {u₃, u₅}) is a $C S_{Θ} (U)$ -rough set of X. Consequently,

u₁, u₂, u₃ and u₅ are possibly electrical discharge machines for the discharge,

u₃ and u₅ are certainly electrical discharge machines for the discharge and

u₁ and u₂ cannot be determined whether two machines are electrical discharge machines for the discharge or not.

The following remark is an immediate consequence of Definition 3.9 and the existence of Example 3.10 with respect to Definition 3.9.

Remark 3.11. Every Mareay’s rough set in [29] is a rough set in Definition 3.9, but the converse is not true in general. Therefore the rough set in Definition 3.9 is considered as a generalization of the Mareay’s rough set whenever U and V are identical.

The existence of Example 3.10 leads to the following definition.

Definition 3.12. Let $(U, V, C S_{Θ} (U))$ be an $C S_{Θ} (U)$ -approximation space and let X be a non-empty subset of U. $\bar{Θ} (X)$ is called a non-empty $C S_{Θ} (U)$ -upper approximation of X in $(U, V, C S_{Θ} (U))$ if $\bar{Θ} (X)$ is a non-empty subset of U. Similarly, we can define non-empty $C S_{Θ} (U)$ -lower approximations. The $C S_{Θ} (U)$ -rough set Θ (X) of X in $(U, V, C S_{Θ} (U))$ is referred to as a non-empty $C S_{Θ} (U)$ -rough set if $\bar{Θ} (X)$ is a non-empty $C S_{Θ} (U)$ -upper approximation and $\underline{Θ} (X)$ is a non-empty $C S_{Θ} (U)$ -lower approximation.

Proposition 3.13. Let $(U, V, C S_{Θ} (U))$ be an $C S_{Θ} (U)$ -approximation space. If X and Y are both non-empty subsets of U, then we have the following statements.

$\bar{Θ} (U) = U$ and $\underline{Θ} (U) = U$ .

$\bar{Θ} (\emptyset) = \emptyset$ and $\underline{Θ} (\emptyset) = \emptyset$ .

$X \subseteq \bar{Θ} (X)$ and $\underline{Θ} (X) \subseteq X$ .

$\bar{Θ} (X \cup Y) = \bar{Θ} (X) \cup \bar{Θ} (Y)$ and $\underline{Θ} (X \cap Y) = \underline{Θ} (X) \cap \underline{Θ} (Y)$ .

$\bar{Θ} (X \cap Y) \subseteq \bar{Θ} (X) \cap \bar{Θ} (Y)$ and $\underline{Θ} (X \cup Y) \supseteq \underline{Θ} (X) \cup \underline{Θ} (Y)$ .

$\underline{Θ} (X^{c}) = (\bar{Θ} (X))^{c}$ , where X^c and $(\bar{Θ} (X))^{c}$ are complements of X and $\bar{Θ} (X)$ , respectively.

$\bar{Θ} (\bar{Θ} (X)) = \bar{Θ} (X)$ and $\underline{Θ} (\underline{Θ} (X)) = \underline{Θ} (X)$ .

$\bar{Θ} ((\bar{Θ} (X))^{c}) = (\bar{Θ} (X))^{c}$ , where $(\bar{Θ} (X))^{c}$ is a complement of $\bar{Θ} (X)$ .

$\underline{Θ} ((\underline{Θ} (X))^{c}) = (\underline{Θ} (X))^{c}$ , where $(\underline{Θ} (X))^{c}$ is a complement of $\underline{Θ} (X)$ .

If X ⊆ Y, then $\bar{Θ} (X) \subseteq \bar{Θ} (Y)$ and $\underline{Θ} (X) \subseteq \underline{Θ} (Y)$ .

Proof. Using the similar method in the proof of Proposition 2.1 in [29], we can prove that (1)-(9) hold.

□

Definition 3.14. Let $(U, V, C S_{Θ} (U))$ be an $C S_{Θ} (U)$ -approximation space and let X be a non-empty subset of U. We call X a set over a non-empty interior set if $\underline{Θ} (X)$ is a non-empty $C S_{Θ} (U)$ -lower approximation of X in $(U, V, C S_{Θ} (U))$ and $\underline{Θ} (X)$ is a proper subset of X.

Proposition 3.15. Let $(U, V, C S_{Θ} (U))$ be an $C S_{Θ} (U)$ -approximation space and let X be a non-empty subset of U. If X is a set over a non-empty interior set, then Θ (X) is a non-empty $C S_{Θ} (U)$ -rough set of X in $(U, V, C S_{Θ} (U))$ .

Proof Suppose that X is a set over a non-empty interior set. Then we have that $\underline{Θ} (X)$ is a non-empty $C S_{Θ} (U)$ -lower approximation and $\underline{Θ} (X) \subset X$ . By Proposition 3.13 (3), we obtain that $\emptyset \neq X \subseteq \bar{Θ} (X)$ . Thus we get $\bar{Θ} (X)$ is a non-empty $C S_{Θ} (U)$ -upper approximation. We shall verify that _{Θ
bnd} (X)≠ ∅. Suppose that _{Θ
bnd} (X) =∅. Then we have $\bar{Θ} (X) = \underline{Θ} (X)$ . From Proposition 3.13 (3), once again, it follows that $\underline{Θ} (X) = X$ , a contradiction. Thus _{Θ
bnd} (X)≠ ∅. As a consequence, Θ (X) is a non-empty $C S_{Θ} (U)$ -rough set of X. □

Proposition 3.16. Let $(U, C S_{Θ} (U))$ be an $C S_{Θ} (U)$ -approximation space. If Θ is reflexive and anti-symmetric, then X is a $(U, C S_{Θ} (U))$ -definable set for every non-empty subset X of U.

Proof. By Propositions 3.7 and 3.13 (3), we can prove that the statement holds. □

Proposition 3.17. Let $(U, C S_{Θ} (U))$ be an $C S_{Θ} (U)$ -approximation space and let $(U, C S_{Ψ} (U))$ be an $C S_{Ψ} (U)$ -approximation space. If Θ ⊆ Ψ where Θ is reflexive and Ψ is transitive, then $\bar{Θ} (X) \subseteq \bar{Ψ} (X)$ for every non-empty subset X of U.

Proof. Let X be a non-empty subset of U. Then we prove that $\bar{Θ} (X) \subseteq \bar{Ψ} (X)$ . Indeed, let $u_{1} \in \bar{Θ} (X)$ . Then CS_Θ (u₁)∩ X ≠ ∅. Thus there exists u₂ ∈ U such that u₂ ∈ CS_Θ (u₁) ∩ X. Hence S_Θ (u₁) = S_Θ (u₂). Since Θ is reflexive, we have (u₂, u₂) ∈ Θ. Whence u₂ ∈ S_Θ (u₂) = S_Θ (u₁). Thus we have (u₁, u₂) ∈ Θ. Since Θ ⊆ Ψ, we have (u₁, u₂) ∈ Ψ. Similary, we get (u₂, u₁) ∈ Ψ. We shall show that S_Ψ (u₁) = S_Ψ (u₂). Now, let u₃ ∈ S_Ψ (u₂). Then (u₂, u₃) ∈ Ψ. Since Ψ is a transitive relation, we have (u₁, u₃) ∈ Ψ. Thus u₃ ∈ S_Ψ (u₁), which yields S_Ψ (u₂) ⊆ S_Ψ (u₁). Similary, we can prove that S_Ψ (u₁) ⊆ S_Ψ (u₂). Whence S_Ψ (u₁) = S_Ψ (u₂), and so u₂ ∈ CS_Ψ (u₁). Thus we have u₂ ∈ CS_Ψ (u₁) ∩ X. Hence CS_Ψ (u₁)∩ X ≠ ∅, which yields $u_{1} \in \bar{Ψ} (X)$ . Therefore, $\bar{Θ} (X) \subseteq \bar{Ψ} (X)$ . □

Proposition 3.18. Let $(U, C S_{Θ} (U))$ be an $C S_{Θ} (U)$ -approximation space and let $(U, C S_{Ψ} (U))$ be an $C S_{Ψ} (U)$ -approximation space. If Θ ⊆ Ψ where Θ is reflexive and Ψ is transitive, then $\underline{Ψ} (X) \subseteq \underline{Θ} (X)$ for every non-empty subset X of U.

Proof. Let X be a non-empty subset of U. Then we prove that $\underline{Ψ} (X) \subseteq \underline{Θ} (X)$ . Indeed, wesuppose that $u_{1} \in \underline{Ψ} (X)$ . Then CS_Ψ (u₁) ⊆ X. We shall show that CS_Θ (u₁) ⊆ CS_Ψ (u₁). Let u₂ ∈ CS_Θ (u₁). Then S_Θ (u₁) = S_Θ (u₂). Since Θ is a reflexive relation, we have (u₁, u₁) ∈ Θ. Hence we get u₁ ∈ S_Θ (u₁), and so u₁ ∈ S_Θ (u₂). Thus we get (u₂, u₁) ∈ Θ. By the assumption, we obtain that (u₂, u₁) ∈ Ψ. Similary, we get that (u₁, u₂) ∈ Ψ. Now, we shall prove that S_Ψ (u₁) = S_Ψ (u₂). Let u₃ ∈ S_Ψ (u₂). Then (u₂, u₃) ∈ Ψ. Since Ψ is a transitive relation, we have (u₁, u₃) ∈ Ψ. Whence u₃ ∈ S_Ψ (u₁). Hence S_Ψ (u₂) ⊆ S_Ψ (u₁). Similary, we can prove that S_Ψ (u₁) ⊆ S_Ψ (u₂). Hence S_Ψ (u₁) = S_Ψ (u₂). Thus we have u₂ ∈ CS_Ψ (u₁). Whence we get that CS_Θ (u₁) ⊆ CS_Ψ (u₁) ⊆ X. Therefore $u_{1} \in \underline{Θ} (X)$ . This means that $\underline{Ψ} (X) \subseteq \underline{Θ} (X)$ . □

4. Roughness in semigroups

In this section we develop the use of the novel rough set of Section 3 in a semigroup under a preorder and compatible relation. We establish the notions of rough semigroups, rough ideals and rough completely prime ideals. Then we provide sufficient conditions of rough semigroups, rough ideals and rough completely prime ideals.

Definition 4.1. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space. $(S, C S_{Θ} (S))$ is called an $C S_{Θ} (S)$ -approximation space type PCR if Θ is a preorder and compatible relation.

Proposition 4.2. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type PCR. Then $(C S_{Θ} (s_{1})) (C S_{Θ} (s_{2})) \subseteq C S_{Θ} (s_{1} s_{2})$ for all for all s₁, s₂ ∈ S

Proof. Let s₁ and s₂ be two elements in S. Suppose that s₃ ∈ (CS_Θ (s₁)) (CS_Θ (s₂)). Then there exist s₄ ∈ CS_Θ (s₁), s₅ ∈ CS_Θ (s₂) such that s₃ = s₄s₅. Thus S_Θ (s₁) = S_Θ (s₄) and S_Θ (s₂) = S_Θ (s₅). Hence we get that S_Θ (s₁s₂) = S_Θ (s₄s₅). Indeed, we suppose that s₆ ∈ S_Θ (s₄s₅). Then (s₄s₅, s₆) ∈ Θ. Since Θ is reflexive, we have (s₄, s₄) and (s₅, s₅) are in Θ, and so s₄ ∈ S_Θ (s₄) and s₅ ∈ S_Θ (s₅). Whence s₄ ∈ S_Θ (s₁) and s₅ ∈ S_Θ (s₂). Thus (s₁, s₄) ∈ Θ and (s₂, s₅) ∈ Θ. Since Θ is compatible, we have (s₁s₂, s₄s₅) ∈ Θ. Since Θ is transitive, we have (s₁s₂, s₆) ∈ Θ, and so s₆ ∈ S_Θ (s₁s₂). Hence S_Θ (s₄s₅) ⊆ S_Θ (s₁s₂). Similarly, we can show that S_Θ (s₁s₂) ⊆ S_Θ (s₄s₅). Thus S_Θ (s₁s₂) = S_Θ (s₄s₅), which yields s₃ = s₄s₅ ∈ CS_Θ (s₁s₂). Therefore (CS_Θ (s₁)) (CS_Θ (s₂)) ⊆ CS_Θ (s₁s₂). □

In what follows, we give Example 4.3 to illustrate that the property in Proposition 4.2 is indispensable.

Example 4.3. Let S : = {s₁, s₂, s₃, s₄, s₅} be the semigroup with multiplication table on S as Table 2.

Table 2
The multiplication table on S

·	s ₁	s ₂	s ₃	s ₄	s ₅
s ₁	s ₁	s ₂	s ₃	s ₂	s ₅
s ₂	s ₂	s ₂	s ₃	s ₂	s ₅
s ₃	s ₃	s ₃	s ₃	s ₃	s ₃
s ₄	s ₂	s ₂	s ₃	s ₂	s ₅
s ₅	s ₅	s ₅	s ₃	s ₅	s ₅

Define Θ : = {(s₁, s₁), (s₂, s₂), (s₂, s₅), (s₃, s₂), (s₃, s₃), (s₃, s₅), (s₄, s₄), (s₅, s₂), (s₅, s₅)}. Then it is easy to check that Θ is a preorder and compatible relation. Thus successor classes of the elements in S induced by Θ are S_Θ (s₁) : = {s₁},

S_Θ (s₂) : = {s₂, s₅},

S_Θ (s₃) : = {s₂, s₃, s₅},

S_Θ (s₄) : = {s₄} and

S_Θ (s₅) : = {s₂, s₅}.

Hence cores of successor classes of the elements in S induced by Θ are

CS_Θ (s₁) : = {s₁},

CS_Θ (s₂) : = {s₂, s₅},

CS_Θ (s₃) : = {s₃},

CS_Θ (s₄) : = {s₄} and

CS_Θ (s₅) : = {s₂, s₅}. Here it is straightforward to verify that

(CS_Θ (s)) (CS_Θ (s′)) ⊆ CS_Θ (ss′)

for all s, s′ ∈ S.

Observe that, in Example 4.3, it does not true in general for an equality case. We consider the following example.

Example 4.4. Let S : = {s₁, s₂, s₃, s₄, s₅} be the semigroup with multiplication table on S as Table 3.

Table 3

The multiplication table on S

·	s ₁	s ₂	s ₃	s ₄	s ₅
s ₁	s ₁	s ₁	s ₃	s ₁	s ₅
s ₂	s ₁	s ₂	s ₃	s ₁	s ₅
s ₃	s ₃	s ₃	s ₃	s ₃	s ₃
s ₄	s ₁	s ₁	s ₃	s ₄	s ₅
s ₅	s ₅	s ₅	s ₃	s ₅	s ₅

Define Θ : = {(s₁, s₁), (s₁, s₂), (s₁, s₄), (s₂, s₁), (s₂, s₂), (s₂, s₄), (s₃, s₃), (s₃, s₅), (s₄, s₁), (s₄, s₂), (s₄, s₄), (s₅, s₅)}. Then it is easy to check that Θ is a preorder and compatible relation. Thus successor classes of the elements in S induced by Θ are S_Θ (s₁) : = {s₁, s₂, s₄}, S_Θ (s₂) : = {s₁, s₂, s₄}, S_Θ (s₃) : = {s₃, s₅}, S_Θ (s₄) : = {s₁, s₂, s₄} and S_Θ (s₅) : = {s₅}.

Hence cores of successor classes of the elements in S induced by Θ are CS_Θ (s₁) : = {s₁, s₂, s₄}, CS_Θ (s₂) : = {s₁, s₂, s₄}, CS_Θ (s₃) : = {s₃}, CS_Θ (s₄) : = {s₁, s₂, s₄} and CS_Θ (s₅) : = {s₅}. Here it is straightforward to check that $(C S_{Θ} (s)) (C S_{Θ} (s^{'})) = C S_{Θ} (s s^{'})$ for all s, s′ ∈ S. Based on this point, the property can be considered as a special case of Proposition 4.2. This important example leads to the following definition.

Definition 4.5. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type PCR. The collection $C S_{Θ} (S)$ is called a complete collection induced by Θ (briefly, Θ-complete) if for all s₁, s₂ ∈ S, $(C S_{Θ} (s_{1})) (C S_{Θ} (s_{2})) = C S_{Θ} (s_{1} s_{2}) .$

Definition 4.6. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type PCR. If $C S_{Θ} (S)$ is a complete collection induced by Θ, then Θ is called a complete relation. $(S, C S_{Θ} (S))$ is called an $C S_{Θ} (S)$ -approximation space type CR if Θ is complete.

Proposition 4.7. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type PCR. If Θ is anti-symmetric, then Θ is complete.

Proof. We only need to prove that CS_Θ (s₁s₂) ⊆ (CS_Θ (s₁)) (CS_Θ (s₂)) holds for all s₁, s₂ ∈ S. In fact, let s₁ and s₂ be two elements in S. Suppose that s₃ ∈ CS_Θ (s₁s₂). Then by Proposition 3.7, we obtain that s₃ = s₁s₂. From Proposition 3.4 (1), it follows that s₃ = s₁s₂ ∈ (CS_Θ (s₁)) (CS_Θ (s₂)), which yields CS_Θ (s₁s₂) ⊆ (CS_Θ (s₁)) (CS_Θ (s₂)).

On the other hand, by Proposition 4.2, it follows that (CS_Θ (s₁)) (CS_Θ (s₂)) ⊆ CS_Θ (s₁s₂). Therefore (CS_Θ (s₁)) (CS_Θ (s₂)) = CS_Θ (s₁s₂), and so $C S_{Θ} (S)$ is Θ-complete. This means that Θ is complete.□

Proposition 4.8. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type PCR. Then $(\bar{Θ} (X)) (\bar{Θ} (Y)) \subseteq \bar{Θ} (XY)$ for every non-empty subsets X, Y of S.

Proof. Let X and Y be two non-empty subsets of S. Suppose that $s_{1} \in (\bar{Θ} (X)) (\bar{Θ} (Y))$ . Then there exist $s_{2} \in \bar{Θ} (X)$ and $s_{3} \in \bar{Θ} (Y)$ such that s₁ = s₂s₃. Thus CS_Θ (s₂)∩ X ≠ ∅ and CS_Θ (s₃)∩ Y ≠ ∅. Then there exist s₄, s₅ ∈ S such that s₄ ∈ CS_Θ (s₂) ∩ X and s₅ ∈ CS_Θ (s₃) ∩ Y. From Proposition 4.2, it follows that s₄s₅ ∈ (CS_Θ (s₂)) (CS_Θ (s₃)) ⊆ CS_Θ (s₂s₃) and s₄s₅ ∈ XY. Thus CS_Θ (s₂s₃)∩ XY ≠ ∅, which yields $s_{1} = s_{2} s_{3} \in \bar{Θ} (XY)$ . Therefore we get $(\bar{Θ} (X)) (\bar{Θ} (Y)) \subseteq \bar{Θ} (XY)$ . □

Proposition 4.9. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type CR. Then $(\underline{Θ} (X)) (\underline{Θ} (Y)) \subseteq \underline{Θ} (XY)$ for every non-empty subsets X, Y of S.

Proof. Let X and Y be two non-empty subsets of S. Suppose that $s_{1} \in (\underline{Θ} (X)) (\underline{Θ} (Y))$ . Then there exist $s_{2} \in \underline{Θ} (X)$ and $s_{3} \in \underline{Θ} (Y)$ such that s₁ = s₂s₃, and so CS_Θ (s₂) ⊆ X and CS_Θ (s₃) ⊆ Y. Since Θ is complete, we get $C S_{Θ} (s_{2} s_{3}) = C S_{Θ} (s_{2}) C S_{Θ} (s_{3}) \subseteq X Y .$ Whence we have CS_Θ (s₂s₃) ⊆ XY. It follows that $s_{1} = s_{2} s_{3} \in \underline{Θ} (XY)$ . Therefore $(\underline{Θ} (X)) (\underline{Θ} (Y)) \subseteq \underline{Θ} (XY)$ .

We consider the following example.

Example 4.10. According to Example 4.4, we let X : = {s₃, s₄, s₅} be a subset of S. Then we have $\bar{Θ} (X) = S$ and $\underline{Θ} (X) : = {s_{3}, s_{5}}$ . Here it is easy to verify that $\bar{Θ} (X)$ and $\underline{Θ} (X)$ are subsemigroups, ideals and completely prime ideals of S. Moreover, we also have _{Θ
bnd} (X)≠ ∅. Based on the preorder and compatible relation Θ, we see that the semigroup S contains the set X in which the $C S_{Θ} (U)$ -upper and $C S_{Θ} (U)$ -lower approximations of X are semigroups, ideals and completely prime ideals. In the sequel, existences of subsemigroups, ideals and completely prime ideals of S induced by the preorder and compatible relation lead to the following definition.

Definition 4.11. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type PCR and let X be a non-empty subset of S. We call a non-empty $C S_{Θ} (S)$ -upper approximation $\bar{Θ} (X)$ an $C S_{Θ} (S)$ -upper approximation semigroup if it is a subsemigroup of S. We call a non-empty $C S_{Θ} (S)$ -lower approximation $\underline{Θ} (X)$ a $C S_{Θ} (S)$ -lower approximation semigroup if it is a subsemigroup of S. A non-empty $C S_{Θ} (S)$ -rough set Θ (X) is said to be a $C S_{Θ} (S)$ -rough semigroup if $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation semigroup and $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation semigroup.

Similarly, we can define $C S_{Θ} (S)$ -rough (completely prime) ideals.

Theorem 4.12 Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type PCR. If X is a subsemigroup of S, then $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation semigroup.□

Proof. Suppose that X is a subsemigroup of S. Then XX ⊆ X. By Proposition 3.13 (3), we obtain that $\emptyset \neq X \subseteq \bar{Θ} (X) .$

Hence we get $\bar{Θ} (X)$ is a non-empty $C S_{Θ} (S)$ -upper approximation. From Proposition 3.13 (9), it follows that $\bar{Θ} (XX) \subseteq \bar{Θ} (X)$ . By Proposition 4.8, we get $(\bar{Θ} (X)) (\bar{Θ} (X)) \subseteq \bar{Θ} (XX) \subseteq \bar{Θ} (X) .$

Hence $\bar{Θ} (X)$ is a subsemigroup of S. Thus $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation semigroup.

Theorem 4.13. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type CR. If X is a subsemigroup of S with $\underline{Θ} (X) \neq \emptyset$ , then $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation semigroup.

Proof. Suppose that X is a subsemigroup of S. Then XX ⊆ X. Obviously, $\underline{Θ} (X)$ is a non-empty $C S_{Θ} (S)$ -lower approximation. From Proposition 3.13 (9), it follows that $\underline{Θ} (XX) \subseteq \underline{Θ} (X)$ . By Proposition 4.9, we obtain that $(\underline{Θ} (X)) (\underline{Θ} (X)) \subseteq \underline{Θ} (XX) \subseteq \underline{Θ} (X) .$

Thus $\underline{Θ} (X)$ is a subsemigroup of S. Therefore $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation semigroup. □

The following corollary is an immediate consequence of Proposition 3.15, Theorem 4.12 and Theorem 4.13.

Corollary 4.14. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type CR. If X is a subsemigroup of S over a non-empty interior set, then Θ (X) is a $C S_{Θ} (S)$ -rough semigroup.

Observe that, in Corollary 4.14, the converse is not true in general. We present an example as the following.

Example 4.15. According to Example 4.4, suppose that X : = {s₂, s₄, s₅} is a subset of S, then we have that $\bar{Θ} (X) : = {s_{1}, s_{2}, s_{4}, s_{5}}$ and $\underline{Θ} (X) : = {s_{5}}$ . Thus _{Θ
bnd} (X)≠ ∅. Hence it is straightforward to check that $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation semigroup and $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation semigroup. However, X is not a subsemigroup of S. Consequently, Θ (X) is a $C S_{Θ} (S)$ -rough semigroup, but X is not a subsemigroup of S.

Theorem 4.16. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type PCR. If X is an ideal of S, then $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation ideal.

Proof. Suppose that X is an ideal of S. Then SX ⊆ X. From Proposition 3.13 (9), we get $\bar{Θ} (SX) \subseteq \bar{Θ} (X)$ . By Proposition 3.13 (1), we obtain that $\bar{Θ} (S) = S$ . From Proposition 4.8, it follows that $S (\bar{Θ} (X)) = (\bar{Θ} (S)) (\bar{Θ} (X)) \subseteq \bar{Θ} (SX) \subseteq \bar{Θ} (X) .$

Hence $\bar{Θ} (X)$ is a left ideal of S.

Similarly, we can prove that $\bar{Θ} (X)$ is a right ideal of S. Therefore we get that $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation ideal.

□

Theorem 4.17. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type CR. If X is an ideal of S with $\underline{Θ} (X) \neq \emptyset$ , then $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation ideal.

Proof. Suppose that X is an ideal of S. Then SX ⊆ X. From Proposition 3.13 (9), we get $\underline{Θ} (SX) \subseteq \underline{Θ} (X)$ . By Proposition 3.13 (1), we obtain that $\underline{Θ} (S) = S$ . From Proposition 4.9, it follows that $S (\underline{Θ} (X)) = (\underline{Θ} (S)) (\underline{Θ} (X)) \subseteq \underline{Θ} (SX) \subseteq \underline{Θ} (X) .$

Thus $\underline{Θ} (X)$ is a left ideal of S.

Similarly, we can prove that $\underline{Θ} (X)$ is a right ideal of S. Thus $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation ideal. □

The following corollary is an immediate consequence of Proposition 3.15, Theorem 4.16 and Theorem 4.17.

Corollary 4.18. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type CR. If X is an ideal of S over a non-empty interior set, then Θ (X) is a $C S_{Θ} (S)$ -rough ideal.

Observe that, in Corollary 4.18, the converse is not true in general. We present an example as the following.

Example 4.19. According to Example 4.4, if X : = {s₂, s₃, s₅} is a subset of S, then we have $\bar{Θ} (X) = S$ and $\underline{Θ} (X) : = {s_{3}, s_{5}}$ . Thus we get _{Θ
bnd} (X)≠ ∅. Obviously, $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation ideal, and it is straightforward to check that $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation ideal. However, X is not an ideal of S. Consequently, Θ (X) is a $C S_{Θ} (S)$ -rough ideal, but X is not an ideal of S.

Theorem 4.20. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type CR. If X is a completely prime ideal of S, then $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation completely prime ideal.

Proof. Suppose that X is a completely prime ideal of S. Then we prove that $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation completely prime ideal. In fact, since X is an ideal of S, by Theorem 4.16, we have that $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation ideal. Let s₁, s₂ ∈ S be such that $s_{1} s_{2} \in \bar{Θ} (X)$ . Then, by the Θ-complete property of $C S_{Θ} (S)$ , we get that $(C S_{Θ} (s_{1})) (C S_{Θ} (s_{2})) \cap X = C S_{Θ} (s_{1} s_{2}) \cap X \neq \emptyset .$

Thus there exist s₃ ∈ CS_Θ (s₁), s₄ ∈ CS_Θ (s₂) such that s₃s₄ ∈ X. Since X is a completely prime ideal, we have s₃ ∈ X or s₄ ∈ X. Thus we have that CS_Θ (s₁)∩ X ≠ ∅ or CS_Θ (s₂) ∩ X ≠ ∅, and so $s_{1} \in \bar{Θ} (X)$ or $s_{2} \in \bar{Θ} (X)$ . Therefore $\bar{Θ} (X)$ is a completely prime ideal of S. As a consequence, $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation completely prime ideal. □

Theorem 4.21. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type CR. If X is a completely prime ideal of S with $\underline{Θ} (X) \neq \emptyset$ , then $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation completely prime ideal.

Proof. Suppose that X is a completely prime ideal of S with $\underline{Θ} (X) \neq \emptyset$ . Then X is an ideal of S. Thus by Theorem 4.17, we have $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation ideal. Let s₁, s₂ ∈ S be such that $s_{1} s_{2} \in \underline{Θ} (X)$ . Since Θ is complete, we have $(C S_{Θ} (s_{1})) (C S_{Θ} (s_{2})) = C S_{Θ} (s_{1} s_{2}) \subseteq X .$

Now, we suppose that $s_{1} \notin \underline{Θ} (X)$ . Then CS_Θ (s₁) is not a subset of X. Thus there exists s₃ ∈ CS_Θ (s₁) but s₃ ∉ X. For each s₄ ∈ CS_Θ (s₂), $s_{3} s_{4} \in (C S_{Θ} (s_{1})) (C S_{Θ} (s_{2})) \subseteq X .$

Whence s₃s₄ ∈ X. Since X is a completely prime ideal and s₃ ∉ X, we have s₄ ∈ X. Thus we get CS_Θ (s₂) ⊆ X, which yields $s_{2} \in \underline{Θ} (X)$ . Hence we get $\underline{Θ} (X)$ is a completely prime ideal of S. Therefore $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation completely prime ideal. □

The following corollary is an immediate consequence of Proposition 3.15, Theorem 4.20 and Theorem 4.21.

Corollary 4.22. Let $(S, C S_{Θ} (S))$ be an $C S_{Θ} (S)$ -approximation space type CR. If X is a completely prime ideal of S over a non-empty interior set, then Θ (X) is a $C S_{Θ} (S)$ -rough completely prime.

Observe that, in Corollary 4.22, the converse is not true in general. We present an example as the following.

Example 4.23. According to Example 4.4, if X : = {s₁, s₃, s₅} is a subset of S, then we have $\bar{Θ} (X) = S$ and $\underline{Θ} (X) : = {s_{3}, s_{5}}$ . Hence we get _{Θ
bnd} (X)≠ ∅. Obviously, $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation completely prime ideal, and it is straightforward to check that $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation completely prime ideal. Here we can verify that X is an ideal of S, but it is not a completely prime ideal of S since s₂s₄ = s₁ ∈ X but s₂ ∉ X and s₄ ∉ X. As a consequence, Θ (X) is a $C S_{Θ} (S)$ -rough completely prime ideal, but X is not a completely primeideal of S.

5. Homomorphic images of roughness in semigroups

In this section we study the relationships between rough semigroups (resp. rough ideals, rough completely prime ideals) and their homomorphic images. Throughout this section we assume that T is a semigroup.

Proposition 5.1. Let f be an epimorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ , where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. Then the following statements hold.

For all s₁, s₂ ∈ S, s₁ ∈ CS_Θ (s₂) if and only if f (s₁) ∈ CS_Ψ (f (s₂)).

$f (\bar{Θ} (X)) = \bar{Ψ} (f (X))$ for every non-empty subset X of S.

$f (\underline{Θ} (X)) \subseteq \underline{Ψ} (f (X))$ for every non-empty subset X of S.

If f is injective, then $f (\underline{Θ} (X)) = \underline{Ψ} (f (X))$ for every non-empty subset X of S.

If Ψ is a preorder and compatible relation, then Θ is a preorder and compatible relation.

Proof. (1) Let s₁, s₂ ∈ S be such that s₁ ∈ CS_Θ (s₂). Then f (s₁), f (s₂) ∈ T and S_Θ (s₁) = S_Θ (s₂). We shall prove that S_Ψ (f (s₁)) = S_Ψ (f (s₂)). Suppose that t₁ ∈ S_Ψ (f (s₁)). Then (f (s₁), t₁) ∈ Ψ. Since f is surjective, there exists s₃ ∈ S such that f (s₃) = t₁. Whence (f (s₁), f (s₃)) ∈ Ψ, and so (s₁, s₃) ∈ Θ. Thus s₃ ∈ S_Θ (s₁). Whence we have s₃ ∈ S_Θ (s₂). Hence we have (s₂, s₃) ∈ Θ, and so (f (s₂), f (s₃)) ∈ Ψ. Thus t₁ = f (s₃) ∈ S_Ψ (f (s₂)). Then we have S_Ψ (f (s₁)) ⊆ S_Ψ (f (s₂)). Similarly, we can show that S_Ψ (f (s₂)) ⊆ S_Ψ (f (s₁)). Therefore we get S_Ψ (f (s₁)) = S_Ψ (f (s₂)). Consequently, we get f (s₁) ∈ CS_Ψ (f (s₂)).

Conversely, it is easy to verify that s₁ ∈ CS_Θ (s₂) whenever f (s₁) ∈ CS_Ψ (f (s₂)) for all s₁, s₂ ∈ S.

(2) Let X be a non-empty subset of S. We verify firstly that $f (\bar{Θ} (X)) = \bar{Ψ} (f (X))$ . Let $t_{1} \in f (\bar{Θ} (X))$ . Then there exists $s_{1} \in \bar{Θ} (X)$ such that f (s₁) = t₁. Therefore CS_Θ (s₁)∩ X ≠ ∅. Thus there exists s₂ ∈ S such that s₂ ∈ CS_Θ (s₁) and s₂ ∈ X. By the argument (1), we obtain that f (s₂) ∈ CS_Ψ (f (s₁)) and f (s₂) ∈ f (X). Then CS_Ψ (f (s₁))∩ f (X) ≠ ∅, and so $t_{1} = f (s_{1}) \in \bar{Ψ} (f (X))$ . Thus we have $f (\bar{Θ} (X)) \subseteq \bar{Ψ} (f (X))$ .

On the other hand, we let $t_{2} \in \bar{Ψ} (f (X))$ . Then there exists s₃ ∈ S such that f (s₃) = t₂, and so CS_Ψ (f (s₃))∩ f (X) ≠ ∅. Thus there exists s₄ ∈ X such that f (s₄) ∈ f (X) and f (s₄) ∈ CS_Ψ (f (s₃)). By the argument (1), we get that s₄ ∈ CS_Θ (s₃), and so we have CS_Θ (s₃)∩ X ≠ ∅. Hence $s_{3} \in \bar{Θ} (X)$ , and so $t_{2} = f (s_{3}) \in f (\bar{Θ} (X))$ . Thus $\bar{Ψ} (f (X)) \subseteq f (\bar{Θ} (X))$ . This implies that $f (\bar{Θ} (X)) = \bar{Ψ} (f (X))$ .

(3) Let X be a non-empty subset of S. Suppose that $t_{1} \in f (\underline{Θ} (X))$ . Then there exists $s_{1} \in \underline{Θ} (X)$ such that f (s₁) = t₁. Thus we get CS_Θ (s₁) ⊆ X. We shall prove that CS_Ψ (t₁) ⊆ f (X). Let t₂ ∈ CS_Ψ (t₁). Then there exist s₂ ∈ S such that f (s₂) = t₂. Thus we have f (s₂) ∈ CS_Ψ (f (s₁)). By the argument (1), we obtain that s₂ ∈ CS_Θ (s₁), and so s₂ ∈ X. Hence t₂ = f (s₂) ∈ f (X), and thus, CS_Ψ (t₁) ⊆ f (X).Therefore we have $t_{1} \in \underline{Ψ} (f (X))$ . As a consequence, $f (\underline{Θ} (X)) \subseteq \underline{Ψ} (f (X))$ .

(4) Let X be a non-empty subset of S. We only need to prove that $\underline{Ψ} (f (X)) \subseteq f (\underline{Θ} (X))$ . Suppose that $t_{1} \in \underline{Ψ} (f (X))$ . Then there exists s₁ ∈ S such that f (s₁) = t₁. Thus CS_Ψ (f (s₁)) ⊆ f (X). We shall show that CS_Θ (s₁) ⊆ X. Let s₂ ∈ CS_Θ (s₁). Then, by the argument (1), we have f (s₂) ∈ CS_Ψ (f (s₁)). Hence f (s₂) ∈ f (X). Thus there exists s₃ ∈ X such that f (s₃) = f (s₂). By the assumption, s₂ ∈ X, and so CS_Θ (s₁) ⊆ X. Hence $s_{1} \in \underline{Θ} (X)$ , and so $t_{1} = f (s_{1}) \in f (\underline{Θ} (X))$ . Thus $\underline{Ψ} (f (X)) \subseteq f (\underline{Θ} (X))$ .

By the argument (3), we get $f (\underline{Θ} (X)) \subseteq \underline{Ψ} (f (X))$ . Consequently, $f (\underline{Θ} (X)) = \underline{Ψ} (f (X))$ .

(5) The proof is straightforward, so we omit it. □

Proposition 5.2. Let f be an isomorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ , where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If Ψ is complete, then Θ is complete.

Proof. Let s₁, s₂ be two elements in S. Suppose that s₃ ∈ CS_Θ (s₁s₂). Then, by Proposition 5.1 (1), we get that f (s₃) ∈ CS_Ψ (f (s₁s₂)). Since f is a homomorphism and Ψ is complete, we have $\begin{matrix} f (s_{3}) \in & {CS}_{Ψ} (f (s_{1} s_{2})) = {CS}_{Ψ} (f (s_{1}) f (s_{2})) \\ = & ({CS}_{Ψ} (f (s_{1}))) ({CS}_{Ψ} (f (s_{2}))) . \end{matrix}$ Thus there exist t₁ ∈ CS_Ψ (f (s₁)), t₂ ∈ CS_Ψ (f (s₂)) such that f (s₃) = t₁t₂. Since f is surjective, there exist s₄, s₅ ∈ S such that f (s₄) = t₁ and f (s₅) = t₂. From $f (s_{4}) f (s_{5}) = f (s_{3}) \in (C S_{Ψ} (f (s_{1}))) (C S_{Ψ} (f (s_{2}))),$ we get f (s₄) ∈ CS_Ψ (f (s₁)) and f (s₅) ∈ CS_Ψ (f (s₂)). By Proposition 5.1 (1), we obtain that s₄ ∈ CS_Θ (s₁) and s₅ ∈ CS_Θ (s₂). Since f is a homomorphism, we have f (s₃) = f (s₄) f (s₅) = f (s₄s₅). Since f is injective, we get s₃ = s₄s₅. Thus s₃ ∈ CS_Θ (s₁) CS_Θ (s₂). Therefore we have CS_Θ (s₁s₂) ⊆ CS_Θ (s₁) CS_Θ (s₂). On the other hand, we can immediately conclude by Propositions 4.2 and 5.1 (5) that CS_Θ (s₁) CS_Θ (s₂) ⊆ CS_Θ (s₁s₂). Thus CS_Θ (s₁) CS_Θ (s₂) = CS_Θ (s₁s₂). Hence $C S_{Θ} (S)$ is Θ-complete. It follows that Θ is complete. □

Theorem 5.3. Let f be an epimorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type PCR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximationsemigroup if and only if $\bar{Ψ} (f (X))$ is an $C S_{Ψ} (T)$ -upper approximation semigroup.

Proof. Suppose that $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation semigroup. Then, by Proposition 5.1 (2), we obtain that $\begin{array}{l} (\bar{ψ} (f (X))) (\bar{ψ} (f (X))) = (f (\bar{Θ} (X))) (f (\bar{Θ} (X))) \\ = f ((\bar{Θ} (X)) (\bar{Θ} (X))) \subseteq f (\bar{Θ} (X)) = \bar{ψ} (f (X)) . \end{array}$ Hence $\bar{Ψ} (f (X))$ is a subsemigroup of T. Thus we get $\bar{Ψ} (f (X))$ is an $C S_{Ψ} (T)$ -upper approximation semigroup.

Conversely, let $s_{1} \in (\bar{Θ} (X)) (\bar{Θ} (X))$ . From Proposition 5.1 (2), it follows that $\begin{matrix} f (s_{1}) \in & f ((\bar{Θ} (X)) (\bar{Θ} (X))) \\ = & (f (\bar{Θ} (X))) (f (\bar{Θ} (X))) \\ = & (\bar{Ψ} (f (X))) (\bar{Ψ} (f (X))) \\ \subseteq & \bar{Ψ} (f (X)) = f (\bar{Θ} (X)) . \end{matrix}$

Thus there exists $s_{2} \in \bar{Θ} (X)$ such that f (s₁) = f (s₂). Hence we have CS_Θ (s₂)∩ X ≠ ∅. From Proposition 3.4 (1), it follows that f (s₁) ∈ CS_Ψ (f (s₂)). By Proposition 5.1 (1), we obtain that s₁ ∈ CS_Θ (s₂). From Proposition 3.4 (2), it follows that CS_Θ (s₁) = CS_Θ (s₂). Thus we have CS_Θ (s₁)∩ X ≠ ∅, and so $s_{1} \in \bar{Θ} (X)$ . Hence $(\bar{Θ} (X)) (\bar{Θ} (X)) \subseteq \bar{Θ} (X)$ . Thus $\bar{Θ} (X)$ is a subsemigroup of S. Therefore $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation semigroup. □

Theorem 5.4. Let f be an isomorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type PCR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation semigroup if and only if $\underline{Ψ} (f (X))$ is a $C S_{Ψ} (T)$ -lower approximation semigroup.

Proof. By Proposition 5.1 (4) and using the similar method in the proof of Theorem 5.3, we can prove that the statement holds. □

The following corollary is an immediate consequence of Theorems 5.3 and 5.4.

Corollary 5.5. Let f be an isomorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type PCR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then Θ (X) is a $C S_{Θ} (S)$ -rough semigroup if and only if Ψ (f (X)) is a $C S_{Ψ} (T)$ -roughsemigroup.

Theorem 5.6. Let f be an epimorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type PCR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation ideal if and only if $\bar{Ψ} (f (X))$ is an $C S_{Ψ} (T)$ -upper approximation ideal.

Proof. Suppose that $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation ideal. Then we have $S \bar{Θ} (X) \subseteq \bar{Θ} (X)$ . Whence we have $f (S \bar{Θ} (X)) \subseteq f (\bar{Θ} (X))$ . By Proposition 5.1 (2), we obtain that $T \bar{Ψ} (f (X)) = f (S \bar{Θ} (X)) \subseteq f (\bar{Θ} (X)) = \bar{Ψ} (f (X)) .$

Hence $\bar{Ψ} (f (X))$ is a left ideal of T. Similarly, we can prove that $\bar{Ψ} (f (X))$ is a right ideal of T. Thus $\bar{Ψ} (f (X))$ is an $C S_{Ψ} (T)$ -upper approximation ideal. Conversely, we suppose that $\bar{Ψ} (f (X))$ is an $C S_{Ψ} (T)$ -upper approximation ideal. Then $T \bar{Ψ} (f (X)) \subseteq \bar{Ψ} (f (X))$ . Now, let $s_{1} \in S \bar{Θ} (X)$ . From Proposition 5.1 (2), it follows that $\begin{matrix} f (s_{1}) \in & f (S \bar{Θ} (X)) = T \bar{Ψ} (f (X)) \\ \subseteq & \bar{Ψ} (f (X)) = f (\bar{Θ} (X)) . \end{matrix}$

Thus there exists $s_{2} \in \bar{Θ} (X)$ such that f (s₁) = f (s₂), and so CS_Θ (s₂)∩ X ≠ ∅. By Proposition 3.4 (1), we obtain that f (s₁) ∈ CS_Ψ (f (s₂)). By Proposition 5.1 (1), we obtain s₁ ∈ CS_Θ (s₂). From Proposition 3.4 (2), it follows that CS_Θ (s₁) = CS_Θ (s₂). Hence we have CS_Θ (s₁)∩ X ≠ ∅, and so $s_{1} \in \bar{Θ} (X)$ . Thus $S \bar{Θ} (X) \subseteq \bar{Θ} (X)$ . Then $\bar{Θ} (X)$ is a left ideal of S. Similarly, we can prove that $\bar{Θ} (X)$ is a right ideal of S. Therefore $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation ideal.□

Theorem 5.7. Let f be an isomorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type PCR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation ideal if and only if $\underline{Ψ} (f (X))$ is a $C S_{Ψ} (T)$ -lower approximation ideal.

Proof. By Proposition 5.1 (4) and using the similar method in the proof of Theorem 5.6, we can prove that the statement holds. □

The following corollary is an immediate consequence of Theorems 5.6 and 5.7.

Corollary 5.8. Let f be an isomorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type PCR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then Θ (X) is a $C S_{Θ} (S)$ -rough ideal if and only if Ψ (f (X)) is a $C S_{Ψ} (T)$ -rough ideal.

Theorem 5.9. Let f be an epimorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type CR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation completely prime ideal if and only if $\bar{Ψ} (f (X))$ is an $C S_{Ψ} (T)$ -upper approximation completely prime ideal.

Proof. Assume that $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation completely prime ideal. Let t₁, t₂ ∈ T be such that $t_{1} t_{2} \in \bar{Ψ} (f (X))$ . Thus there exist s₁, s₂ ∈ S such that f (s₁) = t₁ and f (s₂) = t₂. Hence we have CS_Ψ (f (s₁) f (s₂))∩ f (X) ≠ ∅. Since Ψ is complete, we have $(C S_{ψ} (f (s_{1}))) (C S_{ψ} (f (s_{2}))) \cap f (X) \neq \emptyset .$

Then there exist f (s₃) ∈ CS_Ψ (f (s₁)) and f (s₄) ∈ CS_Ψ (f (s₂)) such that f (s₃) f (s₄) ∈ f (X), and so f (s₃s₄) ∈ f (X). Then there exists s₅ ∈ X such that f (s₃s₄) = f (s₅). By Proposition 5.1 (1), we obtain that s₃ ∈ CS_Θ (s₁) and s₄ ∈ CS_Θ (s₂). From Proposition 4.2 and Proposition 5.1 (5), it follows that s₃s₄ ∈ CS_Θ (s₁s₂). By Proposition 3.4 (2), we obtain that CS_Θ (s₁s₂) = CS_Θ (s₃s₄). Note that f (s₃s₄) ∈ CS_Ψ (f (s₃s₄)) since Proposition 3.4 (1). Then f (s₅) ∈ CS_Ψ (f (s₃s₄)). By Proposition 5.1 (1), once again, we get that s₅ ∈ CS_Θ (s₃s₄) = CS_Θ (s₁s₂). Thus CS_Θ (s₁s₂)∩ X ≠ ∅, and so $s_{1} s_{2} \in \bar{Θ} (X)$ . Since $\bar{Θ} (X)$ is a completely prime ideal of S, we have $s_{1} \in \bar{Θ} (X)$ or $s_{2} \in \bar{Θ} (X)$ . Hence we have $f (s_{1}) \in f (\bar{Θ} (X))$ or $f (s_{2}) \in f (\bar{Θ} (X))$ . From Proposition 5.1 (2), we get $f (s_{1}) \in \bar{Ψ} (f (X))$ or $f (s_{2}) \in \bar{Ψ} (f (X))$ , which yields $t_{1} \in \bar{Ψ} (f (X))$ or $t_{2} \in \bar{Ψ} (f (X))$ . Thus $\bar{Ψ} (f (X))$ is a completely prime ideal of T. Therefore $\bar{Ψ} (f (X))$ is an $C S_{Ψ} (T)$ -upper approximation completely prime ideal.

Conversely, we suppose that $\bar{Ψ} (f (X))$ is an $C S_{Θ} (S)$ -upper approximation completely prime ideal. Let s₆, s₇ be two elements in S such that $s_{6} s_{7} \in \bar{Θ} (X)$ . Then $f (s_{6} s_{7}) \in f (\bar{Θ} (X))$ . By Proposition 5.1 (2), we obtain that $f (s_{6}) f (s_{7}) = f (s_{6} s_{7}) \in f (\bar{Θ} (X)) = \bar{Ψ} (f (X))$ .

Thus $f (s_{6}) \in \bar{Ψ} (f (X))$ or $f (s_{7}) \in \bar{Ψ} (f (X))$ . Now, we consider the following two cases.

Case 1. If $f (s_{6}) \in \bar{Ψ} (f (X))$ , then we have that $f (s_{6}) \in f (\bar{Θ} (X))$ since Proposition 5.1 (2). Thus there exists $s_{8} \in \bar{Θ} (X)$ such that f (s₆) = f (s₈). Whence CS_Θ (s₈)∩ X ≠ ∅. By Proposition 3.4 (1), we obtain that f (s₈) ∈ CS_Ψ (f (s₈)). Thus we get f (s₆) ∈ CS_Ψ (f (s₈)). By Proposition 5.1 (1), we have s₆ ∈ CS_Θ (s₈). From Proposition 3.4 (2), it follows that CS_Θ (s₆) = CS_Θ (s₈). Thus we have CS_Θ (s₆)∩ X ≠ ∅, and so $s_{6} \in \bar{Θ} (X)$ .

Case 2. If $f (s_{7}) \in \bar{Ψ} (f (X))$ , then $s_{7} \in \bar{Θ} (X)$ since the proof is similar to that the first case.

As a consequence, $\bar{Θ} (X)$ is an $C S_{Θ} (S)$ -upper approximation completely prime ideal. □

Theorem 5.10. Let f be an isomorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type CR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then $\underline{Θ} (X)$ is a $C S_{Θ} (S)$ -lower approximation completely prime ideal if and only if $\underline{Ψ} (f (X))$ is a $C S_{Ψ} (T)$ -lower approximation completely prime ideal.

Proof. By Proposition 5.1 (4) and using the similar method in the proof of Theorem 5.9, we can prove that the statement holds. □

The following corollary is an immediate consequence of Theorems 5.9 and 5.10.

Corollary 5.11 Let f be an isomorphism from S in $(S, C S_{Θ} (S))$ to T in $(T, C S_{Ψ} (T))$ type CR, where the binary relation Θ : = {(s₁, s₂) ∈ S × S : (f (s₁), f (s₂)) ∈ Ψ}. If X is a non-empty subset of S, then Θ (X) is a $C S_{Θ} (S)$ -rough completely prime ideal if and only if Ψ (f (X)) is a $C S_{Ψ} (T)$ -rough completely prime ideal.

6. Discussion

In this section we discuss to the important relationships of this work and approximation processing models in [1 , 29].

In general, notions of generalizations of rough sets in approximation spaces have been developed as the following the diagram.

Pawlak′sroughsets [1] → Mareay′sroughsets [29] → Ourroughsets

The Mareay’s rough set is a generalization of the Pawlak’s rough set if the equivalence property of a relation is contained in the approximation processing model of Mareay’s rough set theory. Our rough set is a generalization of the Mareay’s rough set if our rough set is considered under the single universe.

In a semigroup, we discuss to main results of this work (Sections 4 and 5), Kuroki [11], Xiao and Zhang [12], and Wang and Zhan [14] by using Tables 4, 5 and 6 below.

Table 4
The results of upper approximations in semigroups

	Model in [11]	Model in [12]	Model in [14]	Our Model
UAS (SC)	✓		✓	✓
UAI (SC)	✓		✓	✓
UAC (SC)		✓	✓	✓
UAS (HP)				✓
UAI (HP)		✓		✓
UAC (HP)		✓		✓

Table 5

The results of lower approximations in semigroups

	Model in [11]	Model in [12]	Model in [14]	Our Model
LAS (SC)	✓		✓	✓
LAI (SC)	✓		✓	✓
LAC (SC)		✓	✓	✓
LAS (HP)				✓
LAI (HP)		✓		✓
LAC (HP)		✓		✓

Table 6

The results of rough sets in semigroups

	Model in [11]	Model in [12]	Model in [14]	Our Model
RS (SC)				✓
RI (SC)				✓
RC (SC)				✓
RS (HP)				✓
RI (HP)				✓
RC (HP)				✓

In the following Tables 4, 5 and 6, the symbol ✓ denotes two arguments as the following.

The sufficient condition (briefly, SC) of an upper approximation semigroup (briefly, UAS) (resp. a lower approximation semigroup (briefly, LAS) and a rough semigroup (briefly, RS)) is provided in [11 , 14], or this paper. Similarly, if sufficient conditions of an upper approximation ideal (briefly, UAI) (resp. a lower approximation ideal (briefly, LAI) and a rough ideal (briefly, RI)) and an upper approximation completely prime ideal (briefly, UAC) (resp. a lower approximation completely prime ideal (briefly, LAC) and a rough completely prime ideal (briefly, RC)) are provided.

The relationship between the UAS (resp. LAS and RS) and the homomorphic image of the UAS (resp. LAS and RS) is demonstrated under homomorphism problems (briefly, HP) in [11 , 14], or this paper. Similarly, if UAI (resp. LAI and RI) and UAC (resp. LAC and RC) are examined under HP.

From Tables 4, 5 and 6, we observe that sufficient conditions are totally obtained in this work. Also, we have relationships under homomorphism problems which are completely examined in this paper.

7. Conclusion

Based on the approximation processing model of Mareay’s rough set theory induced by an arbitrary binary relation on the single universe, a generalization of the Mareay’s rough set was made in an approximation space based on cores of successor classes induced by an arbitrary binary relation between two universes, and a corresponding example was gave. Furthermore, several algebraic properties of the new approximation processing model were verified in detail. Based on a preorder and compatible relation, approximation processings in semigroups were used from the new approximation processing model. Section 6 indicate that sufficient conditions of rough semigroups, rough ideals and rough completely prime ideals are entirely obtained, and relationships under homomorphism problems are perfectly verified. Hence the approximation processing model under the novel generalization can be used in algebraic data fields. However, when we consider other types of semigroups, the corresponding issues need to be further investigated.

Finally, we hope that a new approximation processings model in this paper may provide a powerful tool for assessment and decision problems in pure and applied sciences, information sciences, computer sciences, and new trends of intelligent technology under uncertainty information and knowledge.

Footnotes

Acknowledgement

The authors would like to exhibit their sincere thanks to the anonymous referees for their considerable ideas. This work was supported by a grant from the Faculty of Science and Technology, Nakhon Sawan Rajabhat University of Nakhon Sawan Province and the Research Center for Academic Excellence in Mathematics, Faculty of Science, Naresuan University of Phitsanulok Province in Thailand.

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Generalizations of rough sets induced by binary relations approach in semigroups

Abstract

Keywords

1. Introduction

2. Preliminaries

3. Generalizations of rough sets induced by binary relations

Table 1 The information table of damage values

Table 2 The multiplication table on S

6. Discussion

Table 4 The results of upper approximations in semigroups

Footnotes

Acknowledgement

References

Table 1
The information table of damage values

Table 2
The multiplication table on S

Table 4
The results of upper approximations in semigroups