Rough Prime Ideals in Rough Semigroups 1

International Mathematical Forum, Vol. 11, 2016, no. 8, 369 - 377 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6114

Rough Prime Ideals in Rough Semigroups Nurettin Ba˘ gırmaz Mardin Artuklu University, Mardin, Turkey c 2016 Nurettin Ba˘gırmaz. This article is distributed under the Creative Copyright Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper, we introduced the notion of rough prime ideals and we studied some properties of them on approximation spaces. Finally, rough image and rough inverse image of rough prime ideal were discussed.

Mathematics Subject Classification: 03E99, 20M99 Keywords: Rough set, rough semigroup, rough prime ideal

1

Introduction

The notion of rough sets was introduced by Z. Pawlak [17] in 1982 as an powerful mathematical tool for uncertain data while modeling the problems in computer science, medical science, data analysis and many other diverse fields [18, 7, 8]. Some authors, for example, Bonikowaski [16], Iwinski [14], and Pomykala and Pomykala [6] studied algebraic properties of rough sets. In 1994, Biswas and Nanda [13] introduced the notion of rough group and rough subgroups that their notion depends on the upper approximation and does not depend on the lower approximation. Miao et al. [4] improved definitions of rough group and rough subgroup, and proved their new properties. On the other hand, Kuroki and Wang [10] presented some properties of the lower and upper approximations with respect to the normal subgroups in 1996. In addition, some properties of the lower and the upper approximations with respect to the normal subgroups were studied in [15, 5, 3, 19, 2]. Also, Kuroki [11], introduced the notion of a rough ideal in a semigroup. Davvaz [1], introduced the notion of rough subring with respect to an ideal of a ring. Xiao and Zhang

370

Nurettin Ba˘gırmaz

[12], studied the notions of rough prime ideals and rough fuzzy prime ideals in ¨ a semigroup. Ba˘gırmaz and Ozcan [9], studied the notion of rough semigroup on approximation space. In this paper, we define the rough prime ideals, then we discuss some properties of them. Finally, we study the rough image and the rough inverse image of the rough prime ideal. Our definition of rough prime ideal is similar to the definition of rough ideal [9].

2

Preliminaries

In this section, we are going to list some definitions about rough sets and rough semigroups. Definition 2.1 [17]Let U be a finite non-empty set called universe and R be a family equivalence relation on U . The pair (U, R) is called an approximation space. Definition 2.2 [17]Let U be a universe and R be an equivalence relation on U . We denote the equivalence class of object x in R by [x]R . Definition 2.3 [17]Let (U, R) be an approximation space and X be a subset of U . The sets (1) R (X) = {x | [x]R ∩ X 6= ∅} , (2) R (X) = {x | [x]R ⊆ X} are called upper approximation and lower approximation of X in (U, R), respectively. Proposition 2.4 [17] Let X, Y ⊂ U , where U is a universe. Then, the approximations have the following properties: (1) R (X) ⊂ X ⊂ R (X) , (2) R (∅) = ∅ = ∅, R (U ) = R (U ) = U, (3) R (X) ∩ R (X) = R (X ∩ Y ) (4) R (X ∩ Y ) ⊆ R (X) ∩ R (Y ) , (5) R (X) ∪ R (Y ) ⊆ R (X ∪ Y ) ,

Rough prime ideals in rough semigroups

371

(6) R (X) ∪ R (Y ) = R (X ∪ Y ) , (7) X ⊂ Y if and only if R (X) ⊂ R (Y ) , R (X) ⊂ R (Y ) , (8) RR (X) = X and RR (X) = X. Let (U, R) be an approximation space and (·) be a binary operation defined on U . In the present paper onwards, we shall write xy instead of x·y, ∀ x, y ∈ U. If X and Y are two subsets of U , we denote by XY the subset composed of all the elements of the form xy, where x ∈ X, y ∈ Y . Proposition 2.5 [10] Let (U, R) be an approximation space. Let X and Y be nonempty subsets of U. Then R (X) R (Y ) = R (XY ) . Definition 2.6 [9] Let (U, R) be an approximation space and (·) be a binary operation defined on U. A subset S of U is called a rough semigroup on approximation space, provided the following properties are satisfied: (1) For all x, y ∈ S, x · y ∈ R (S) , (2) For all x, y, z ∈ S, (x · y) · z = x · (y · z) property holds in R (S) . Example 2.7 Let U = {1, 2, 3, 4, 5} be a universe with the following multiplication table: · 1 2 3 4 1 1 2 1 1 2 2 2 3 4 3 1 3 3 1 4 4 3 4 4 5 3 3 4 4 A classification

5 5 5 5 5 5 of U is U/R = {E1 , E2 , E3 }, where E1 = {1, 2, 3} , E2 = {4} , E3 = {5} .

Let S1 = {2, 3, 4} , then R (S1 ) = {1, 2, 3, 4} . From Definition 2.6, S1 ⊆ U is a rough semigroup. Let S2 = {2, 3, 5} , then R (S2 ) = {1, 2, 3, 5} . Because 5 · 3 = 4 ∈ / R (S2 ) , we have S2 is not a rough semigroup. Proposition 2.8 [9] Let (U, R) be an approximation space and S ⊆ U . Then

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(1) If S is a semigroup, then S is a rough semigroup on approximation space. (2) If H is a subsemigroup of semigroup S, then H is a rough subsemigroup of rough semigroup S. (3) If I is a left (right, two-sided) ideal of semigroup S, then I is a rough left (right, two-sided) ideal of rough semigroup S. Definition 2.9 [9] Let (U, R) be an approximation space and (·) be a binary operation defined on U. A nonempty subset I of a rough semigroup S is said to be a rough left (resp. right) ideal of S if SI ⊆ R (I) (resp. IS ⊆ R (I))

A semigroup is an algebraic structure on a nonempty set S together with an associative binary operation. That means, a semigroup is a set S together with a binary operation “·” that satisfies: (1) For all x, y ∈ S, x · y ∈ S, (2) For all x, y, z ∈ S, (x · y) · z = x · (y · z) property holds in S. An ideal H of a semigroup S is a prime ideal of S such that xy ∈ H for some x, y ∈ S implies x ∈ H or y ∈ H.

3

Rough prime ideals

In this section we introduce the notion of rough prime ideals on an approximation space and study some of its properties. Definition 3.1 Let (U, R) be an approximation space and (·) be a binary operation defined on U . An rough ideal H is called rough prime ideal of a rough semigroup S if for a, b ∈ S, ab ∈ R (H) implies a ∈ H or b ∈ H. The following example shows that a prime ideal is not a rough prime ideal in general on the same condition of Proposition 2.8 item (3). Example 3.2 Let U = {1, 2, 3, 4, 5} be a universe with the following multiplication table: · 1 2 3 4 5

1 1 1 1 4 3

2 1 2 2 3 3

3 1 3 3 4 4

4 1 4 1 4 4

5 5 5 5 5 5

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A classification of U is U/R = {E1 , E2 , E3 }, where E1 = {1, 3} , E2 = {2, 4} , E3 = {5} . Let S = {1, 2, 3} be a semigroup and H = {1} . Then R (S) = {1, 2, 3, 4} and R (H) = {1, 3} . From Definition 2.6, S ⊆ U is a rough semigroup. It is clear that H is a prime ideal of S. The set H is not a rough prime ideal for 2.3 = 3 ∈ R (H) but 2 ∈ / R (H) and 3 ∈ / R (H) . The following proposition can be obtained if we strengthen on the above conditions of Proposition 2.8 item (3). Proposition 3.3 Let (U, R) be an approximation space and S ⊆ U . Let R (H) = HH. Then, If H is a prime ideal of semigroup S, then H is a rough prime ideal of rough semigroup S. Proof. Let H be a prime ideal of semigroup S, that is, a, b ∈ S, ab ∈ H implies a ∈ H or b ∈ H. By Proposition 2.4. (1), we have that H ⊆ R (H) . Thus ab ∈ R (H) = HH. Hence, a ∈ H or b ∈ H. This means that H is a rough prime ideal of rough semigroup S. Proposition 3.4 Let (U, R) be an approximation space and S ⊆ U. Then, the union of a set rough prime ideals of a rough semigroup S is a rough prime ideal of rough semigroup S. Proof. Let [Ai ] be a collection of rough prime ideals of S , where i ranges over index set I. Then, by Proposition 2.4. (6), we have that an arbitrary −

R

−

∪ Ai

i∈I −

−

= ∪ R (Ai ) . Thus, if a, b ∈ S and ab ∈ R i∈I

∪ Ai

i∈I

we have that

−

ab ∈ R (Ai ), for some i ∈ I. But R (Ai ) is prime, that is, a ∈ Ai or b ∈ Ai . Thus, a ∈ ∪ Ai or b ∈ ∪ Ai . Hence, ∪ Ai is rough prime ideal of rough i∈I

i∈I

i∈I

semigroup S. In general, the intersection of rough prime ideals is not a rough prime ideal, as is shown in the following: Example 3.5 Let U = {1, 2, 3, 4, 5, 6} be a universe with the following multiplication table: · 1 2 3 4 5 6 1 1 1 1 2 2 1 2 1 2 2 3 4 6 3 3 3 3 2 4 6 4 2 3 1 4 6 6 5 1 2 3 4 5 6 6 2 6 2 6 6 6

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A classification of U is U/R = {E1 , E2 , E3 , E4 }, where E1 E2 E3 E4

= {1, 2} , = {3, 5} , = {4} , = {6} .

Let S = {1, 3, 4, 6} , then R (S) = {1, 2, 3, 4, 5, 6} . From Definition 2.6, S ⊆ U is a rough semigroup. Let A = {1, 4} and B = {1, 3} , then R (A) = {1, 2, 4} and R (B) = {1, 2, 3, 5} . From Definition 3.1, A and B are rough prime ideals of rough semigroup S. Then A ∩ B = {1} and R (A ∩ B) = {1, 2} . Thus, the set A ∩ B / A∩B and 4 ∈ / A∩B. is not a rough prime ideal for 3.4 = 2 ∈ R (A ∩ B) but 3 ∈ Proposition 3.6 Let (U, R) be an approximation space and S ⊆ U. Let [Ai ] ∩ Ai

be a set of rough prime ideals of rough semigroup S and R

i∈I

= ∩ R (Ai ) , i∈I

where i ranges over an arbitrary index set I.Then, the intersection of a set rough prime ideals of a rough semigroup S is a rough prime ideal of S if and only if it is a rough prime ideal of the union of the given ideals. Proof. The necessity being obvious , we proceed to prove the sufficiency. Let [Ai ] be a set of rough prime ideals of S , where i ranges over an arbitrary index set I. By hypothesis, ∩ Ai is a rough prime of ∪ Ai and hence is non i∈I i∈I - empty. Let x, y ∈ S, xy ∈ R ∩ Ai . Since R ∩ Ai = ∩ R (Ai ) , then i∈I

i∈I

i∈I

/ R (Aj ) ; and if if x ∈ / ∩ R (Ai ) there is some R (Aj ) (j ∈ I) such that x ∈ i∈I

y∈ / ∩ R (Ai ) then there is some R (Ak ) (k ∈ I) such that y ∈ / R (Ak ) . Hence i∈I

x, y ∈ / R (Aj ∩ Ak ) . But xy ∈ ∩ R (Ai ) ⊆ R (Aj ∩ Ak ) , and both Aj and Ak i∈I

are rough prime, whence either x ∈ Aj or y ∈ Aj , and either x ∈ Ak or y ∈ Ak . Therefore , we have either x ∈ Aj and y ∈ Ak or else y ∈ Aj and x ∈ Ak . In either case, x, y ∈ Aj ∪ Ak ⊆ ∪ Ai . But, by hypothesis, ∩ Ai is a rough prime i∈I

i∈I

in ∪ Ai , whence either x ∈ ∩ Ai or y ∈ ∩ Ai . Hence, ∩ Ai is rough prime i∈I

i∈I

i∈I

i∈I

ideal of rough semigroup S.

4

Rough image and rough inverse image of rough prime ideals

Let (U1 , R1 ), (U2 , R2 ) be two approximation spaces, and (·) , (◦) be binary operations over universes U1 and U2 , respectively.


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Definition 4.1 Let S1 ⊂ U1 , S2 ⊂ U2 be rough semigroups. If there exists a surjection ϕ : R (S1 ) → R (S2 ) such that ϕ(x · y) = ϕ(x) ◦ ϕ(y) for all x, y ∈ R (S1 ) then ϕ is called a rough homomorphism and S1 , S2 are called rough homomorphic semigroups. Proposition 4.2 Let S1 ⊂ U1 , S2 ⊂ U2 be rough homomorphic semigroups and let H be a rough prime ideal of S1 . If ϕ R (H) = R (ϕ (H)) , then ϕ (H) is a rough prime ideal of S2 . Proof. Let ϕ(x), ϕ(y) ∈ S2 , ϕ(x) ◦ ϕ(y) = ϕ(x · y) ∈ R (ϕ (H)) .Since ϕ R (H) = R (ϕ (H)) we have x · y ∈ R (H). Since H is a rough prime ideal of S1 , we have x ∈ H or y ∈ H. Thus ϕ(x) ∈ ϕ (H) or ϕ(y) ∈ ϕ (H). Hence ϕ (H) is a rough prime ideal of S2 . Proposition 4.3 Let S1 ⊂ U1 , S2 ⊂ U2 be rough homomorphic semigroups and let H2 be a rough prime ideal of S2 . If ϕ R (H1 ) = R (ϕ (H1 )) , then ϕ−1 (H2 ) = H1 is a rough subsemigroup of S1 . Proof. Let x, y ∈ S1 , x · y ∈ R (H1 ) ,then we have ϕ(x), ϕ(y) ∈ S2 . Since ϕ−1 (H2 ) = H1 , we have ϕ (H1 ) = H2 , and so R (H2 ) = R (ϕ (H1 )) = ϕ R (H1 ) . Thus ϕ(x · y) = ϕ(x) ◦ ϕ(y) ∈ ϕ R (H1 ) = R (H2 ) . Since H2 is a rough prime ideal, ϕ(x) ∈ H2 or ϕ(y) ∈ H2 . Since ϕ (H1 ) = H2 , we have ϕ(x) ∈ ϕ (H1 ) or ϕ(y) ∈ ϕ (H1 ) . Thus x ∈ H1 or y ∈ H1 . Hence ϕ−1 (H2 ) = H1 is a rough prime ideal of S1 .

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