Soft ideals, Soft Quasi Ideals and Soft Bi-ideals with

Soft ideals, Soft Quasi Ideals and Soft Bi-ideals with Applications in Semigroups M. Irfan Ali and M. Shabir Department of Mathematics,

Quaid-i-Azam University, Islamabad, Pakistan

K. P. Shum Department of Mathematics , The University of Hong Kong , Pokfulam Road, Hong Kong, China (SAR) [email protected]

Abstract A collection of ideals of a given semigroup S is called a soft ideal over S. It is easy to see that the concept of soft ideals is more general than the usual concept of ideals . Similarly, we can define soft quasi-ideals and soft bi-ideals over a given semigroup S and obtain some interesting properties. In this paper, we shall characterize regular semigroups by using their soft ideals, soft quasi ideals and soft bi-ideals. As a consequence, soft regular semigroups are also characterized by their soft right ideals and soft left ideals.

Keywords: Soft sets; Regular semigroups; Soft quasi ideals; Soft bi-ideals.

1

Introduction

It is well known that many problems in different disciplines such as economics, engineering science, social sciences, medical and artificial intelligence are usually not precise. There are always many types of uncertainties involved in the data. The classical tools used to deal with 1

all these uncertainties are applicable only under certain environment. In particular, modeling, reasoning and calculations performed are always exact and precise, uncertainties cannot simply be handled by these tools. In dealing with uncertainties, many theories have been recently developed , including theory of Probability , theory of Fuzzy sets, theory of Intuitionistic Fuzzy sets and theory of Rough sets. . . etc. Although many new techniques have been developed as a result of these theories , yet difficulties are still there. The major difficulties posed by these theories are probably due to the inadequacy of parameters (see[13]). In 1999, Molodtsov first initiated the theory of soft sets involving enough parameters so that many difficulties that we are facing become easier by applying soft sets . In fact, when dealing with uncertainties, we find that the theory of soft sets is quite useful. On the other hand, algebraic structures of groups, rings and semigroups have been recently studied by using fuzzy sets , for instance ,the structure of fuzzy groups were first initiated by Rosenfeld in 1971 [16]. Later on, Kuroki applied the theory of fuzzy sets to study fuzzy semigroups. He first characterized the fuzzy semigroups by using fuzzy ideals [7]. Moreover, Ahsan, Khan and Shabir characterized semigroups by their fuzzy bi-ideals [1]. In addition, rough algebraic structures have also been studied by other authors, for instance, Nanda and Biswas studied the structure of rough groups and rough subgroups by using rough sets in [3] and Kuroki first discussed the properties of rough ideals in semigroups [8]. Recently, the properties of rough prime ideals of semigroups have been further investigated by Xiao and Zhang in [18]. The study of soft groups was initiated by Akta¸s and Ca˘gman [2]. It has been shown by some authors that soft sets and soft groups have applications in decision making problems and in information science ,for example, see [10], [11] and [2]. By a soft semigroup, we mean a collection of subsemigroups of a semigroup, whereas a soft ideal (quasi-ideal, bi-ideal) is a collection of ideals (quasi-ideals, bi-ideals) of a given semigroup. In general, when we talk about a soft ideal (quasi-ideal, bi-ideal), we actually mean that we are considering a collection of ideals (quasi-ideals,bi-ideals) over a semigroup. Thus, the concept of a soft ideal (quasi-ideal, bi-ideal) is a more general concept than the concept of ideal (quasi-ideal, bi-ideal). In this paper, we will characterize regular semigroups by using their soft ideals, soft quasi-ideals and soft bi-ideals. As a result, soft regular semigroups can also be similarly characterized. 2

2

Some basic definitions

Let S be a semigroup. Then an element 1 of a semigroup S is called an identity element of S if x1 = 1x = x for all x in S. If a semigroup S does not contain an identity element, then we simply adjoin 1 to S by defining x1 = 1x = x for all x ∈ S ∪ {1} . We usually write S 1 with the following meaning ½ S if S has an identity 1 S = S ∪ {1} otherwise A non-empty subset I of S is called a subsemigroup of S if I 2 ⊆ I. The subsemigroup I is called a left(resp. right) ideal of S if SI ⊆ I (resp. IS ⊆ I) and a two-sided ideal ( or simply ideal) of S if it is both a left and a right ideal of S. If S and T are semigroups, then their Cartesian product S × T is still a semigroup if we define (s, t) (s0 , t0 ) = (ss0 , tt0 ) for all s,s0 ∈ S and t,t0 ∈ T . A non empty subset Q of S is called a quasi-ideal if QS ∩ SQ ⊆ Q. It is clear that every one sided ideal is a quasi-ideal however, its converse is not true. We can easily observe that the intersection of quasi-ideals is either empty or a quasi-ideal. A subsemigroup B of S is called a bi-ideal if BSB ⊆ B. It is clear that the intersection of bi-ideals of a semigroup S is either empty or a bi-ideal of S. Also, the product of bi-ideals of S is still a bi-ideal of S . For other definitions and terminologies of semigroups not given in this paper, the reader is referred to Howie [5]. Let E be a set . Then, we call a pair (F, E) a soft set (over U ) if and only if F is a mapping of E which maps into the set of all subsets of U [13]. Denote the set of all soft sets by S (U ) . It has been interpreted by Molodtsov in [13] that a soft set is indeed a parameterized family of subsets of U , and thus E is always referred to as a set of parameters. We now formulate the following definitions.

Definition 1 Let (F, A) and (G, B) be two soft sets over a semigroup S. Then (G, B) is called a soft subset over (F, A) if B ⊆ A and G (b) ⊆ F (b) , for all b ∈ B. 3

Definition 2 A soft set (F, A) over S is called a soft set with cover if

S

F (α) = S.

α∈A

Two soft sets (F, A) and (G, B) over S are said to be equal if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A) . Throughout this paper, S and T are used to denote semigroups unless otherwise stated.

Definition 3 A soft set (F, A) over S is called a soft semigroup over S if F (α) is a subsemigroup of S when F (α) 6= ∅, for all α ∈ A.

Definition 4 A soft set (F, A) over S is called a soft (left,right) ideal over S if F (α) is an (left,right)ideal of S when F (α) 6= ∅, for all α ∈ A.

The product of two soft sets over a universe S endowed with a binary operation “ ∗ ” is defined by Molodtsov as following :

Definition 5 Let (F, A) and (G, B) be soft sets over S. Then the operation “ ∗ ” for soft sets is defined by (F, A) ∗ (G, B) = (H, A × B), where H (a, b) = F (a) ∗ G (b) , a ∈ A, b ∈ B, and A × B is the Cartesian product of A and B.

If there is no ambiguity then we simply write (F, A) (G, B) instead of (F, A) ∗ (G, B) and F (a) G (b) for F (a) ∗ G (b) . The operations and , ∩, or and ∪ were first given in [11] which were later termed as the basic intersection, intersection, basic union and union, respectively by Pei [15]. We now adopted the following definitions given by Pei in [15].

Definition 6 Let (F, A) and (G, B) be any two soft sets over a semigroup S. Then define the followings:

4

a The basic intersection of two soft sets (F, A) and (G, B) is defined as the soft set (H, C) = (F, A) ∧ (G, B), where C = A × B, and H (a, b) = F (a) ∩ G (b) for all (a, b) ∈ A × B. b The intersection (H, C) of two soft sets (F, A) and (G, B) is defined as the soft set (H, C) = (F, A) ∩ (G, B), where C = A ∩ B and H (c) = F (c) ∩ G (c) for all c ∈ C. c The basic union of two soft sets (F, A) and (G, B) is defined as the soft set (H, C) = (F, A) ∨ (G, B),where C = A × B and H (a, b) = F (a) ∪ G (b) for all (a, b) ∈ A × B. d The union (H, C) of two soft sets (F, A) and (G, B) is defined as the soft set (H, C) = (F, A) ∪ (G, B) , where C = A ∪ B and  F (c) , if c ∈ A\B  G (c) , if c ∈ B\A H (c) =  F (c) ∪ G (c) , if c ∈ A ∩ B The Cartesian product of two soft groups was defined by Aktas and Cagman in [2]. Now , we can similarly define the Cartesian product of soft semigroups as following: Definition 7 Let (F, A) and (G, B) be two soft semigroups over S. Then their Cartesian product is defined as (F, A) × (G, B) = (H, A × B), where H (a, b) = F (a) × G (b), for all (a, b) ∈ A × B.

Definition 8 The restricted union (H, C) of two soft sets (F, A) and (G, B) is defined as the b (G, B) , where C = A ∪ B and soft set(H, C) = (F, A) ∪  F (c) , if c ∈ A\B  G (c) , if c ∈ B\A H (c) =  F (c) ∩ G (c) , if c ∈ A ∩ B Let (F, A) and (G, B) be any two soft ideals over S. Then the following properties hold :

1. (F, A) ∗ (G, B) is still a soft ideal over S. 2. If A ∩ B 6= ∅, then (F, A) ∩ (G, B) is a soft ideal over S. 5

3. (F, A) ∪ (G, B) is also a soft ideal over S. 4. (F, A) ∧ (G, B) is also a soft ideal over S. 5. (F, A) ∨ (G, B) is also a soft ideal over S.

Let (F, A) and (G, B) be two soft semigroups(ideals) over S and T respectively. Then (F, A) × (G, B) is also a soft semigroup(ideal) over S × T.

3

Properties of soft quasi-ideals and soft bi-ideals

The notion of quasi-ideal in a semigroup was first introduced by Steinfeld in [17]. In fact, the concept of quasi-ideals plays an important role in the characterization of different algebraic structures. In semigroups, it is interesting to note that the intersection and the basic intersection of a soft left ideal and a soft right ideal over a semigroup S is neither a soft left ideal nor a soft right ideal over S. This interesting fact can be illustrated in the following example :

Example 1 Let S = {a, b, c, d, e} be a semigroup with the following Cayley table:

. a b c d e

a a a a a a

b a a b a d

c a a c a e

d a b a d a

Table 1

6

e a c a e a

Let (R, A) and (L, B) be soft sets over S, where A = B = S and R and L are defined as R (a) = {a} , R (b) = {a, b, c} = R (c) , R (d) = R (e) = {a, d, e}, L (a) = {a} , L (b) = {a, b, d} , L (c) = {a, c, e} , L (d) = {a, b, d} , L (e) = {a, c, e} . Then (R, A) is a soft right ideal over S and (L, B) is a soft left ideal over S . Let (Q, C) = (R, A) ∩ (L, B) , where C = A ∩ B = S and Q (x) = R (x) ∩ L (x) , for all x ∈ C. Then Q (a) = {a} , Q (b) = {a, b} , Q (c) = {a, c} , Q (d) = {a, d} , Q (e) = {a, e}. Thus, it is clear that (Q, C) is neither a soft right ideal nor a soft left ideal over S. Similarly, it can be shown that (R, A) ∧ (L, B) is neither a soft left ideal nor a soft right ideal over S.

Definition 9 A soft set (F, A) over a given semigroup S is called a soft quasi-ideal over S if F (α) is a quasi-ideal of S when F (α) 6= ∅, for all α ∈ A.

Soft quasi-ideals over a semigroup S have the following properties :

Proposition 1 Let (R, A) be a soft right ideal over S and (L, B) a soft left ideal over S. Then (R, A) ∩ (L, B) is a soft quasi-ideal over S, where A ∩ B 6= ∅.

Proof. Since (R, A) ∩ (L, B) = (H, C) , where C = A ∩ B 6= ∅, we can easily see that for any c ∈ C, H (c) = R (c) ∩ L (c). Since R (c) is a right ideal and L (c) is a left ideal of S, R (c) ∩ L (c) is a quasi-ideal of S. Hence (R, A) ∩ (L, B) is indeed a soft quasi-ideal over S. ¤ Proposition 2 Let (R, A) be a soft right ideal and (L, B) a soft left ideal over S. Then (R, A) ∧ (L, B) is a soft quasi-ideal over S.

Proof. We can easily see that (R, A) ∧ (L, B) = (H, C),where C = A × B and H (a, b) = R (a) ∩ L (b).Since R (a) is a right ideal of S and L (b) is a left ideal of S, R (a) ∩ L (b) is clearly a quasi-ideal of S.Consequently (R, A) ∧ (L, B) is a soft quasi-ideal over S. ¤ 7

Proposition 3 Let (R, A) be a soft right (left) ideal over S Then (R, A) is a soft quasi-ideal over S. Proof. Let (R, A) be a soft right ideal over S.Then R (a) is a right ideal of S, for all a ∈ A. Since each right(left) ideal of S is a quasi-ideal of S, (R, A)is a soft quasi-ideal over S. ¤ It is easy to see that if (F, A) and (G, B) are two soft quasi-ideals over a semigroup S, then the following statements hold: 1. (F, A) ∩ (G, B) is a soft quasi-ideal over S, where A ∩ B 6= ∅. 2. (F, A) ∧ (G, B) is a soft quasi-ideal over S. 3. (F, A) ∪ (G, B) is a soft quasi-ideal over S, where A ∩ B = ∅. b (G, B) is a soft quasi-ideal over S. 4. (F, A) ∪ Now we define soft quasi-ideal of a soft semigroup over S . Definition 10 Let (G, B) be a soft subset of a soft semigroup (F, A) over S. Then (G, B) is said to be a soft quasi-ideal of (F, A) if G (b) 6= ∅ is a quasi-ideal of F (b) , for all b ∈ B. The soft quasi-ideal of a soft semigroup defined above is different from the usual soft quasiideal over a given semigroup S. The following example is to illustrate this situation. Example 2 Let S = {0, a, b, c} be a semigroup with the following Cayley table :

· 0 a b c

0 0 0 0 0

a 0 0 0 0

b 0 0 0 a

Table 2 8

c 0 0 a b

Consider the soft set (F, S) where F : S → P (S) is defined as F (0) = {0},F (a) = {0, a},F (b) = {0, a, b},F (c) = {0, a, b, c} . It is clear that (F, S) is a soft semigroup over S. Now consider the soft set (G, {b}) where G : {b} → P (S) is defined asG (b) = {0, b} . Since {b} ⊆ S and G (b) is a quasi-ideal of F (b), (G, {b}) is a soft quasi-ideal of (F, S). But {0, b} S ∩ S {0, b} = {0, a} * {0, b} , that is, {0, b} is not a quasi-ideal of S, (G, {b}) is clearly not a soft quasi-ideal over S. Soft quasi-deals also have the following important properties :

Proposition 4 Let (F, A) be a soft semigroup over S and {(Hi , Bi ) ; i ∈ I} be a non empty family of soft quasi-ideals of (F, A) . Then the following statements hold:

1.

T

(Hi , Bi ) is a soft quasi-ideal of (F, A) .

i∈I

2.

V

(Hi , Bi ) is a soft quasi-ideal of

i∈I

3. If Bi ∩ Bj = ∅ for all i, j ∈ I then

V

(F, A) .

i∈I

S

(Hi , Bi ) is a soft quasi-ideal of (F, A) .

i∈I

Proof. The proofs are straightforward and are hence omitted . ¤ Definition 11 A soft set (B, A) over S is called a soft bi-ideal over S if B (a) is a bi-ideal of S, when B (a) 6= ∅, for all α ∈ A.

It is obvious that every soft bi-ideal over S is a soft semigroup over S.

Proposition 5 Every soft quasi-ideal over a semigroup S is a soft bi-ideal over S.

Proof. Let (F, A) be a soft quasi-ideal over a semigroup S. Then F (a) is a quasi-ideal over S for all a ∈ A. Since every quasi-ideal is a bi-ideal of S, (F, A) is a soft bi-ideal over S. ¤ 9

Proposition 6 Let (F, A) and (G, B) be two soft quasi-ideals over a semigroup S. Then (F, A) ∗ (G, B) is a soft bi-ideal over S, where “ ∗ ” is a binary operation defined on S.

Proof. It is known that (F, A) ∗ (G, B) = (H, A × B),where H is a function from A × B to P (S) defined by H (a, b) = F (a) ∗ G (b). Since F (a) and G (b) are quasi-ideals of S, for all a ∈ A, b ∈ B, the following properties hold:

F (a) ∗ G (b) ∗ F (a) ∗ G (b) ⊆ F (a) ∗ G (b) ∗ S ∗ G (b) ⊆ F (a) ∗ G (b) because G (b) is a bi-ideal of S.

Thus F (a) ∗ G (b) is a subsemigroup of S, so (H, A × B) is a soft semigroup over S. Also F (a) ∗ G (b) ∗ S ∗ F (a) ∗ G (b) ⊆ F (a) ∗ G (b) ∗S ∗ S ∗ G (b) ⊆ F (a) ∗ G (b) ∗ S ∗ G (b) ⊆ F (a) ∗ G (b) This shows that F (a) ∗ G (b) is a bi-ideal of S and consequently, (F, A) ∗ (G, B) is a soft bi-ideal over S. ¤

4

Characterizations of Regular and Soft Regular Semigroups

An element a of a semigroup S is called von Neumann regular or simply regular if there exists an element x ∈ S such that a = axa. If every element of a semigroup S is regular then S is called a regular semigroup. It is well known that a completely regular semigroup whose idempotents are central can be expressed as an union of groups, therefore, the class of regular semigroups forms a core class of semigroups because it still has many similar properties of groups. We now classify regular semigroups by considering its soft right ideals and soft left ideals. We obtain the following theorems. 10

Theorem 1 A semigroup S is a regular semigroup if and only if (R, A) ∗ (L, B) = (R, A) ∧ (L, B) , for every soft right ideal (R, A) and soft left ideal (L, B) over S.

Proof. (N ecessity) Since (R, A) ∗ (L, B) = (H, A × B) , where H is a function from A × B to P (S) defined by H (a, b) = R (a) ∗ L (b) , one can easily see that (R, A) ∧ (L, B) = (G, A × B) , where G is a function from A × B to P (S) such that G (a, b) = R (a) ∩ L (b) Now A × B ⊆ A × B and observe that R (a) ∗ L (b) ⊆ R (a) ∗ S ⊆ R (a) and R (a) ∗ L (b) ⊆ S ∗ L (b) ⊆ L (b) Hence, R (a) ∗ L (b) ⊆ R (a) ∩ L (b) for all a ∈ A, b ∈ B, and consequently (H, A × B) ⊆ (G, A × B). In proving the converse inclusion, we suppose that u ∈ R (a) ∩ L (b) . Since S is regular and u ∈ S , there exists v ∈ S such that u = uvu. Now since u ∈ R (a) and vu ∈ L (b), u = uvu ∈ R (a) ∗ L (b). This shows that (R, A) ∧ (L, B) ⊆ (R, A) ∗ (L, B). Therefore, (G, A × B) ⊆ (H, A × B) and so, (R, A) ∗ (L, B) = (R, A) ∧ (L, B) (Suf f iciency) Suppose that A = B = S and R is a function which maps from A to P (S) . Define R (u) = uS 1 , for all u ∈ S and let L be a function which maps from B to P (S) , defined by L (u) = S 1 u, for all u ∈ S. Then (R, S) is a soft right ideal and (L, S) is a soft left ideal over S. Thus by our hypothesis, u ∈ R (u) ∩ L (u) = R (u) L (u) = uS 1 S 1 u ⊆ uS 1 u. This shows that S is indeed a regular semigroup. ¤ Theorem 2 Let S be a semigroup . Then S is regular if and only if (Q, B) ∧ (L, A) ⊆ (Q, B) ∗ (L, A) for every soft left ideal (L, A) and every soft quasi-ideal (Q, B) over S. 11

Proof. (N ecessity) Since (Q, B) ∗ (L, A) = (H, B × A) , where H is a function from B × A to P (S) defined by H (b, a) = Q (b) ∗ L (a) , where b ∈ B, a ∈ A. Also , (Q, B) ∧ (L, A) = (G, B × A), where G is a function from B × A to P (S) defined by G (b, a) = Q (b) ∩ L (a) , where b ∈ B, a ∈ A. Now B×A ⊆ B×A. Let u ∈ Q (b)∩L (a) . Then, u ∈ Q (b) and u ∈ L (a). Since u ∈ S and S is regular, there exists v ∈ S such that u = uvu ∈ Q (b) ∗ S ∗ L (a) ⊆ Q (b) ∗ L (a) . Hence, (Q, B) ∧ (L, A) ⊆ (Q, B) ∗ (L, A) . (Suf f iciency) Suppose that A = B = S, and Q is a function which maps from B to P (S) defined by Q (u) = uS 1 , for all u ∈ S and L is a function which maps from A to P (S) defined by L (u) = S 1 u, for all u ∈ S. Then (Q, S) is a soft quasi-ideal and (L, S) is a soft left ideal over S. Thus, by hypothesis, we have u ∈ Q (u) ∩ L (u) ⊆ Q (u) L (u) = uS 1 S 1 u ⊆ uS 1 u. This shows that S is a regular semigroup. ¤ The following theorem is another characterization theorem of regular semigroups .

Theorem 3 Let S be a semigroup. Then S is regular if and only if (B, B) ∧ (L, A) ⊆ (B, B) ∗ (L, A), for every soft bi-ideal (B, B) and every soft left ideal (L, A) over S .

Proof. The proof is similar to Theorem 2 and is hence omitted . ¤ We now consider soft regular semigroups .

Definition 12 A soft semigroup (F, A) over a semigroup S is called a soft regular semigroup if for each α ∈ A, F (α) is regular .

The following example shows that if S is a regular semigroup then the soft semigroup (F, A) over the semigroup S may not be regular. 12

Example 3 Let S = {a, b, c, d, e} be a semigroup with the binary operation as shown in the Cayley table.

· a b c d e

a a a a a a

b a a b a d

c a a c a e

d a b a d a

e a c a e a

Table 3 We can easily verify that the above semigroup is a regular semigroup but non commutative. Consider S = A and F (a) = {a, b} , F (b) = {a, b, c} = F (c) , F (d) = {a, d} , F (e) = {a, e}.Then (F, S) is a soft semigroup over S but it is not regular because F (a), F (b) and F (c) are not regular subsemigroups of S. Example 4 Consider a semigroup S = {a, b, c, d} with binary operation as defined in the following Cayley table : · a b c d

a a a a a

b a a a a

c a a a a

d a a a d

Table 4 Clearly, S is not a regular semigroup. Let A = {α, β} be a set of parameters such that F (α) = {a} , F (β) = {a, d}. Then (F, A) is a soft regular semigroup over S because F (α) and F (β) are regular subsemigroups of S.

In the above examples, we have demonstrated that the regularity of a semigroup S does not imply the regularity of a soft semigroup over S. Also, the regularity of a soft semigroup 13

over a given semigroup S does not imply the regularity of the semigroup. However, we still have the following theorem.

Theorem 4 Let (F, A) be a soft regular semigroup over a semigroup S with cover. Then S is a regular semigroup.

Proof. Let (F, A) be a soft regular semigroup over S. Then F (α) is regular for each α ∈ A. S Now let x ∈ S. Because S = F (α) , there exists some β ∈ A such that x ∈ F (β). Since α∈A S F (β) is regular, there exists y ∈ F (β) such that x = xyx. Since y ∈ F (β) ⊆ F (α) = S , α∈A

S is regular. ¤

It can be easily seen that every soft semigroup (F, A) over S is regular if S is a band such that every soft semigroup (F, A) over S is regular if x2 = x for all x ∈ S. Soft semigroups also have the following properties:

Theorem 5 Let (F, A) be a soft semigroup over S (not necessarily) regular . Then the following assertions are equivalent .

1. (F, A) is regular. 2. (R, B) b ◦ (L, C) = (R, B) ∩ (L, C) for all soft right ideals (R, B) and soft left ideals (L, C) over F (α) for each α ∈ B ∩ C 6= ∅ .

Proof. (1) ⇒ (2) It can be easily verified that (R, B) b ◦ (L, C) = (H, B ∩ C) , where H is a function from B ∩ C to P (S) defined by H (α) = R (α) L (α) for α ∈ B ∩ C. Also, we have (R, B) ∩ (L, C) = (G, B ∩ C) ,where G is a function from B ∩ C to P (S) such that G (α) = R (α) ∩ L (α) for α ∈ B ∩ C. 14

Because B∩C ⊆ B∩C and also we have R (α) L (α) ⊆ R (α)∗F (α) ⊆ R (α) and R (α) L (α) ⊆ F (α) L (α) ⊆ L (α). These imply that R (α) L (α) ⊆ R (α) ∩ L (α) for all α ∈ B, α ∈ C. Hence (H, B ∩ C) ⊆ (G, B ∩ C). In proving the reverse inclusion, we suppose that u ∈ R (α)∩L (α) . Since u ∈ F (α) and F (α) is regular, there exists v ∈ F (α) such that u = uvu. Now u ∈ R (α) , vu ∈ L (α) , and hence u = uvu ∈ R (α) L (α). This shows that (R, B) ∩ (L, C) ⊆ (R, B) b ◦ (L, C). Therefore (G, B ∩ C) ⊆ (H, B ∩ C). Thus, (R, A) b ◦ (L, B) = (R, A) ∩ (L, B) . (2) ⇒ (1) Suppose that B = C = F (α) and R is a function which maps from F (α) to P (F (α)) defined by R (u) = uF 1 (α) , for all u ∈ F (α) and L is a function from F (α) to P (F (α)) defined by L (u) = F 1 (α) u, for all u ∈ F (α) . Then (R, F (α)) is a soft right ideal and (L, F (α)) is a soft left ideal over F (α). Thus, by hypothesis, u ∈ R (u) ∩ L (u) = R (u) L (u) = uF 1 (α) F 1 (α) u ⊆ uF 1 (α) u. This shows that F (α) is regular. ¤ Conclusion: In the literature, regular semigroups have been characterized by using its ideals, quasi-ideals, bi-ideals, fuzzy ideals, fuzzy quasi-ideals , fuzzy bi-ideals, rough ideals etc. In this paper, we have shown that we can use some very general tools, namely the soft ideals,soft bi-ideals and soft quasi-ideals to characterize the regular semigroups. Thus, the results concerning bi-ideals ,quasi-ideals ,fuzzy quasi-ideals and rough ideals etc. in the literature can be unified under the broad framework “ soft sets”. It is expected that the theory of soft algebraic structure will have some applications in soft computing and information science.

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