Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2017: 0.32

ICV 2017: 161.82

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 16, Issue 1

Issues

Volume 13 (2015)

On rough sets induced by fuzzy relations approach in semigroups

Rukchart Prasertpong / Manoj Siripitukdet
  • Corresponding author
  • Department of Mathematics, Faculty of Science, Naresuan University, Mueang, Phitsanulok, 65000, Thailand
  • Research Center for Academic Excellence in Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/math-2018-0136

Abstract

In this paper, we introduce a rough set in a universal set based on cores of successor classes with respect to level in a closed unit interval under a fuzzy relation, and some interesting properties are investigated. Based on this point, we propose a rough completely prime ideal in a semigroup structure under a compatible preorder fuzzy relation, including the rough semigroup and rough ideal. Then we provide sufficient conditions for them. Finally, the relationships between rough completely prime ideals (rough semigroups and rough ideals) and their homomorphic images are verified.

Keywords: Rough sets; Semigroups; Rough semigroups; Rough ideals; Rough completely prime ideals; Fuzzy relations; Compatible preorder fuzzy relations

MSC 2010: 20M12; 20M99

1 Introduction

The Pawlak’s rough set theory is a classical tool for assessing the problems and decision problems in many fields with respect to informations and technology. This theory was introduced by Pawlak [1] in 1982. He proposed the concept of Pawlak’s rough sets in universal sets based on equivalence classes induced by equivalence relations. For an equivalence relation on a universal set and a non-empty subset of the universal set, the Pawlak’s rough set of the non-empty subset is given by mean of a pair of the Pawlak’s upper approximation and the Pawlak’s lower approximation where the difference between the Pawlak’s upper approximation and the Pawlak’s lower approximation (The Pawlak’s boundary region) is a non-empty set. The Pawlak’s upper approximation is the union of all the equivalence classes which have a non-empty intersection with the non-empty subset. The Pawlak’s lower approximation is the union of all the equivalence classes which are subset of the non-empty subset. As mentioned above, the Pawlak’s rough set model is defined as a mathematical tool with respect to assessments of decisions. This assessment model is an important tool for dealing with algebraic systems [2,3,4,5, 6,7,8,9,10,11,12,13,14], information sciences [15] and computer sciences [16] etc.

From Pawlak’s rough sets induced by equivalence relations, the generalized Pawlak’s rough sets using arbitrary binary relations (briefly, binary relations) were introduced by many researchers. In 1998, Yao [17] introduced roughness models using successor neighborhoods induced by binary relations [SNθ(u) := {u′U : (u, u′) ∈ θ} denotes a successor neighborhood of u induced by a binary relation θ on a universal set U where u is an element in U]. In 2016, Mareay [18] introduced rough sets using cores of successor neighborhoods induced by binary relations [CSNθ(u) := {u′U : SNθ(u) = SNθ(u′)} denotes a core of a successor neighborhood of u induced by a binary relation θ on a universal set U where u is an element in U]. If a binary relation on a universal set is an equivalence relation, then the Yao’s rough set and the Mareay’s rough set are generalizations of the Pawlak’s rough set.

The classical fuzzy set theory was introduced by Zadeh [19] in 1965. Based on this point, Zadeh [20, 21] introduced the concept of fuzzy relations in 1971 which it is researched by many researchers in several fields, such as information sciences [22] and decision systems [23] etc.

The semigroup structure (see [24]) is an algebraic system with respect to wide applications, especially the, notions of Pawlak’s rough sets in semigroups. For combinations of Pawlak’s rough set theory and semigroup theory, Kuroki [4] proposed the notion of rough ideals in semigroups based on congruence classes induced by congruence relations (equivalence relations and compatible relations) in 1997. Thereafter, Xiao and Zhang [7] proposed the notion of rough completely prime ideals in semigroups based on congruence classes induced by congruence relations in 2006. For the combination of Pawlak’s rough set theory, fuzzy set theory and semigroup theory, Wang and Zhan [13] introduced the concept of rough semigroups based on congruence relations with respect to fuzzy ideals of semigroups in 2016.

From an interesting idea about generalized rough set models in the sense of Mareay [18], and after providing some preliminaries about some important definitions of fuzzy relations and semigroups in Section 2, we introduce a rough set in a universal set based on cores of successor classes with respect to level in a closed unit interval under a fuzzy relation, and we verify some interesting properties in Section 3. In Section 4, we introduce a rough completely prime ideal in a semigroup structure under a compatible preorder fuzzy relation, including the rough semigroup and rough ideal. Then we provide sufficient conditions for them. In Section 5, we investigate the relationships between rough completely prime ideals (rough semigroups and rough ideals) and their homomorphic images. Finally, we give a conclusion of the work in Section 6.

2 Preliminaries

In this section, we review some important definitions which will be necessary in the subsequent sections. Throughout this paper, U and V denote two non-empty universal sets.

Definition 2.1

[19] A fuzzy set of U is defined as a function from U to the closed unit interval [0, 1].

Definition 2.2

[22] Let 𝓕(U × V) be a family of all fuzzy sets of U × V. An element in 𝓕(U × V) is referred to as a fuzzy relation from U to V. An element in 𝓕(U × V) is called a fuzzy relation on U if U = V. For a fuzzy relation Θ ∈ 𝓕(U × V) and elements uU, vV, the value of Θ(u, v) in [0, 1] representing the membership grade of relation between u and v under Θ. If Θ ∈ 𝓕(U × V) where U := {u1, u2, u3, …, um} and V := {v1, v2, v3, …, vn}, then the fuzzy relation Θ is represented by the matrix as

Θ(u1,v1)Θ(u1,v2)Θ(u1,v3)Θ(u1,vn)Θ(u2,v1)Θ(u2,v2)Θ(u2,v3)Θ(u2,vn)Θ(u3,v1)Θ(u3,v2)Θ(u3,v3)Θ(u3,vn)Θ(um,v1)Θ(um,v2)Θ(um,v3)Θ(um,vn).

Definition 2.3

[22] Let Θ be a fuzzy relation from U to V. Θ is called serial if for all uU, there exists vV such that Θ(u, v) = 1.

Definition 2.4

[22] Let Θ be a fuzzy relation on U.

  1. Θ is called reflexive if for all uU, Θ(u, u) = 1,

  2. Θ is called symmetric if for all u1, u2U, Θ(u1, u2) = Θ(u2, u1),

  3. Θ is called transitive if for all u1, u2U, Θ(u1, u2) ≥ ∨u3U (Θ(u1, u3) ∧ Θ(u3, u2)),

  4. Θ is called a similarity fuzzy relation if it is reflexive, symmetric and transitive.

A semigroup [24] (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 [25] of S if XXX. A non-empty subset X of S is called a left (right) ideal [25] of S if SXX (XSX), and if it is both a left ideal and a right ideal of S, then it is called an ideal [25]. An ideal X of S is called a completely prime ideal [25] of S if for all s1, s2S, s1s2X implies s1X or s2X.

Definition 2.5

[25] Let Θ be a fuzzy relations on S. Θ is called compatible if for all s1, s2, s3S,

Θ(s1s3,s2s3)Θ(s1,s2)andΘ(s3s1,s3s2)Θ(s1,s2).

3 Rough sets induced by fuzzy relations

In this section, we construct rough sets induced by fuzzy relations. Then we give the real-world example and some interesting properties.

Definition 3.1

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. For an element uU,

SΘ(u;ι):={vV:Θ(u,v)ι}

is called a successor class of u with respect to ι-level under Θ.

Remark 3.2

Let ι ∈ [0, 1]. If Θ is a serial fuzzy relation from U to V, then SΘ(u; ι) ≠ ∅ for all uU.

Definition 3.3

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. For an element u1U,

CSΘ(u1;ι):={u2U:SΘ(u1;ι)=SΘ(u2;ι)}

is called a core of the successor class of u1 with respect to ι-level under Θ.

We denote by 𝓒𝓢Θ(U; ι) the collection of CSΘ(u; ι) for all uU.

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

Proposition 3.4

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. Then the following statements hold.

  1. For all uU, uCSΘ(u; ι).

  2. For all u1, u2U, u2CSΘ(u1; ι) if and only if CSΘ(u1; ι) = CSΘ(u2; ι).

The following remark is an immediate consequence of Proposition 3.4.

Remark 3.5

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. Then 𝓒𝓢Θ(U; ι) is the partition of U.

Proposition 3.6

Let ι ∈ [0, 1] and let Θ be a fuzzy relation on U. Then we have the following statements.

  1. If Θ is reflexive, then CSΘ(u; ι) ⊆ SΘ(u; ι) for all uU.

  2. If Θ is a similarity fuzzy relation, then SΘ(u; ι) and CSΘ(u; ι) are identical classes for all uU.

Proof

The proof is straightforward, so we omit it. □

In the following, we give the concept of rough sets induced by fuzzy relations.

Definition 3.7

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. A triple (U, V, 𝓒𝓢Θ(U; ι)) is called an approximation space based on 𝓒𝓢Θ(U; ι) (briefly, 𝓒𝓢Θ(U; ι)-approximation space). If U = V, then (U, V, 𝓒𝓢Θ(U; ι)) is replaced by a pair (U, 𝓒𝓢Θ(U; ι)).

Definition 3.8

Let (U, V, 𝓒𝓢Θ(U; ι)) be an 𝓒𝓢Θ(U; ι)-approximation space. For a non-empty subset X of U, we define three sets as follows:

Θ(X; ι) := ⋃uU{CSΘ(u; ι) : CSΘ(u; ι) ∩ X ≠ ∅},

Θ(X; ι) := ⋃uU{CSΘ(u; ι) : CSΘ(u; ι) ⊆ X} and

Θbnd(X; ι) := Θ(X; ι) − Θ(X; ι).

Then

  1. Θ(X; ι) is called an upper approximation of X in (U, V, 𝓒𝓢Θ(U; ι))

    (briefly, 𝓒𝓢Θ(U; ι)-upper approximation of X).

  2. Θ(X; ι) is called a lower approximation of X in (U, V, 𝓒𝓢Θ(U; ι))

    (briefly, 𝓒𝓢Θ(U; ι)-lower approximation of X).

  3. Θbnd(X; ι) is called a boundary region of X in (U, V, 𝓒𝓢Θ(U; ι))

    (briefly, 𝓒𝓢Θ(U; ι)-boundary region of X).

  4. If Θbnd(X; ι) ≠ ∅, then Θ (X; ι) := (Θ(X; ι), Θ(X; ι)) is called a rough set of X in (U, V, 𝓒𝓢Θ(U; ι))

    (briefly, 𝓒𝓢Θ(U; ι)-rough set of X).

  5. If Θbnd(X; ι) = ∅, then X is called a definable set in (U, V, 𝓒𝓢Θ(U; ι))

    (briefly, 𝓒𝓢Θ(U; ι)-definable set).

According to Definition 3.8, it is easy to prove that

Θ(X; ι) := {uU : CSΘ(u; ι) ∩ X ≠ ∅} and

Θ(X; ι) := {uU : CSΘ(u; ι) ⊆ X}.

Here we present an example as the following.

Example 3.9

Let U = {u1, u2, u3, u4, u5} be a set of doctoral students in a mathematical business classroom of a university and let V = {v1, v2, v3, v4} be a set of subjects where

v1 is business,

v2 is economics,

v3 is computer sciences and

v4 is mathematics.

For a fuzzy relation Θ ∈ 𝓕(U × V) and elements uU, vV, the number Θ(u, v) in the closed unit interval [0, 1] is defined as the score of the doctoral student u with respect to the subject v under Θ. The scores of all doctoral students in U with respect to subjects in V under Θ are given as the following matrix.

0.70.90.80.90.80.90.70.90.90.80.80.90.50.50.90.90.90.90.60.9

Let ι = 0.9 be a minimal score level. If an educational measurement committee assign X := {u2, u3, u5} which is a set of excellent doctoral students under the global evaluation, then the assessment of X in an 𝓒𝓢Θ(U; 0.9)-approximation space (U, V, 𝓒𝓢Θ(U; 0.9)) is derived by the process as the following.

According to Definition 3.1, it follows that

SΘ(u1; 0.9) := {v2, v4},

SΘ(u2; 0.9) := {v2, v4},

SΘ(u3; 0.9) := {v1, v4},

SΘ(u4; 0.9) := {v3, v4} and

SΘ(u5; 0.9) := {v1, v2, v4}.

According to Definition 3.3, it follows that

CSΘ(u1; 0.9) := {u1, u2},

CSΘ(u2; 0.9) := {u1, u2},

CSΘ(u3; 0.9) := {u3},

CSΘ(u4; 0.9) := {u4} and

CSΘ(u5; 0.9) := {u5}.

According to Definition 3.8, it follows that

Θ(X; 0.9) := {u1, u2, u3, u5},

Θ(X; 0.9) := {u3, u5} and

Θbnd(X; 0.9) := {u1, u2}.

Therefore Θ R̦(X; 0.9) := ({u1, u2, u3, u5}, {u3, u5}) is a 𝓒𝓢Θ(U; 0.9)-rough set of X. Consequently,

  1. u1, u2, u3 and u5 are possibly excellent doctoral students,

  2. u3 and u5 are certainly excellent doctoral students and

  3. for u1 and u2 it cannot be determined whether two students are excellent doctoral students or not.

In what follows, Definition 3.10 follows from the example as the union of upper and lower approximations.

Definition 3.10

Let (U, V, 𝓒𝓢Θ(U; ι)) be an 𝓒𝓢Θ(U; ι)-approximation space and let X be a non-empty subset of U. Θ(X; ι) is called a non-empty 𝓒𝓢Θ(U; ι)-upper approximation of X in (U, V, 𝓒𝓢Θ(U; ι)) if Θ(X; ι) is a non-empty subset of U. Similarly, we can define a non-empty 𝓒𝓢Θ(U; ι)-lower approximation. Θ R̦(X; ι) is referred to as a non-empty 𝓒𝓢Θ(U; ι)-rough set in (U, V, 𝓒𝓢Θ(U; ι)) if Θ(X; ι) is a non-empty 𝓒𝓢Θ(U; ι)-upper approximation and Θ(X; ι) is a non-empty 𝓒𝓢Θ(U; ι)-lower approximation.

Proposition 3.11

Let (U, V, 𝓒𝓢Θ(U; ι)) be an 𝓒𝓢Θ(U; ι)-approximation space. If X and Y are non-empty subsets of U, then we have the following statements.

  1. Θ(U; ι) = U and

    Θ(U; ι) = U.

  2. Θ(∅; ι) = ∅ and

    Θ(∅; ι) = ∅.

  3. XΘ(X; ι) and

    Θ(X; ι) ⊆ X.

  4. Θ(XY; ι) = Θ(X; ι) ∪ Θ(Y; ι) and

    Θ(XY; ι) = Θ(X; ι) ∩ Θ(Y; ι).

  5. Θ(XY; ι) ⊆ Θ(X; ι) ∩ Θ(Y; ι) and

    Θ(XY; ι) ⊇ Θ(X; ι) ∪ Θ(Y; ι).

  6. Θ(Xc; ι) = (Θ(X; ι))c, where Xc and (Θ(X; ι))c are complements of X and Θ(X; ι), respectively.

  7. Θ(Θ(X; ι); ι) = Θ(X; ι) and

    Θ(Θ(X; ι); ι) = Θ(X; ι).

  8. Θ((Θ(X; ι))c; ι) = (Θ(X; ι))c, where (Θ(X; ι))c is a complement of Θ(X; ι) and

    Θ((Θ(X; ι))c; ι) = (Θ(X; ι))c, where (Θ(X; ι))c is a complement of Θ(X; ι).

  9. If XY, then Θ(X; ι) ⊆ Θ(Y; ι) and Θ(X; ι) ⊆ Θ(Y; ι).

Proof

The proof is straightforward, so we omit it. □

Definition 3.12

Let (U, V, 𝓒𝓢Θ(U; ι)) be an 𝓒𝓢Θ(U; ι)-approximation space and let X be a non-empty subset of U. If Θ(X; ι) is a non-empty 𝓒𝓢Θ(U; ι)-lower approximation of X in (U, V, 𝓒𝓢Θ(U; ι)) and Θ(X; ι) is a proper subset of X, then X is called a set over a non-empty interior set.

Proposition 3.13

Let (U, V, 𝓒𝓢Θ(U; ι)) be an 𝓒𝓢Θ(U; ι)-approximation space and let X be a non-empty subset of U. If X is a set over non-empty interior set, then Θ R̦(X; ι) is a non-empty 𝓒𝓢Θ(U; ι)-rough set of X in (U, V, 𝓒𝓢Θ(U; ι)).

Proof

Suppose that X is a set over a non-empty interior set. Then we have that Θ(X; ι) is a non-empty 𝓒𝓢Θ(U; ι)-lower approximation and Θ(X; ι) ⊂ X. By Proposition 3.11 (3), we obtain that ∅ ≠ XΘ(X; ι). Thus we get Θ(X; ι) is a non-empty 𝓒𝓢Θ(U; ι)-upper approximation. We shall verify that Θbnd(X; ι) ≠ ∅. Suppose that Θbnd(X; ι) = ∅. Then we have Θ(X; ι) = Θ(X; ι). From Proposition 3.11 (3), once again, it follows that Θ(X; ι) = X, a contradiction. Therefore Θbnd(X; ι) ≠ ∅. Consequently, Θ(X; ι) is a non-empty 𝓒𝓢Θ(U; ι)-rough set of X. □

Example 3.14

Let U := {u1 = 3, u2 = 1, u3=13,u4=19,u5=127 } and V := {v1 = 2, v2 = 2 3 , v3 = 6, v4 = 6 3 }. Define a fuzzy relation Θ ∈ 𝓕(U × V) by

Θ(u,v)=cosuvifuv1sinuvifu<v

for all (u, v) ∈ U × V. Then we have the following ranges of Θ.

0.994520.819610.690980.482320.965100.939580.895470.819610.988360.979850.965100.939580.996120.993280.988360.979850.998710.997760.996120.99328

Let ι = 0.95 and let X := {u2, u3} be a non-empty subset of U. According to Definition 3.1, it follows that

SΘ(u1; 0.95) := {v1},

SΘ(u2; 0.95) := {v1},

SΘ(u3; 0.95) := {v1, v2, v3},

SΘ(u4; 0.95) := {v1, v2, v3, v4} and

SΘ(u5; 0.95) := {v1, v2, v3, v4}.

According to Definition 3.3, it follows that

CSΘ(u1; 0.95) := {u1, u2},

CSΘ(u2; 0.95) := {u1, u2},

CSΘ(u3; 0.95) := {u3},

CSΘ(u4; 0.95) := {u4, u5} and

CSΘ(u5; 0.95) := {u4, u5}.

Here it is easy to check that Θ(X; 0.95) is a non-empty 𝓒𝓢Θ(U; 95)-lower approximation of X, and also Θ(X; 0.95) ⊂ X. Note that XΘ(X; 0.95). Thus we get Θ(X; 0.95) ≠ ∅ and Θ(X; 0.95) ≠ Θ(X; 0.95). It follows that Θ R̦(X; 0.95) is a non-empty 𝓒𝓢Θ(U; 0.95)-rough set of X.

Proposition 3.15

Let (U, 𝓒𝓢Θ(U; ι)) be an 𝓒𝓢Θ(U; ι)-approximation space and let (U, 𝓒𝓢Ψ(U; κ)) be an 𝓒𝓢Ψ(U; κ)-approximation space. If ικ and ΘΨ where Θ is reflexive and Ψ is transitive, then we have Θ(X; ι) ⊆ Ψ(X; κ) for every non-empty subset X of U.

Proof

Let X be a non-empty subset of U. Then we prove that Θ(X; ι) ⊆ Ψ(X; κ). In fact, let u1Θ(X; ι). Then CSΘ(u1; ι) ∩ X ≠ ∅. Thus there exists u2CSΘ(u1; ι) ∩ X, and so SΘ(u1; ι) = SΘ(u2; ι). Since Θ is reflexive, we have Θ(u2, u2) = 1 ≥ ι. Whence u2SΘ(u2; ι) = SΘ(u1; ι). Thus we have Θ(u1, u2) ≥ ι. Since ικ and ΘΨ, we have Ψ(u1, u2) ≥ Θ(u1, u2) ≥ κ, and so Ψ(u1, u2) ≥ κ. Similary, we have Ψ(u2, u1) ≥ κ. We shall verify that SΨ(u1; κ) = SΨ(u2; κ). Now, let u3SΨ(u2;κ). Then Ψ(u2, u3) ≥ κ. Since Ψ is transitive, we have

Ψ(u1,u3)u4U(Ψ(u1,u4)Ψ(u4,u3))Ψ(u1,u2)Ψ(u2,u3)κκ=κ.

Hence Ψ(u1, u3) ≥ κ. Thus u3SΨ(u1; κ), which yields SΨ(u2; κ) ⊆ SΨ(u1; κ). Similary, we can prove that SΨ(u1; κ) ⊆ SΨ(u2; κ). Whence we get SΨ(u1; κ) = SΨ(u2; κ), and so u2CSΨ(u1; κ). Thus we have that u2CSΨ(u1; κ) ∩ X. Hence CSΨ(u1; κ) ∩ X ≠ ∅, which yields u1Ψ(X; κ). Therefore we get that Θ(X; ι) ⊆ Ψ(X; κ). □

Proposition 3.16

Let (U, 𝓒𝓢Θ(U; ι)) be an 𝓒𝓢Θ(U; ι)-approximation space and let (U, 𝓒𝓢Ψ(U; κ)) be an 𝓒𝓢Ψ(U; κ)-approximation space. If ικ and ΘΨ where Θ is reflexive and Ψ is transitive, then we have Ψ(X; κ) ⊆ Θ(X; ι) for every non-empty subset X of U.

Proof

Let X be a non-empty subset of U. Then we prove that Ψ(X; κ) ⊆ Θ(X; ι). Indeed, let u1Ψ(X; κ). Then CSΨ(u1; ι) ⊆ X. We shall show that CSΘ(u1; ι) ⊆ CSΨ(u1; κ). Let u2CSΘ(u1; ι). Then we have SΘ(u1; ι) = SΘ(u2; ι). Since Θ is reflexive, we have that Θ(u1, u1) = 1 ≥ ι. Hence u1SΘ(u1; ι), and so u1SΘ(u2; ι). Thus Θ(u2, u1) ≥ ι. By the assumption, we have Ψ(u2, u1) ≥ Θ(u2, u1) ≥ κ, and so Ψ(u2, u1) ≥ κ. Similary, we get that Ψ(u1, u2) ≥ κ. We shall prove that SΨ(u1; κ) = SΨ(u2; κ). Let u3SΨ(u2; κ). Then Ψ(u2, u3) ≥ κ. Since Ψ is transitive, we have

Ψ(u1,u3)u4U(Ψ(u1,u4)Ψ(u4,u3))Ψ(u1,u2)Ψ(u2,u3)κκ=κ.

Thus Ψ(u1, u3) ≥ κ, and so u3SΨ(u1; κ). Hence SΨ(u2; κ) ⊆ SΨ(u1; κ). Similary, we can prove that SΨ(u1; κ) ⊆ SΨ(u2; κ), which yields SΨ(u1; κ) = SΨ(u2; κ). Thus we have u2CSΨ(u1; κ), and so CSΘ(u1; ι) ⊆ CSΨ(u1; κ) ⊆ X. Therefore u1Θ(X; ι). This means that Ψ(X; κ) ⊆ Θ(X; ι). □

4 Roughness in semigroups

In this section, we propose the definition of compatible preorder fuzzy relations on semigroups. Then we introduce the roughness in semigroups induced by compatible preorder fuzzy relations. We provide sufficient conditions for them and give some interesting properties and examples.

Definition 4.1

Let Θ be a fuzzy relation on S. Θ is called a compatible preorder fuzzy relation if Θ is reflexive, transitive and compatible. An 𝓒𝓢Θ(S; ι)-approximation space (S, 𝓒𝓢Θ(S; ι)) is called an 𝓒𝓢Θ(S; ι)-approximation space type CPF if Θ is a compatible preorder fuzzy relation.

Proposition 4.2

If (S, 𝓒𝓢Θ(S; ι)) is an 𝓒𝓢Θ(S; ι)-approximation space type CPF, then

(CSΘ(s1;ι))(CSΘ(s2;ι))CSΘ(s1s2;ι)

for all s1, s2S.

Proof

Let s1, s2 be two elements in S and let s3 ∈ (CSΘ(s1; ι)) (CSΘ(s2; ι)). Then there exist s4CSΘ(s1; ι) and s5CSΘ(s2; ι) such that s3 = s4s5. Thus SΘ(s1; ι) = SΘ(s4; ι) and SΘ(s2; ι) = SΘ(s5; ι). Hence we get that SΘ(s1s2; ι) = SΘ(s4s5; ι). Indeed, we suppose that s6SΘ(s4s5; ι). Then we have Θ(s4s5, s6) ≥ ι. Since Θ is reflexive, we have Θ(s4, s4) = Θ(s5, s5) = 1 ≥ ι, and so s4SΘ(s4; ι) and s5SΘ(s5; ι). Whence s4SΘ(s1; ι) and s5SΘ(s2; ι). Thus Θ(s1, s4) ≥ ι and Θ(s2, s5) ≥ ι. Since Θ is transitive and compatible, we have

Θ(s1s2,s4s5)s7S(Θ(s1s2,s7)Θ(s7,s4s5))Θ(s1s2,s4s2)Θ(s4s2,s4s5)Θ(s1,s4)Θ(s2,s5)ιι=ι.

Hence Θ(s1s2, s4s5) ≥ ι. Since Θ is transitive, we have

Θ(s1s2,s6)s8S(Θ(s1s2,s8)Θ(s8,s6))Θ(s1s2,s4s5)Θ(s4s5,s6)ιι=ι.

Thus Θ(s1s2, s6) ≥ ι, and so s6SΘ(s1s2; ι). Hence SΘ(s4s5; ι) ⊆ SΘ(s1s2; ι). Similarly, we can show that SΘ(s1s2; ι) ⊆ SΘ(s4s5; ι). Thus SΘ(s1s2; ι) = SΘ(s4s5; ι), which yields s3CSΘ(s1s2; ι). This implies that (CSΘ(s1; ι)) (CSΘ(s2; ι)) ⊆ CSΘ(s1s2; ι). □

In the following, we give an example to illustrate that the property in Proposition 4.2 is indispensable.

Example 4.3

Let S := {s1, s2, s3, s4, s5} be a semigroup with multiplication rules defined by Table 1.

Table 1

The multiplication table on S

Define the membership grades of relationship between any two elements in S under the fuzzy relation Θ on S as the following.

1010101000001010001000101

Then it is easy to check that Θ is a compatible preorder fuzzy relation. For ι = 0.9, successor classes of each elements in S with respect to 0.9-level under Θ are

SΘ(s1; 0.9) := {s1, s3, s5},

SΘ(s2; 0.9) := {s2},

SΘ(s3; 0.9) := {s3, s5},

SΘ(s4; 0.9) := {s4} and

SΘ(s5; 0.9) := {s3, s5}.

Hence cores of successor classes of each elements in S with respect to 0.9-level under Θ are

CSΘ(s1; 0.9) := {s1},

CSΘ(s2; 0.9) := {s2},

CSΘ(s3; 0.9) := {s3, s5},

CSΘ(s4; 0.9) := {s4} and

CSΘ(s5; 0.9) := {s3, s5}.

Here it is straightforward to verify that (CSΘ(s; 0.9))(CSΘ(s′; 0.9)) ⊆ CSΘ(ss′; 0.9) for all s, s′S.

Observe that, in Example 4.3, it does not hold in general for the equality case. Now, we consider the following example.

Example 4.4

Let S := {s1, s2, s3, s4, s5} be a semigroup with multiplication rules defined by Table 2.

Table 2

The multiplication table on S

Define the membership grades of relationship between any two elements in S under the fuzzy relation Θ on S as the following.

1000101110011100111000001

Then it is easy to check that Θ is a compatible preorder fuzzy relation. For ι = 0.9, successor classes of each elements in S with respect to 0.9-level under Θ are

SΘ(s1; 0.9) := {s1, s5},

SΘ(s2; 0.9) := {s2, s3, s4},

SΘ(s3; 0.9) := {s2, s3, s4},

SΘ(s4; 0.9) := {s2, s3, s4} and

SΘ(s5; 0.9) := {s5}.

Hence cores of successor classes of each elements in S with respect to 0.9-level under Θ are

CSΘ(s1; 0.9) := {s1},

CSΘ(s2; 0.9) := {s2, s3, s4},

CSΘ(s3; 0.9) := {s2, s3, s4},

CSΘ(s4; 0.9) := {s2, s3, s4} and

CSΘ(s5; 0.9) := {s5}.

Here it is straightforward to check that (CSΘ(s; 0.9))(CSΘ(s′; 0.9)) = CSΘ(ss′; 0.9) for all s, s′S. Based on this point, the property can be considered as a special case of Proposition 4.2. This example leads to the following definition.

Definition 4.5

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CPF. The collection 𝓒𝓢Θ(S; ι) is called complete induced by Θ (briefly, Θ-complete) if for all s1, s2S,

(CSΘ(s1;ι))(CSΘ(s2;ι))=CSΘ(s1s2;ι).

Definition 4.6

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CPF. If 𝓒𝓢Θ(S; ι) is complete induced by Θ, then Θ is called a complete fuzzy relation. (S, 𝓒𝓢Θ(S; ι)) is called an 𝓒𝓢Θ(S; ι)-approximation space type CF if Θ is complete.

Proposition 4.7

If (S, 𝓒𝓢Θ(S; ι)) is an 𝓒𝓢Θ(S; ι)-approximation space type CPF, then

(Θ¯(X;ι))(Θ¯(Y;ι))Θ¯(XY;ι),

for every non-empty subsets X, Y of S.

Proof

Let X and Y be two non-empty subsets of S. Suppose that s1 ∈ (Θ(X; ι))(Θ(Y; ι)). Then there exist s2Θ(X; ι) and s3Θ(Y; ι) such that s1 = s2s3. Thus we have that CSΘ(s2; ι) ∩ X ≠ ∅ and CSΘ(s3; ι) ∩ Y ≠ ∅. Then there exist s4, s5S such that s4CSΘ(s2; ι) ∩ X and s5CSΘ(s3; ι) ∩ Y. From Proposition 4.2, it follows that s4s5 ∈ (CSΘ(s2; ι))(CSΘ(s3; ι)) ⊆ CSΘ(s2s3; ι) and s4s5XY. Thus CSΘ(s2s3; ι) ∩ XY ≠ ∅, which yields s1 = s2s3Θ(XY; ι). Therefore (Θ(X; ι))( Θ(Y; ι)) ⊆ Θ(XY; ι). □

Proposition 4.8

If (S, 𝓒𝓢Θ(S; ι)) is an 𝓒𝓢Θ(S; ι)-approximation space type CF, then

(Θ_(X;ι))(Θ_(Y;ι))Θ_(XY;ι),

for every non-empty subsets X, Y of S.

Proof

Let X and Y be two non-empty subsets of S and let s1 ∈ (Θ(X; ι))(Θ(Y; ι)). Then there exist s2Θ(X; ι) and s3Θ(Y; ι) such that s1 = s2s3, and so CSΘ(s2; ι) ⊆ X and CSΘ(s3; ι) ⊆ Y. Since Θ is complete, we get CSΘ(s2s3; ι) = CSΘ(s2; ι)CSΘ(s3; ι) ⊆ XY. Thus CSΘ(s2s3; ι) ⊆ XY. Hence s1 = s2s3Θ(XY; ι). Therefore (Θ(X; ι))(Θ(Y; ι)) ⊆ Θ(XY; ι). □

We consider the following example.

Example 4.9

According to Example 4.4, suppose that X := {s1, s4, s5} is a subset of S. Then we have Θ(X; ι) = S and Θ(X; ι) := {s1, s5}. Here it is easy to verify that Θ(X; ι) and Θ(X; ι) are subsemigroups, ideals and completely prime ideals of S. Moreover, we also have Θbnd(X; ι) is a non-empty set. For the existence of subsemigroups, ideals and completely prime ideals of S under compatible preorder fuzzy relations in this example, we give the following definition.

Definition 4.10

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CPF and let X be a non-empty subset of S. A non-empty 𝓒𝓢Θ(S; ι)-upper approximation Θ(X; ι) of X in (S, 𝓒𝓢Θ(S; ι)) is called an 𝓒𝓢Θ(S; ι)-upper approximation semigroup if it is a subsemigroup of S. A non-empty 𝓒𝓢Θ(S; ι)-lower approximation Θ(X; ι) of X in (S, 𝓒𝓢Θ(S; ι)) is called a 𝓒𝓢Θ(S; ι)-lower approximation semigroup if it is a subsemigroup of S. A non-empty 𝓒𝓢Θ(S; ι)-rough set Θ R̦(X; ι) of X in (S, 𝓒𝓢Θ(S; ι)) is called a 𝓒𝓢Θ(S; ι)-rough semigroup if Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation semigroup and Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation semigroup.

Similarly, we can define 𝓒𝓢Θ(S; ι)-rough (completely prime) ideals.

Theorem 4.11

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CPF. If X is a subsemigroup of S, then Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation semigroup.

Proof

Suppose that X is a subsemigroup of S. Then XXX. By Proposition 3.11 (3), we obtain that ∅ ≠ XΘ(X; ι). Hence Θ(X; ι) is a non-empty 𝓒𝓢Θ(S; ι)-upper approximation. From Proposition 3.11 (9), it follows that Θ(XX; ι) ⊆ Θ(X; ι). By Proposition 4.7, we obtain that

(Θ¯(X;ι))(Θ¯(X;ι))Θ¯(XX;ι)Θ¯(X;ι).

Hence Θ(X; ι) is a subsemigroup of S. Thus Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation semigroup. □

Theorem 4.12

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CF. If X is a subsemigroup of S with Θ(X; ι) ≠ ∅, then Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation semigroup.

Proof

Suppose that X is a subsemigroup of S. Then XXX. Obviously, Θ(X; ι) is a non-empty 𝓒𝓢Θ(S; ι)-lower approximation. From Proposition 3.11 (9), it follows that Θ(XX; ι) ⊆ Θ(X; ι). By Proposition 4.8, we obtain that

(Θ_(X;ι))(Θ_(X;ι))Θ_(XX;ι)Θ_(X;ι).

Thus Θ(X; ι) is a subsemigroup of S. Therefore Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation semigroup. □

The following corollary is an immediate consequence of Proposition 3.13, Theorem 4.11 and Theorem 4.12.

Corollary 4.13

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CF. If X is a subsemigroup of S over a non-empty interior set, then Θ R̦(X; ι) is a 𝓒𝓢Θ(S; ι)-rough semigroup.

Observe that, in Corollary 4.13, the converse is not true in general. We present an example as the following.

Example 4.14

According to Example 4.4, suppose that X := {s3, s4, s5} is a subset of S, then we have Θ(X; 0.9) := {s2, s3, s4, s5} and Θ(X; 0.9) := {s5}. Thus we see that Θbnd(X; 0.9) ≠ ∅. Hence it is straightforward to check that Θ(X; 0.9) is an 𝓒𝓢Θ(S; 0.9)-upper approximation semigroup and Θ(X; 0.9) is a 𝓒𝓢Θ(S; 0.9)-lower approximation semigroup. However, X is not a subsemigroup of S. Consequently, Θ R̦(X; 0.9) is a 𝓒𝓢Θ(S; 0.9)-rough semigroup, but X is not a subsemigroup of S.

Theorem 4.15

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CPF. If X is an ideal of S, then Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation ideal.

Proof

Suppose that X is an ideal of S. Then SXX. From Proposition 3.11 (9), it follows that Θ(SX; ι) ⊆ Θ(X; ι). By Proposition 3.11 (1), we obtain that Θ(S; ι) = S. From Proposition 4.7, it follows that

S(Θ¯(X;ι))=(Θ¯(S;ι))(Θ¯(X;ι))Θ¯(SX;ι)Θ¯(X;ι).

Hence Θ(X; ι) is a left ideal of S.

Similarly, we can prove that Θ(X; ι) is a right ideal of S. Therefore we have Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation ideal. □

Theorem 4.16

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CF. If X is an ideal of S with Θ(X; ι) ≠ ∅, then Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation ideal.

Proof

Suppose that X is an ideal of S. Then SXX. From Proposition 3.11 (9), it follows that Θ(SX; ι) ⊆ Θ(X; ι). By Proposition 3.11 (1), we obtain that Θ(S; ι) = S. From Proposition 4.8, it follows that

S(Θ_(X;ι))=(Θ_(S;ι))(Θ_(X;ι))Θ_(SX;ι)Θ_(X;ι).

Thus Θ(X; ι) is a left ideal of S.

Similarly, we can prove that Θ(X; ι) is a right ideal of S. Thus Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation ideal. □

The following corollary is an immediate consequence of Proposition 3.13, Theorem 4.15 and Theorem 4.16.

Corollary 4.17

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CF. If X is an ideal of S over a non-empty interior set, then Θ R̦(X; ι) is a 𝓒𝓢Θ(S; ι)-rough ideal.

Observe that, in Corollary 4.17, the converse is not true in general. We present an example as the following.

Example 4.18

According to Example 4.4, if X := {s1, s3, s5} is a subset of S, then we have Θ(X; 0.9) = S and Θ(X; 0.9) := {s1, s5}. Thus we see that Θbnd(X; 0.9) ≠ ∅. Obviously, Θ(X; 0.9) is an 𝓒𝓢Θ(S; 0.9)-upper approximation ideal, and it is straightforward to check that Θ(X; 0.9) is a 𝓒𝓢Θ(S; 0.9)-lower approximation ideal. However, X is not an ideal of S. Consequently, Θ R̦(X; 0.9) is a 𝓒𝓢Θ(S; 0.9)-rough ideal, but X is not an ideal of S.

Theorem 4.19

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CF. If X is a completely prime ideal of S, then Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation completely prime ideal.

Proof

We prove that Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation completely prime ideal. In fact, since X is an ideal of S, by Theorem 4.15, we have that Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation ideal. Let s1, s2S such that s1s2Θ(X; ι). Then by the Θ-complete property of 𝓒𝓢Θ(S; ι), we get

(CSΘ(s1;ι))(CSΘ(s2;ι))X=CSΘ(s1s2;ι)X.

Thus there exist s3CSΘ(s1; ι) and s4CSΘ(s2; ι) such that s3s4X. Since X is a completely prime ideal, we have s3X or s4X. Hence we have CSΘ(s1; ι) ∩ X ≠ ∅ or CSΘ(s2; ι) ∩ X ≠ ∅, and so s1Θ(X; ι) or s2Θ(X; ι). Therefore Θ(X; ι) is a completely prime ideal of S. As a consequence, Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation completely prime ideal. □

Theorem 4.20

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CF. If X is a completely prime ideal of S with Θ(X; ι) ≠ ∅, then Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation completely prime ideal.

Proof

Since X is an ideal of S, by Theorem 4.16, Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation ideal. Let s1, s2S such that s1s2Θ(X; ι). Since Θ is complete, we have

(CSΘ(s1;ι))(CSΘ(s2;ι))=CSΘ(s1s2;ι)X.

Now, we suppose that s1Θ(X; ι). Then CSΘ(s1; ι) is not a subset of X. Thus there exists s3CSΘ(s1; ι) but s3X. For each s4CSΘ(s2; ι),

s3s4(CSΘ(s1;ι))(CSΘ(s2;ι))X.

Whence s3s4X. Since X is a completely prime ideal and s3X, we have s4X. Thus CSΘ(s2; ι) ⊆ X, which yields s2Θ(X; ι). Hence we get Θ(X; ι) is a completely prime ideal of S. Therefore Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation completely prime ideal. □

The following corollary is an immediate consequence of Proposition 3.13, Theorem 4.19 and Theorem 4.20.

Corollary 4.21

Let (S, 𝓒𝓢Θ(S; ι)) be an 𝓒𝓢Θ(S; ι)-approximation space type CF. If X is a completely prime ideal of S over a non-empty interior set, then Θ R̦(X; ι) is a 𝓒𝓢Θ(S; ι)-rough completely prime.

Observe that, in Corollary 4.21, the converse is not true in general. We present an example as the following.

Example 4.22

According to Example 4.4, if X := {s1, s2, s5} is a subset of S, then we have Θ(X; 0.9) = S and Θ(X; 0.9) := {s1, s5}. Thus we see that Θbnd(X; 0.9) ≠ ∅. Obviously, Θ(X; 0.9) is an 𝓒𝓢Θ(S; 0.9)-upper approximation completely prime ideal, and it is straightforward to check that Θ(X; 0.9) is a 𝓒𝓢Θ(S; 0.9)-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 s3s4 = s2X but s3X and s4X. As a consequence, Θ R̦(X; 0.9) is a 𝓒𝓢Θ(S; 0.9)-rough completely prime ideal, but X is not a completely prime ideal of S.

5 Homomorphic images of roughness in semigroups

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

Proposition 5.1

Let f be an epimorphism from S in (S, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)), where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). Then the following statements hold.

  1. For all s1, s2S, s1CSΘ(s2; ι) if and only if f(s1) ∈ CSΨ(f(s2); ι).

  2. f(Θ(X; ι)) = Ψ(f(X); ι) for every non-empty subset X of S.

  3. f(Θ(X; ι)) ⊆ Ψ(f(X); ι) for every non-empty subset X of S.

  4. If f is injective, then f(Θ(X; ι)) = Ψ(f(X); ι) for every non-empty subset X of S.

  5. If Ψ is a compatible preorder fuzzy relation, then Θ is a compatible preorder fuzzy relation.

Proof

  1. Let s1, s2S be such that s1CSΘ(s2; ι). Then f(s1), f(s2) ∈ T and SΘ(s1; ι) = SΘ(s2; ι). In the following, we shall prove that SΨ(f(s1); ι) = SΨ(f(s2); ι). Let t1SΨ(f(s1); ι). Then Ψ(f(s1), t1) ≥ ι. Since f is surjective, there exists s3S such that f(s3) = t1. Whence Ψ(f(s1), f(s3)) ≥ ι, and so Θ(s1, s3) ≥ ι. Thus s3SΘ(s1; ι). Whence we have s3SΘ(s2; ι). Hence Θ(s2, s3) ≥ ι, and so Ψ(f(s2), f(s3)) ≥ ι. Thus t1 = f(s3) ∈ SΨ(f(s2); ι). Then we have SΨ(f(s1); ι) ⊆ SΨ(f(s2); ι). Similarly, we can show that SΨ(f(s2); ι) ⊆ SΨ(f(s1); ι). Therefore SΨ(f(s1); ι) = SΨ(f(s2); ι). As a consequence, f(s1) ∈ CSΨ(f(s2); ι).

    Conversely, it is easy to verify that s1CSΘ(s2; ι) whenever f(s1) ∈ CSΨ(f(s2); ι) for all s1, s2S.

  2. Let X be a non-empty subset of S. We verify firstly that f(Θ(X; ι)) = Ψ(f(X); ι). Suppose that t1f(Θ(X; ι)). Then there exists s1Θ(X; ι) such that f(s1) = t1. Therefore we have CSΘ(s1; ι) ∩ X ≠ ∅. Thus there exists s2S such that s2CSΘ(s1; ι) and s2X. By the argument (1), we obtain that f(s2) ∈ CSΨ(f(s1); ι) and f(s2) ∈ f(X). Then we have CSΨ(f(s1); ι) ∩ f(X) ≠ ∅, and so t1 = f(s1) ∈ Ψ(f(X); ι). Thus we have f(Θ(X; ι)) ⊆ Ψ(f(X); ι).

    On the other hand, let t2Ψ(f(X); ι). Then there exists s3S such that f(s3) = t2, and so CSΨ(f(s3); ι) ∩ f(X) ≠ ∅. Thus there exists s4X such that f(s4) ∈ f(X) and f(s4) ∈ CSΨ(f(s3); ι). By the argument (1), we get that s4CSΘ(s3; ι), and so we have CSΘ(s3; ι) ∩ X ≠ ∅. Hence s3Θ(X; ι), and so t2 = f(s3) ∈ f(Θ(X; ι)). Thus we get Ψ(f(X); ι) ⊆ f(Θ(X; ι)). This implies that f(Θ(X; ι)) = Ψ(f(X); ι).

  3. Let X be a non-empty subset of S. Let t1f(Θ(X; ι)). Then there exists s1Θ(X; ι) such that f(s1) = t1. Thus we get CSΘ(s1; ι) ⊆ X. We shall prove that CSΨ(t1; ι) ⊆ f(X). Let t2CSΨ(t1; ι). Then there exist s2S such that f(s2) = t2. Thus we have f(s2) ∈ CSΨ(f(s1); ι). By the argument (1), we obtain that s2CSΘ(s1; ι), and so s2X. Hence we have t2 = f(s2) ∈ f(X), and Thus CSΨ(t1; ι) ⊆ f(X). Therefore we have t1Ψ(f(X); ι). As a consequence, f(Θ(X; ι)) ⊆ Ψ(f(X); ι).

  4. Let X be a non-empty subset of S. We only need to prove that Ψ(f(X); ι) ⊆ f(Θ(X; ι)). Suppose that t1Ψ(f(X); ι). Then there exists s1S such that f(s1) = t1. Thus we have CSΨ(f(s1); ι) ⊆ f(X). We shall show that CSΘ(s1; ι) ⊆ X. Let s2CSΘ(s1; ι). Then by the argument (1), we have f(s2) ∈ CSΨ(f(s1); ι). Hence f(s2) ∈ f(X). Thus there exists s3X such that f(s3) = f(s2). By the assumption, we have s2X, and so CSΘ(s1; ι) ⊆ X. Hence s1Θ(X; ι), and so t1 = f(s1) ∈ f(Θ(X; ι)). Thus Ψ(f(X); ι) ⊆ f(Θ(X; ι)).

    By the argument (3), we get f(Θ(X; ι)) ⊆ Ψ(f(X); ι). Consequently, f(Θ(X; ι)) = Ψ(f(X); ι).

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

Proposition 5.2

Let f be an isomorphism from S in (S, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)), where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If Ψ is complete, then Θ is complete.

Proof

Let s1, s2 be two elements in S and let s3CSΘ(s1s2; ι). Then by Proposition 5.1 (1), we get that f(s3) ∈ CSΨ(f(s1s2); ι). Since f is a homomorphism and Ψ is complete, we have

f(s3)CSΨ(f(s1s2);ι)=CSΨ(f(s1)f(s2);ι)=(CSΨ(f(s1);ι))(CSΨ(f(s2);ι)).

Thus there exist t1CSΨ(f(s1); ι) and t2CSΨ(f(s2); ι) such that f(s3) = t1t2. Since f is surjective, there exist s4, s5S such that f(s4) = t1 and f(s5) = t2. From

f(s4)f(s5)=f(s3)(CSΨ(f(s1);ι))(CSΨ(f(s2);ι)),

it follows that f(s4) ∈ CSΨ(f(s1); ι) and f(s5) ∈ CSΨ(f(s2); ι). By Proposition 5.1 (1), we obtain that s4CSΘ(s1; ι) and s5CSΘ(s2; ι). Since f is a homomorphism, we have f(s3) = f(s4)f(s5) = f(s4s5). Since f is injective, we get s3 = s4s5. Thus we get that s3CSΘ(s1; ι)CSΘ(s2; ι). Therefore we have CSΘ(s1s2; ι) ⊆ CSΘ(s1; ι)CSΘ(s2; ι).

On the other hand, by Proposition 4.2 and Proposition 5.1 (5), CSΘ(s1; ι)CSΘ(s2; ι) ⊆ CSΘ(s1s2; ι). Thus CSΘ(s1; ι)CSΘ(s2; ι) = CSΘ(s1s2; ι). Hence 𝓒𝓢Θ(S; ι) is Θ-complete. Therefore Θ is complete. □

Theorem 5.3

Let f be an epimorphism from S in (S, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CPF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation semigroup if and only if Ψ(f(X); ι) is an 𝓒𝓢Ψ(T; ι)-upper approximation semigroup.

Proof

Suppose that Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation semigroup. Then by Proposition 5.1 (2),

(Ψ¯(f(X);ι))(Ψ¯(f(X);ι))=(f(Θ¯(X;ι)))(f(Θ¯(X;ι)))=f((Θ¯(X;ι))(Θ¯(X;ι)))f(Θ¯(X;ι))=Ψ¯(f(X);ι).

Hence Ψ(f(X); ι) is a subsemigroup of T. Thus Ψ(f(X); ι) is an 𝓒𝓢Ψ(T; ι)-upper approximation semigroup.

Conversely, let s1 ∈ (Θ(X; ι))(Θ(X; ι)). From Proposition 5.1 (2), it follows that

f(s1)f((Θ¯(X;ι))(Θ¯(X;ι)))=(f(Θ¯(X;ι)))(f(Θ¯(X;ι)))=(Ψ¯(f(X);ι))(Ψ¯(f(X);ι))Ψ¯(f(X);ι)=f(Θ¯(X;ι)).

Thus there exists s2Θ(X; ι) such that f(s1) = f(s2). Hence we have CSΘ(s2; ι) ∩ X ≠ ∅. From Proposition 3.4 (1), it follows that f(s1) ∈ CSΨ(f(s2); ι). By Proposition 5.1 (1), we obtain that s1CSΘ(s2; ι). From Proposition 3.4 (2), it follows that CSΘ(s1; ι) = CSΘ(s2; ι). Thus we have CSΘ(s1; ι) ∩ X ≠ ∅, and so s1Θ(X; ι). Hence we have that (Θ(X; ι))(Θ(X; ι)) ⊆ Θ(X; ι). Thus Θ(X; ι) is a subsemigroup of S. Therefore Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation semigroup. □

Theorem 5.4

Let f be an isomorphism from S in (S, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CPF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation semigroup if and only if Ψ(f(X); ι) is a 𝓒𝓢Ψ(T; ι)-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, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CPF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ R̦(X; ι) is a 𝓒𝓢Θ(S; ι)-rough semigroup if and only if Ψ (f(X); ι) is a 𝓒𝓢Ψ(T; ι)-rough semigroup.

Theorem 5.6

Let f be an epimorphism from S in (S, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CPF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation ideal if and only if Ψ(f(X); ι) is an 𝓒𝓢Ψ(T; ι)-upper approximation ideal.

Proof

Suppose that Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation ideal. Then we have SΘ(X; ι) ⊆ Θ(X; ι). Whence we have f(SΘ(X; ι)) ⊆ f(Θ(X; ι)). By Proposition 5.1 (2), we obtain that

TΨ¯(f(X);ι)=f(SΘ¯(X;ι))f(Θ¯(X;ι))=Ψ¯(f(X);ι).

Hence Ψ(f(X); ι) is a left ideal of T. Similarly, we can prove that Ψ(f(X); ι) is a right ideal of T. Thus Ψ(f(X); ι) is an 𝓒𝓢Ψ(T; ι)-upper approximation ideal.

Conversely, we suppose that Ψ(f(X); ι) is an 𝓒𝓢Ψ(T; ι)-upper approximation ideal. Then we have Ψ(f(X); ι) ⊆ Ψ(f(X); ι). Now, let s1SΘ(X; ι). From Proposition 5.1 (2), it follows that

f(s1)f(SΘ¯(X;ι))=TΨ¯(f(X);ι)Ψ¯(f(X);ι)=f(Θ¯(X;ι)).

Thus there exists s2Θ(X; ι) such that f(s1) = f(s2), and so CSΘ(s2; ι) ∩ X ≠ ∅. By Proposition 3.4 (1), we obtain that f(s1) ∈ CSΨ(f(s2); ι). By Proposition 5.1 (1), we obtain s1CSΘ(s2; ι). From Proposition 3.4 (2), it follows that CSΘ(s1; ι) = CSΘ(s2; ι). Hence we have CSΘ(s1; ι) ∩ X ≠ ∅, and so s1Θ(X; ι). Thus (X; ι) ⊆ Θ(X; ι). Whence Θ(X; ι) is a left ideal of S. Similarly, we can prove that Θ(X; ι) is a right ideal of S. Therefore Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation ideal. □

Theorem 5.7

Let f be an isomorphism from S in (S, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CPF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation ideal if and only if Ψ(f(X); ι) is a 𝓒𝓢Ψ(T; ι)-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, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CPF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ R̦(X; ι) is a 𝓒𝓢Θ(S; ι)-rough ideal if and only if Ψ (f(X); ι) is a 𝓒𝓢Ψ(T; ι)-rough ideal.

Theorem 5.9

Let f be an epimorphism from S in (S, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation completely prime ideal if and only if Ψ(f(X); ι) is an 𝓒𝓢Ψ(T; ι)-upper approximation completely prime ideal.

Proof

Assume that Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation completely prime ideal. Let t1, t2T be such that t1t2Ψ(f(X); ι). Thus there exist s1, s2S such that f(s1) = t1 and f(s2) = t2. Hence we have CSΨ(f(s1)f(s2); ι) ∩ f(X) ≠ ∅. Since Ψ is complete, we have

(CSΨ(f(s1);ι))(CSΨ(f(s2);ι))f(X)=CSΨ(f(s1)f(s2);ι)f(X).

Then there exist f(s3) ∈ CSΨ(f(s1); ι) and f(s4) ∈ CSΨ(f(s2); ι) such that f(s3)f(s4) ∈ f(X), and so f(s3s4) ∈ f(X). Then there exists s5X such that f(s3s4) = f(s5). By Proposition 5.1 (1), we obtain that s3CSΘ(s1; ι) and s4CSΘ(s2; ι). From Propositions 4.2 and 5.1 (5), we get that s3s4CSΘ(s1s2; ι). By Proposition 3.4 (2), we obtain that CSΘ(s1s2; ι) = CSΘ(s3s4; ι). Note that f(s3s4) ∈ CSΨ(f(s3s4); ι). Then f(s5) ∈ CSΨ(f(s3s4); ι). By Proposition 5.1 (1), once again, we get that s5CSΘ(s3s4; ι) = CSΘ(s1s2; ι). Thus CSΘ(s1s2; ι) ∩ X ≠ ∅, and so s1s2Θ(X; ι). Since Θ(X; ι) is a completely prime ideal of S, we have s1Θ(X; ι) or s2Θ(X; ι). Hence we have f(s1) ∈ f(Θ(X; ι)) or f(s2) ∈ f(Θ(X; ι)). From Proposition 5.1 (2), we get f(s1) ∈ Ψ(f(X); ι) or f(s2) ∈ Ψ(f(X); ι), which yields t1Ψ(f(X); ι) or t2Ψ(f(X); ι). Thus Ψ(f(X); ι) is a completely prime ideal of T. Therefore Ψ(f(X); ι) is an 𝓒𝓢Ψ(T; ι)-upper approximation completely prime ideal.

Conversely, we suppose that Ψ(f(X); ι) is an 𝓒𝓢Θ(S; ι)-upper approximation completely prime ideal. Now, let s6, s7 be elements in S such that s6s7Θ(X; ι). Then f(s6s7) ∈ f(Θ(X; ι)). By Proposition 5.1 (2), we obtain that

f(s6)f(s7)=f(s6s7)f(Θ¯(X;ι))=Ψ¯(f(X);ι).

Thus f(s6) ∈ Ψ(f(X); ι) or f(s7) ∈ Ψ(f(X); ι). Now, we consider the following two cases.

  • Case 1

    If f(s6) ∈ Ψ(f(X); ι), then we have f(s6) ∈ f(Θ(X; ι)) since Proposition 5.1 (2). Thus there exists s8Θ(X; ι) such that f(s6) = f(s8). Whence CSΘ(s8; ι) ∩ X ≠ ∅. By Proposition 3.4 (1), we obtain that f(s8) ∈ CSΨ(f(s8); ι). Thus f(s6) ∈ CSΨ(f(s8); ι). By Proposition 5.1 (1), we have s6CSΘ(s8; ι). From Proposition 3.4 (2), it follows that CSΘ(s6; ι) = CSΘ(s8; ι). Thus we have CSΘ(s6; ι) ∩ X ≠ ∅, and so s6Θ(X; ι).

  • Case 2

    If f(s7) ∈ Ψ(f(X); ι), then s7Θ(X; ι) since the proof is similar to that the case above.

As a consequence, Θ(X; ι) is an 𝓒𝓢Θ(S; ι)-upper approximation completely prime ideal. □

Theorem 5.10

Let f be an isomorphism from S in (S, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ(X; ι) is a 𝓒𝓢Θ(S; ι)-lower approximation completely prime ideal if and only if Ψ(f(X); ι) is a 𝓒𝓢Ψ(T; ι)-lower approximation completely prime ideal.

Proof

By Proposition 5.1 (4) and using the similar method as 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, 𝓒𝓢Θ(S; ι)) to T in (T, 𝓒𝓢Ψ(T; ι)) type CF, where Θ is defined by for all s1, s2S, Θ(s1, s2) = Ψ(f(s1), f(s2)). If X is a non-empty subset of S, then Θ R̦(X; ι) is a 𝓒𝓢Θ(S; ι)-rough completely prime ideal if and only if Ψ (f(X); ι) is a 𝓒𝓢Ψ(T; ι)-rough completely prime ideal.

6 Conclusions

In the present paper, we proposed rough sets in universal sets based on cores of successor classes with respect to level in closed unit intervals under fuzzy relations. Then we gave the real world example and proved some interesting properties. Based on this point, we gave a definition of a non-empty rough set in a universal set. Then we derived a sufficient condition of the such set. We introduced concepts of rough semigroups, rough ideals and rough completely prime ideals in semigroups under compatible preorder fuzzy relations. Then we derived sufficient conditions for them. We proved the relationships between rough semigroups (resp. rough ideals and rough completely prime ideals) and their homomorphic images.

Finally, we hope that the definitions and results of rough sets in universal sets and semigroup structures using fuzzy relations under mathematical principles in this paper may provide a powerful tool for assessment problems and decision problems in several fields with respect to informations and technology.

Acknowledgement

The authors would like to indicate their sincere thanks to the anonymous referees for their important 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.

References

  • [1]

    Pawlak Z., Rough sets, Int. J. Inf. Comp. Sci., 1982, 11, 341–356. CrossrefGoogle Scholar

  • [2]

    Biswas R., Nanda S., Rough groups and rough subgroups, Bull. Pol. Acad. Sci. Math., 1994, 42, 251–254. Google Scholar

  • [3]

    Kuroki N., Mordeson J. N., Structure of rough sets and rough groups, J. Fuzzy Math., 1997, 5, 183–191. Google Scholar

  • [4]

    Kuroki N., Rough ideals in semigroups, Inform. Sci., 1997, 100, 139–163. CrossrefGoogle Scholar

  • [5]

    Jun Y. B., Roughness of ideals in BCK-algebras, Sci. Math. Jpn., 2003, 57(1), 165–169. Google Scholar

  • [6]

    Davvaz B., Roughness in rings, Inform. Sci., 2004, 164(1), 147–163. CrossrefGoogle Scholar

  • [7]

    Xiao Q. M., Zhang Z. L., Rough prime ideals and rough fuzzy prime ideals in semigroups, Inform. Sci., 2006, 176, 725–733. CrossrefGoogle Scholar

  • [8]

    Davvaz B., Roughness based on fuzzy ideals, Inform. Sci., 176, 2006, 2417–2437. CrossrefGoogle Scholar

  • [9]

    Davvaz B., Mahdavipour M., Roughness in modules, Inform. Sci., 176, 2006, 3658–3674. CrossrefGoogle Scholar

  • [10]

    Ali M. I., Shabir M., Tanveer S., Roughness in hemirings, Neural. Comput. Appl., 2012, 21, 171–180. CrossrefGoogle Scholar

  • [11]

    Yaqoob N., Aslam M., Chinram R., Rough prime bi-ideals and rough fuzzy prime bi-ideals in semigroups, Ann. Fuzzy Math. Inform., 2012, 3, 203–211. Google Scholar

  • [12]

    Yang L., Xu L., Roughness in quantales, Inform. Sci., 2013, 220, 568–579. CrossrefGoogle Scholar

  • [13]

    Wang Q., Zhan J., Rough semigroups and rough fuzzy semigroups based on fuzzy ideals, Open Math., 2016, 14, 1114–1121. Web of ScienceGoogle Scholar

  • [14]

    Rehman N., Park C., Ali Shah S.I., Ali A., On generalized roughness in LA-semigroups, Mathematics, 2018, 6(7), 1–8. Web of ScienceGoogle Scholar

  • [15]

    Yao Y.Y., The superiority of three-way decisions in probabilistic rough set models, Inform. Sci., 2011, 181, 1080–1096. Web of ScienceCrossrefGoogle Scholar

  • [16]

    Zhang J.B., Li T.R., Chen H.M., Composite rough sets for dynamic data mining, Inform. Sci., 2014, 257, 81–100. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    Yao Y. Y., Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 1998, 111, 239–259. CrossrefGoogle Scholar

  • [18]

    Mareay R., Generalized rough sets based on neighborhood systems and topological spaces, J. Egyptian Math. Soc, 2016, 24, 603–608. CrossrefGoogle Scholar

  • [19]

    Zadeh L. A., Fuzzy sets, Inform. Control, 1965, 8, 338–353. CrossrefGoogle Scholar

  • [20]

    Zadeh L. A., Similarity relations and fuzzy orderings, Inform. Sci., 1971, 3, 117–200. Google Scholar

  • [21]

    Zadeh L. A., Towards a theory of fuzzy systems, In: R. E. Kalman, R. N. De Clairis (Ed.), Aspects of networks and systems theory, Holt, Rinehart and Winston, New York, 1971, 469–490. Google Scholar

  • [22]

    Wu W.Z., Mi J.S., Zhang W.X., Generalized fuzzy rough sets, Inform. Sci., 2003, 151, 263–282. CrossrefGoogle Scholar

  • [23]

    Pan W., She K., Wei P., Multi-granulation fuzzy preference relation rough set for ordinal decision system, Fuzzy Sets and Systems, 2017, 312, 87–108. Web of ScienceCrossrefGoogle Scholar

  • [24]

    Howie J.M., An introduction to semigroup theory, Academic Press, 1976. Google Scholar

  • [25]

    Mordeson J. N., Malik D. S., Kuroki N., Fuzzy semigroups, Springer-Verlag, Berlin, Heidelberg, New York, 2010. Google Scholar

About the article

Received: 2018-07-17

Accepted: 2018-11-29

Published Online: 2018-12-31


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1634–1650, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0136.

Export Citation

© 2018 Prasertpong and Siripitukdet, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in