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Soft Intersection Interior Ideals, Quasi-ideals and Generalized Bi-Ideals; A New Approach to Semigroup Theory II 2 ˘ A SLIHAN S EZGIN S EZER1 , NAIM C¸ A GMAN 3 ¨ AND A KIN O SMAN ATAG UN 1
Department of Mathematics, Amasya University, 05100 Amasya, Turkey E-mail:
[email protected] Department of Mathematics, Gaziosmanpas¸a University, Tokat, Turkey 3 Department of Mathematics, Bozok University, 66100 Yozgat, Turkey
2
Received: June 14, 2012. Accepted: May 18, 2013.
In this paper, we define soft intersection interior ideals, quasi-ideals and generalized bi-ideals of semigroups, obtain their properties and give the interrelations of them. Moreover, we characterize regular, intra-regular, completely regular, weakly regular and quasi-regular semigroups in terms of these ideals. Keywords: Soft set, soft intersection interior ideals, soft intersection quasiideals, soft intersection generalized bi-ideals, regular semigroups
1 INTRODUCTION The complexities of modeling uncertain data in economics, engineering, environmental science, sociology, medical science and many other fields can not be successfully dealt with by classical methods. While probability theory, fuzzy set theory,rough set theory, vague set theory and the interval mathematics are useful approaches to describing uncertainty, each of these theories has its inherent difficulties. Consequently, Molodtsov [1] proposed a completely new approach for modeling vagueness and uncertainty, which is called soft set theory. Since then, especially soft set operations, have undergone tremendous studies. Maji et al. [2] presented some definitions on soft sets and Ali et al. [3] introduced several operations of soft sets and Sezgin and Atag¨un
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[4] studied on soft set operations, as well. Soft set theory have found its wideranging applications in the mean of algebraic structures such as groups [5,6], semirings [7], rings [8], BCK/BCI-algebras [9–11], BL-algebras [12], fuzzy soft semigroups [13], near-rings [14], soft substructures and union soft substructures [15, 16]. In [17], by introducing soft intersection-union product and soft characteristic function, she made a new approach to the classical ring theory via soft set theory, with the concept of soft union rings. The concepts of soft union rings, soft union left (right, two-sided), soft union bi-ideals and soft union semiprime ideals were defined and studied with respect to soft set operations and soft intersection-union product. Moreover, regular, regular duo, intraregular and strongly regular rings were characterized by the properties of these soft union ideals, respectively. And in [18], Sezer et al. made a new approach to semigroup theory via soft set theory with the concept of soft intersection semigroups. They defined soft intersection semigroups, soft intersection left (right, two-sided) ideals, bi-ideals and soft semiprime ideals of semigroups and obtained their basic properties. Moreover, they characterized regular, intra-regular, completely regular, weakly regular and quasi-regular semigroups by the properties of these ideals. And in [?], the author gave a thorough description of the possible intersection ideals in rings of two first-order operators in two or three variables. Furthermore, it was shown how software may be applied for solving concrete problems. This paper is a following study of [18] and reads as follows: In Section 2, we remind some basic definitions about soft sets, semigroups and soft intersection ideals of semigroups defined in [18]. In Section 3, soft intersection interior ideals of semigroups, Section 4, soft intersection quasi-ideals of semigroups and Section 5, soft intersection generalized bi-ideals of semigroups are defined and their basic properties with respect to soft set operations are obtained. Moreover, the interrelations of them are investigated in each section. In Section 6, regular semigroups, Section 7, intra-regular semigroups, Section 8, completely regular semigroups, Section 9, weakly regular semigroups and section 10, quasi-regular semigroups are characterized by the properties of these ideals.
2 PRELIMINARIES In this section, we recall some notions relevant to semigroups and soft sets. A semigroup S is a nonempty set with an associative binary operation. Throughout this paper, S denotes a semigroup. A nonempty subset A of S is called a right ideal of S if AS ⊆ A and is called a left ideal of S if S A ⊆ A. By
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two-sided ideal (or simply ideal), we mean a subset of S, which is both a left and right ideal of S. A subsemigroup X of S is called a bi-ideal of S if X S X ⊆ X. A nonempty subset A of S is called an interior ideal of S if S AS ⊆ A. A nonempty subset Q of S is called a quasi-ideal of S if Q S ∩ S Q ⊆ Q. We denote by L[a](R[a], J [a], B[a]Q[a], I [a]), the principal left (right, two-sided, bi-ideal, quasi-ideal, interior ideal) of a semigroup S generated by a ∈ S, that is, L[a] R[a] J [a] B[a] Q[a] I [a]
= = = = = =
{a} ∪ Sa, {a} ∪ aS, {a} ∪ Sa ∪ aS ∪ SaS {a} ∪ {a 2 } ∪ aSa {a} ∪ (aS ∩ Sa) {a} ∪ {a 2 } ∪ SaS.
A subset P of a semigroup S is called semiprime if ∀a ∈ S, a 2 ∈ P implies that a ∈ P. A semilattice is a structure S = (S, .), where “.” is an infix binary operation, called the semilattice operation, such that “." is associative, commutative and idempotent. A semigroup S is called regular if for every element a of S there exists an element x in S such that a = axa or equivalently a ∈ aSa. For all undefined concepts and notions about semigroups, we refer to [19–21]. Definition 1. ( [1, 23]) A soft set f A over U is a set defined by / A. f A : E → P(U ) such that f A (x) = ∅ if x ∈ Here f A is also called an approximate function. A soft set over U can be represented by the set of ordered pairs f A = {(x, f A (x)) : x ∈ E, f A (x) ∈ P(U )}. It is clear to see that a soft set is a parametrized family of subsets of the set U . Note that the set of all soft sets over U will be denoted by S(U ). Definition 2. [23] Let f A , f B ∈ S(U ). Then, f A is called a soft subset of f B ˜ f B , if f A (x) ⊆ f B (x) for all x ∈ E. and denoted by f A ⊆ Definition 3. [23] Let f A , f B ∈ S(U ). Then, union of f A and f B , denoted by f A ∪ f B , is defined as f A ∪ f B = f A∪ B , where f A∪ B (x) = f A (x) ∪ f B (x) for all x ∈ E.
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Definition 4. [23] Let f A , f B ∈ S(U ). Then, intersection of f A and f B , denoted by f A ∩ f B , is defined as f A ∩ f B = f A∩ B , where f A∩ B (x) = f A (x) ∩ f B (x) for all x ∈ E. Definition 5. [23] Let f A , f B ∈ S(U ). Then, ∧-product of f A and f B , denoted by f A ∧ f B , is defined as f A ∧ f B = f A∧B , where f A∧B (x, y) = f A (x) ∩ f B (y) for all (x, y) ∈ E × E. Definition 6. [24] Let f A and f B be soft sets over the common universe U and be a function from A to B. Then, soft image of f A under , denoted by ( f A ), is a soft set over U by (( f A ))(b) =
{ f A (a) | a ∈ A and (a) = b}, ∅,
if −1 (b) = ∅, other wise
for all b ∈ B. And soft pre-image (or soft inverse image) of f B under , denoted by −1 ( f B ), is a soft set over U by ( −1 ( f B ))(a) = f B ((a)) for all a ∈ A. Definition 7. [25] Let f A be a soft set over U and α ⊆ U . Then, upper αinclusion of f A , denoted by U( f A ; α), is defined as U( f A : α) = {x ∈ A | f A (x) ⊇ α}. Definition 8. [18] Let S be a semigroup and f S and g S be soft sets over the common universe U . Then, soft intersection product f S ◦ g S is defined by ( f S ◦ g S )(x) =
∅,
x=yz { f S (y)
∩ g S (z)},
if ∃y, z ∈ S such that x = yz, other wise
for all x ∈ S. Theorem 1. [18] Let f S , g S , h S ∈ S(U ). Then, ( f S ◦ g S ) ◦ h S = f S ◦ (g S ◦ h S ). ∪( f S ◦ h S ) and ( f S f S ◦ (g S ∪h S ) = ( f S ◦ g S ) ∪g S ) ◦ h S = ( f S ◦ h S ) ∪(g S ◦ h S ). ∩( f S ◦ h S ) and ( f S (iii) f S ◦ (g S ∩h S ) = ( f S ◦ g S ) ∩g S ) ◦ h S = ( f S ◦ h S ) ∩(g S ◦ h S ). ˜ S , then f S ◦ h S ⊆g ˜ S ◦ h S and h S ◦ f S ⊆h ˜ S ◦ gS . (iv) If f S ⊆g (i) (ii)
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Definition 9. [18] Let X be a subset of S. We denote by S X the soft characteristic function of X and define as S X (x) =
U, ∅,
if x ∈ X, if x ∈ / X
Theorem 2. [18] Let X and Y be nonempty subsets of a semigroup S. Then, the following properties hold: ˜ Y. (i) If X ⊆ Y , then S X ⊆S (ii) S X ∩SY = S X ∩Y , S X ∪SY = S X ∪Y . (iii) S X ◦ SY = S X Y . Definition 10. [18] Let S be a semigroup and f S be a soft set over U . Then, f S is called a soft intersection semigroup of S, if f S (x y) ⊇ f S (x) ∩ f S (y) for all x, y ∈ S. Definition 11. [18] A soft set over U is called a soft intersection left (right) ideal of S over U if f S (ab) ⊇ f S (b) ( f S (ab) ⊇ f S (a)) for all a, b ∈ S. A soft set over U is called a soft intersection two-sided ideal (soft intersection ideal) of S if it is both soft intersection left and soft intersection right ideal of S over U . Definition 12. [18] A soft intersection semigroup f S over U is called a soft intersection bi-ideal of S over U if f S (x yz) ⊇ f S (x) ∩ f S (z) for all x, y, z ∈ S. For the sake of brevity, soft intersection semigroup, soft intersection right (left, two-sided) ideal and soft intersection bi-ideal are abbreviated by S I semigroup, S I -right (left, two sided) ideal and S I -bi-ideal, respectively. It is easy to see that if f S (x) = U for all x ∈ S, then f S is an S I -semigroup (right ideal, left ideal, ideal, bi-ideal) of S over U . We denote such a kind of S I -semigroup (right ideal, left ideal, ideal, bi-ideal) by S [18].
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Lemma 1. Let f S be any S I -semigroup over U . Then, we have the followings: (i) S ◦ S⊆ S. (If S is regular, then S ◦ S = S). (ii) f S ◦ S⊆S and S ◦ f S ⊆S. ∪ S = S and f S ∩ S = fS. (iii) f S Theorem 3. [18] Let X be a nonempty subset of a semigroup S. Then, X is a subsemigroup (left, right, two-sided ideal, bi-ideal) of S if and only if S X is an S I -semigroup (left, right, two-sided ideal, bi-ideal) of S. Proposition 1. [18] Let f S be a soft set over U . Then, fS. f S is an S I -semigroup over U if and only if f S ◦ f S ⊆ fS ( fS ◦ S ◦ fS⊆ f S is an S I -left (right) ideal of S over U if and only if f S ). S⊆ f S and f S ◦ S◦ (iii) f S is an S I -bi-ideal of S over U if and only if f S ◦ f S ⊆ fS. fS⊆ (i) (ii)
Theorem 4. [18] For a semigroup S the following conditions are equivalent: 1. 2.
S is regular. f S ◦ gS = f S ∩g S for every S I -right ideal f S of S over U and S I -left ideal g S of S over U .
3 SOFT INTERSECTION INTERIOR IDEALS OF SEMIGROUPS In this section, we define soft intersection interior ideals of semigroups, study their basic properties with respect to soft operations and soft intersection product. Definition 13. Let f S be an S I -semigroup over U . Then, f S is called a soft intersection interior ideal of S, if f S (xay) ⊇ f S (a) for all x, y, a ∈ S. For the sake of brevity, soft intersection interior ideal is abbreviated by S I -interior ideal in what follows.
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Example 1. Consider the semigroup S = {a, b, c, d} with the following operation table: . a b c d a a a a a b a a a a c a a b a d a a b b Let U = D3 = {< x, y >: x 3 = y 2 = e, x y = yx 2 } = {e, x, x 2 , y, yx, yx 2 } be the universal set and f S be soft set over U such that f S (a) = {e, x, y, yx}, f S (b) = {e}, f S (c) = {e, x}, f S (d) = {e}. Then, one can easily show that f S is an S I -interior ideal over U . Now, let U = S3 be the symmetric group. If we construct a soft set g S over U such that g S (a) = =
{(1), (123)}, g S (b) = {(1), (12), (123)}, g S (c) {(1), (12), (123)}, g S (d) = {(123)}
then, since g S (dcb) = g S (a) g S (c), g S is not an S I -interior ideal over U . It is easy to see that if f S (x) = U for all x ∈ S, then f S is an S I -interior ideal over U . We denote such a kind of S I -interior ideal by S. It is obvious S(x) = U for all x ∈ S. It is known that a nonempty subset that S = S S , i.e. A of S is an interior ideal if and only if S AS ⊆ A. It is natural to extend this property to S I -interior ideals with the following: Theorem 5. Let f S be a soft set over U . Then, f S is an S I -interior ideal over U if and only if fS S ◦ fS ◦ S⊆ Proof. Assume that f S is an S I -interior ideal over U . Let a ∈ S. If ( S ◦ fS ◦ S)(a) = ∅, then it is obvious that fS. ( S ◦ fS ◦ S)(a) ⊆ f S (a), thus S ◦ fS ◦ S⊆
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Otherwise, if there exist elements y, z, u and v of S such that x = yz and y = uv, then, since f S is an S I -interior ideal of S, we have f S (x) = f S (yz) = f S (uvz) ⊇ f S (v). Thus, S)(x) = ( S ◦ fS ◦
(( S ◦ fS) ◦ S)(x) = { ( S ◦ f S )(y) ∩ S(z)} x=yz
=
x=yz
=
{(
(U ∩ f S (v))) ∩ U }
y=uv
{(
x=yz
=
( S(u) ∩ f S (v))) ∩ S(z)}
y=uv
x=yz
⊆
{(
(U ∩ f S (uvz))) ∩ U }
y=uv
f S (x)
f S . Note that if y = uv, then ( S⊆ S ◦ f S )(y) = ∅, and so ( S◦ Thus, S ◦ fS ◦ S)(x) = ∅ ⊆ f S (x). fS ◦ f S . Let x, a, y be any element of S. S⊆ Conversely, assume that S ◦ fS ◦ Then, we have: f S (xay)
⊇ =
( S ◦ fS ◦ S)(xay) {( S ◦ f S )( p) ∩ S(q)} xay= pq
⊇
( S ◦ f S )(xa) ∩ S(y)
=
( S ◦ f S )(xa) ∩ U { S(m) ∩ f S (n)}
=
xa=mn
⊇ S(x) ∩ f S (a) =
f S (a)
Hence, f S is an S I -interior ideal over U . This completes the proof. Theorem 6. Let X be a nonempty subset of a semigroup S. Then, X is an interior ideal of S if and only if S X is an S I -interior ideal of S.
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Proof. Assume that X is an interior ideal of S, that is, S X S ⊆ X . Then, we have: S X (by T heor em 2 − (i) and T heor em 2 − (iii)) S = SS ◦ S X ◦ SS = SS X S ⊆ S ◦ SX ◦
and so S X is an S I -interior ideal over U by Theorem 5. Conversely, let x ∈ S X S and S X be an S I -interior ideal of S. Then, by Theorem 5, S ◦ SX ◦ S)(x) = (S S ◦ S X ◦ S S )(x) = S S X S (x) = U S X (x) ⊇ ( and so x ∈ X . Thus, S X S ⊆ X and X is an interior ideal of S. It is obvious that every two-sided ideal of S is an interior ideal of S. Moreover, we have the following: Proposition 2. Let f S be a soft set over U . Then, if f S is an S I -ideal of S over U , f S is an S I -interior ideal of S over U . Proof. Let f S be an S I -ideal of S over U and x, y ∈ S. Then, f S (x yz) = f S ((x y)z) ⊇ f S (x y) ⊇ f S (y). Hence, f S is an S I -interior ideal of S over U . The following example shows that the converse of this property does not hold in general: Example 2. Consider the S I -interior ideal f S in Example 1. Since f S (dc) = f S (b) f S (c) f S is not an S I -left ideal of S, that is, it is not an S I -ideal of S. The following theorem shows that the converse of Proposition 2 holds for a regular semigroup. Theorem 7. Let f S be a soft set over U , where S is a regular semigroup. Then, the following conditions are equivalent: 1. 2.
f S is an S I -ideal of S over U . f S is an S I -interior ideal of S over U .
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Proof. By Proposition 2, it suffices to prove that (2) implies (1). Assume that (2) holds. Let a, b be any elements of S. Then, since S is regular, there exist elements x and y in S such that a = axa and b = byb. Then, since f S is an interior ideal of S, we have f S (ab) = f S ((axa)b) = f S ((ax)a(b)) ⊇ f S (a), and f S (ab) = f S (a(byb)) = f S ((a)b(yb)) ⊇ f S (b). This means that f S is an S I -ideal of S. Thus, (2) implies (1). Proposition 3. Let S be a monoid and f S be a soft set over U . Then, f S is an S I -ideal of S if and only if f S is an S I -interior ideal of S. Proof. The necessity is clear by Proposition 2. Now let us show the sufficiency. For x, y ∈ S, f S (x y) = f S (x ye) ⊇ f S (y) and f S (x y) = f S (ex y) ⊇ f S (x). Thus, f S is an S I -ideal of S. It is known that a semigroup S is called left (right) simple if it contains no proper left (right) ideal of S and is called simple if it contains no proper ideal. Definition 14. [18] A semigroup S is called soft left (right) simple if every S I -left (right) ideal of S is a constant function and is called soft simple if every S I -ideal of S is a constant function. Theorem 8. [22] For a semigroup S, the following conditions are equivalent: 1. 2.
S is simple. S is soft simple.
Theorem 9. For a semigroup S, the following conditions are equivalent: 1. 2. 3.
S is simple. S is soft simple. Every S I -interior ideal of S is constant function.
Proof. The equivalence of (1) and (2) follows from Theorem 8. Assume that (2) holds. Let f S be any S I -interior ideal of S and a and b be any element
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of S. Then, since S is simple, it follows from [ [21], Lemma 1.3.9] that there exist elements x and y in S such that a = xby. Then, since f S is an S I -interior ideal of S, we have f S (a) = f S (xby) ⊇ f S (b). One can similarly show that f S (b) ⊇ f S (a). Thus, f S (a) = f S (b). Since a and b be any elements of S, f S is a constant function and so (2) implies (3). Since every S I -ideal of S is an S I -interior ideal of S, it is clear that (3) implies (2). Definition 15. [18] A soft set f S over U is called soft semiprime if for all a ∈ S, f S (a) ⊇ f S (a 2 ). Proposition 4. Let f S be a soft semiprime S I -interior ideal of a semigroup S. Then, f S (a n ) ⊇ f S (a n+1 ) for all positive integers n. Proof. Let n be any positive integer. Then, f S (a n ) ⊇ f S (a 2n ) ⊇ f S (a 4n ) = f S (a 3n−2 a n+1 a) ⊇ f S (a n+1 ). Definition 16. [21] A semigroup S is called archimedean if for all a, b ∈ S, there exists a positive integer n such that a n ∈ SbS. Proposition 5. Let S be an archimedean semigroup. Then, every soft semiprime S I -interior ideal of S is a constant function. Proof. Let f S be any soft semiprime S I -interior ideal of S and a, b any element of S. Since S is archimedean, there exist elements x, y ∈ S such that a n = xby. Thus, we have f S (a) ⊇ f S (a n ) = f S (xby) ⊇ f S (b). Similarly, we have f S (b) ⊇ f S (a) and so f S (a) = f S (b). Since a and b be any elements of S, f S is a constant function. Proposition 6. Let f S and f T be S I -interior ideals over U . Then, f S ∧ f T is an S I -interior ideal of S × T over U .
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Proof. Let (x1 , y1 ), (x2 , y2 ), (x3 , y2 ) ∈ S × T . Then, f S∧T ((x1 , y1 )(x2 , y2 )(x3 , y3 )) = = ⊇ =
f S∧T (x1 x2 x3 , y1 y2 y3 ) f S (x1 x2 x3 ) ∩ f T (y1 y2 y3 ) f S (x2 ) ∩ f T (y2 ) f S∧T (x2 , y2 )
Therefore, f S ∧ f T is an S I -interior ideal of S × T over U . Definition 17. Let f S , f T be soft intersection interior ideals of S and T , respectively. Then, the product of soft intersection interior ideal f S and f T is defined as f S × f T = f S×T , where f S×T (x, y) = f S (x) × f T (y) for all (x, y) ∈ S × T . Proposition 7. If f S and f T are S I -interior ideals of S and T , respectively, then so is f S × f T of S × T over U × U . Proof. By Definition 17, let f S × f T = f S×T , where f S×T (x, y) = f S (x) × f T (y) for all (x, y) ∈ S × T . Then, for all (x1 , y1 ), (x2 , y2 ), (x3 , y3 ) ∈ S × T , f S×T ((x1 , y1 )(x2 , y2 )(x3 , y3 ))
=
f S×T (x1 x2 x3 , y1 y2 y3 , )
=
f S (x1 x2 x3 ) × f T (y1 y2 y3 )
⊇
f S (x2 ) × f T (y2 )
=
f S×T (x2 , y2 )
Hence, f S × f T = f S×T is an S I -interior ideal of S × T over U × U . Proposition 8. If f S and h S are S I -interior ideals of S over U , then so is f S ∩h S . Proof. Let x, y, z ∈ S. Then, we have ( f S ∩h S )(x yz)
=
f S (x yz) ∩ h S (x yz)
f S (y) ∩ h S (y) = ( f S ∩h S )(y) ⊇
Therefore, f S ∩h S is an S I -interior ideal of S over U .
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Proposition 9. Let f S be a soft set over U and α be a subset of U such that α ∈ I m( f S ), where I m( f S ) = {α ⊆ U : f S (x) = α, f or x ∈ S}. If f S is an S I -interior ideal over U , then U( f S ; α) is an interior ideal of S. Proof. Since f S (x) = α for some x ∈ S, then ∅ = U( f S ; α) ⊆ S. Let a ∈ U( f S ; α) and x, y ∈ S, then f S (a) ⊇ α. We need to show that xay ∈ U( f S ; α) for all a ∈ U( f S ; α) and x, y ∈ S. Since f S is an S I -interior ideal of S over U , it follows that f S (xay) ⊇ f S (a) ⊇ α implying that xay ∈ U( f S ; α). Thus, the proof is completed. Definition 18. Let f S be an S I -interior ideal over U . Then, the interior ideals U( f S ; α) are called upper α-interior ideals of f S . Proposition 10. Let f S be a soft set over U , U( f S ; α) be upper α-interior ideals of f S for each α ⊆ U . Then, f S is an S I -interior ideal of S over U . Proof. Let x, y ∈ S and f S (a) = α1 for some a ∈ S. It follows that a ∈ U( f S ; α1 ). Since U( f S ; α) is an interior ideal of S for all α ⊆ U , it follows that xay ∈ U( f S ; α1 ). Hence, f S (xay) ⊇ α1 = f S (a). Thus, f S is an S I -interior ideal of S over U . In order to show Proposition 9, we have the following example: Example 3. Consider the semigroup in Example 1. Define a soft set f S over U = S3 such that f S (a) = {(1), (12), (13), (23)}, f S (b) = {(1)}, f S (c) = {(1), (13), (123)}, f S (d) = {(1), (12)}. Then, one can easily show that f S is an S I -interior ideal of S over U . By taking into account I m( f S ), we have: U( f S ; {(1), (12), (13), (23)}) = {a}, U( f S ; {(1), (13), (123)}) = {a, c}, U( f S ; {(1), (12)}) = {a, d}, U( f S ; {(1)}) = {a, b, c, d} One can easily show that {a}, {a, c}, {a, d} and {a, b, c, d} are interior ideals of S.
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In order to show Proposition 10, we have the following example: Example 4. Consider the semigroup in Example 1. Define a soft set f S over U = Z such that f S (a) = {0, 1, 2, 3}, f S (b) = {0, 2, 3}, f S (c) = {0, 2}, f S (d) = {0}. By taking into account I m( f S ) = {{0, 1, 2, 3}, {0, 2, 3}, {2, 3}, {2}} we have: ⎧ {a, b, c, d}, ⎪ ⎪ ⎨ {a, b, c}, U( f S ; α) = {a, b}, ⎪ ⎪ ⎩ {a},
if α if α if α if α
= {0} = {0, 2} = {0, 2, 3} = {0, 1, 2, 3}
Since {a}, {a, b}, {a, b, c} and {a, b, c, d} are interior ideals of S, f S is an S I -interior ideal of S over U . Now we define a soft set h S over U = Z such that h S (a) = {0, 2, 4}, h S (b) = {0, 2, 4, 6}, h S (c) = {0, 4}, h S (d) = {0, 2, 4, 6, 8}. By taking into account I m( f S ) = {{0, 2, 4, 6, 8}, {0, 2, 4, 6}, {0, 2, 4}, {0, 4}}, we have ⎧ {a, b, c, d}, ⎪ ⎪ ⎨ {a, b, d}, U( f S ; α) = {b, d}, ⎪ ⎪ ⎩ {d},
if α if α if α if α
= {0, 4} = {0, 2, 4} = {0, 2, 4, 6} = {0, 2, 4, 6, 8}
Since S{b, d}S = {a} ∈ / {b, d}, {b, d} is not an interior ideal of S. Moreover, since; h S (dbb) = h S (a) h S (b) h S is not an S I -interior ideal of S over U . Proposition 11. Let f S and f T be soft sets over U and be a semigroup isomorphism from S to T . If f S is an S I -interior ideal of S over U , then ( f S ) is an S I -interior ideal of T over U .
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Proof. Let t1 , t2 , t3 ∈ T . Since is surjective, then there exist s1 , s2 , s3 ∈ S such that (s1 ) = t1 , (s2 ) = t2 , (s3 ) = t3 . Then, (( f S ))(t1 t2 t3 ) = { f S (s) : s ∈ S, (s) = t1 t2 t3 } = { f S (s) : s ∈ S, s = −1 (t1 t2 t3 )} = { f S (s) : s ∈ S, s = −1 ((s1 s2 s3 )) = s1 s2 s3 } = { f S (s1 s2 s3 ) : si ∈ S, (si ) = ti , i = 1, 2, 3} ⊇ ( { f S (s2 ) : s2 ∈ S, (s2 ) = t2 }) = (( f S ))(t2 ) Hence, ( f S ) is an S I -interior ideal of S over U . Proposition 12. Let f S and f T be soft sets over U and be a semigroup homomorphism from S to T . If f T is an S I -interior ideal of T over U , then −1 ( f T ) is an S I -interior ideal of S over U . Proof. Let s1 , s2 , s3 ∈ S. Then, ( −1 ( f T ))(s1 s2 s3 ) = = ⊇ =
f T ((s1 s2 s3 )) f T ((s1 )(s2 )(s3 )) f T ((s2 )) ( −1 ( f T ))(s2 )
Hence, −1 ( f T ) is an S I -interior ideal over U .
4 SOFT INTERSECTION QUASI-IDEALS OF SEMIGROUPS In this section, we define soft intersection quasi-ideals and study their properties as regards soft set operations, soft intersection product and certain kinds of soft intersection ideals. Definition 19. A soft set over U is called a soft intersection quasi-ideal of S over U if fS. S) ∩( S ◦ f S )⊆ ( fS ◦ For the sake of brevity, soft intersection quasi-ideal is abbreviated by S I quasi-ideal in what follows.
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Proposition 13. Every S I -quasi ideal of S is an S I -semigroup of S. Proof. Let f S be any S I -quasi-ideal of S. Then, since f S ⊆ S, fS ◦ S. S ◦ f S and f S ◦ f S ⊆ fS ◦ fS⊆ Hence, fS ( ∩( f S ◦ S)⊆ S ◦ f S ) fS ◦ fS⊆ as f S is an S I -quasi-ideal of S. That is, f S is an S I -semigroup over U . Proposition 14. Each one-sided S I -ideal of S is an S I -quasi-ideal of S. f S , we have S ◦ fS⊆ Proof. Let f S be an S I -left ideal of S. Then, since fS. ∩( f S ◦ S)⊆ S ◦ fS⊆ ( S ◦ f S ) Thus, f S is an S I -quasi-ideal of S. The converse of Proposition 14 does not hold in general as shown in the following example: Example 5. Consider the semigroup S = {0, a, b, c} with the following operation table: . 0 a b c
0 0 0 0 0
a 0 a 0 c
b 0 b 0 0
c 0 0 0 0
Let U = D3 = {< x, y >: x 3 = y 2 = e, x y = yx 2 } = {e, x, x 2 , y, yx, yx 2 } be the universal set and f S be the soft set over U such that f S (0) = {e, x, y, yx}, f S (a) = {e, x, y, yx}, f S (b) = {yx}, f S (c) = {yx}. Then, one can easily show that f S is an S I -quasi-ideal of S, but since f S (ca) = f S (c) f S (a) f S is not an S I -left ideal and so S I -ideal of S.
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Proposition 15. Every S I -quasi-ideal of S is an S I -bi-ideal of S. Proof. Let f S be an S I -quasi-ideal of S. Then, fS ◦ fS ◦ fS ◦ S ◦ fS⊆ S ◦ fS⊆ S ◦ S⊆ S S ◦ S ◦ fS⊆ S ◦ f S and f S ◦ ( f S , as f S is an S I -quasi-ideal of S. and so f S ◦ S ◦ f S ) S ◦ fS⊆ ∩( f S ◦ S)⊆ Hence, fS. fS ◦ S ◦ fS⊆ Thus, f S is an S I -bi-ideal of S. The converse of Proposition 15 does not hold in general as shown in the following example: Example 6. Consider the semigroup S = {0, 1, 2, 3} with the following table: . 0 1 2 3
0 0 0 0 0
1 0 0 0 0
2 0 0 0 1
3 0 0 1 2
Let U = S3 be the universal set and f S be the soft set over U such that f S (0) = {(1), (12), (123), (132)}, f S (1) = {(1), (132)}, f S (2) = {(1), (123), (132)}, f S (3) = {(1)}. Then, f S is an S I -bi-ideal of S. In fact; ( f S ◦ f S )(0) = {(1), (12), (123), (132)}, ( f S ◦ f S )(1) = {(1)}, ( f S ◦ f S )(2) = {(1)}, ( f S ◦ f S )(3) = ∅ f S . Moreover; and so f S ◦ f S ⊆ S ◦ f S )(0) = {(1), (12), (123), (132)}, ( f S ◦ S ◦ f S )(1) = ( fS ◦ S ◦ f S )(2) = ∅, ( f S ◦ S ◦ f S )(3) = ∅ ∅, ( f S ◦ fS. S ◦ fS⊆ and so f S ◦
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However, f S is not an S I -quasi-ideal of S. In fact; ( fS ◦ S)(1) = {(1), (123), (132)} ( S ◦ f S )(1) = {(1), (123), (132)} and so (( f S ◦ S) ∩( S ◦ f S ))(1) = {(1), (123), (132)} f S (1) = {(1), (132)}. Hence, f S is not an S I -quasi-ideal of S. The following theorem shows that the converse of Proposition 15 holds for a regular semigroup. Theorem 10. Let f S be a soft set over U , where S is a regular semigroup. Then, the following conditions are equivalent: 1. 2.
f S is an S I -quasi-ideal of S over U . f S is an S I -bi-ideal of S over U .
Proof. By Proposition 15, it suffices to prove that (2) implies (1). Assume S ◦ f S (resp. f S ◦ S) is an that (2) holds. Let f S be an S I -bi-ideal of S. Then, S I -left (resp. right) ideal of S. By it follows by Theorem 4 that fS ( fS ◦ S) ∩( S ◦ fS) = ( fS ◦ S) ◦ ( S ◦ f S ) = f S ◦ ( S ◦ S) ◦ f S = f S ◦ S ◦ fS⊆
since f S is an S I -bi-ideal of S. Thus, f S is an S I -quasi-ideal of S and (2) implies (1). Theorem 11. Let X be a nonempty subset of a semigroup S. Then, X is a quasi-ideal of S if and only if S X is an S I -quasi-ideal of S over U . Proof. Assume that X is a quasi-ideal of S, that is, X S ∩ S X ⊆ X . Then, we have: (S X ◦ S) ∩( S ◦ S X ) = (S X ◦ S S ) ∩(S S ◦ S X ) = S X S ∩S X S = S X S∩X S ⊆S X (since X S ∩ S X ⊆ X ) This means that S X is a quasi-ideal of S.
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Conversely, let S X be an S I -quasi-ideal of S over U and x ∈ X S ∩ S X . Thus, S) ∩( S ◦ S X ))(x) = ((S X ◦ S S ) ∩(S S ◦ S X ))(x) = S X (x) ⊇ ((S X ◦ (S X S ∩S S X )(x) = S X S∩S X = U and so x ∈ X . Thus, X S ∩ S X ⊆ X and X is a quasi-ideal of S. Theorem 12. Let f S and g S be any S I -quasi-ideal of S over U . Then, the soft intersection product f S ◦ g S is an S I -bi-ideal of S over U . Proof. Let f S be an S I -quasi-ideal of S. Then, fS ◦ fS ◦ S ◦ fS⊆ S ◦ fS⊆ S ◦ S⊆ S fS ◦ S ◦ S ◦ fS⊆ S ◦ f S and f S ◦ f S , as f S is an S I -quasi-ideal of S. ( S ◦ fS⊆ ∩( f S ◦ S)⊆ and so f S ◦ S ◦ f S ) Hence, fS. S ◦ fS⊆ fS ◦ Then, we have ( f S ◦ f S ◦ gS S ◦ f S ) ◦ gS ⊆ ( f S ◦ gS ) ◦ ( f S ◦ gS ) = ( f S ◦ gS ◦ f S ) ◦ gS ⊆ and ( f S ◦ ( S ◦ ( f S ◦ g S ) = ( f S ◦ (g S ◦ S) ◦ f S ) ◦ g S ⊆ S ◦ S) ◦ f S )◦ ( f S ◦ gS ) ◦ g S ⊆( f S ◦ S ◦ f S ) ◦ g S ⊆ f S ◦ g S . Thus, it follows that f S ◦ g S is an S I -bi-ideal of S over U . Corollary 1. Let S be a regular semigroup and f S , g S be any S I -quasi-ideals of S over U . Then, f S ◦ g S is an S I -quasi-ideal of S over U . Proof. Follows from Theorem 10 and Theorem 12. Proposition 16. Let f S be any S I -right ideal of S and g S be any S I -left ideal of S. Then, f S ∩g S is an S I -quasi-ideal of S. Proof. Let f S be any S I -right ideal of S and g S be any S I -left ideal of S. Then, ( f S ◦ f S S) ∩( S ◦ ( f S S) ∩( S ◦ g S )⊆ (( f S ∩g S ) ◦ ∩g S ))⊆ ∩g S .
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Proposition 17. Let S be a regular semigroup, f S be any S I -right ideal of S and g S be any S I -left ideal of S. Then, f S ◦ g S is an S I -quasi-ideal of S. Proof. Let S be a regular semigroup and f S be an S I -right ideal of S and g S be an S I -left ideal of S. It follows by Proposition 16 that f S ∩g S is an S I -quasi-ideal of S. Since S is regular, f S ◦ gS = f S ∩g S by Theorem 4. Thus, f S ◦ g S is an S I -quasi-ideal of S. Proposition 18. Let f S and g S be any S I -quasi-ideals of S. Then, f S ∩g S is an S I -quasi-ideal of S. Proof. Let f S and g S be any S I -quasi-ideals of S. Then, ( f S ◦ fS ∩g S ) ◦ ∩g S ))⊆ S) ∩( S ◦ ( f S S) ∩( S ◦ f S )⊆ (( f S and (g S ◦ gS . S) ∩( S ◦ ( f S S) ∩( S ◦ g S )⊆ (( f S ∩g S ) ◦ ∩g S ))⊆ Thus, f S S) ∩( S ◦ ( f S (( f S ∩g S ) ◦ ∩g S ))⊆ ∩g S . Proposition 19. Let f S be a soft set over U and α be a subset of U such that α ∈ I m( f S ). If f S is an S I -quasi-ideal of S over U , then U( f S ; α) is a quasi-ideal of S. Proof. Since f S (x) = α for some x ∈ S, then ∅ = U( f S ; α) ⊆ S. Let a ∈ (S · U( f S ; α) ∩ U( f S ; α) · S). Then, there exist x, y ∈ U( f S ; α) and s, r ∈ S such that a = sx = yr. Thus, f S (x) ⊇ α and f S (y) ⊇ α. Since ( S ◦ f S )(a) = {
{ S(c) ∩ f S (d)}
a=cd
⊇ S(s) ∩ f S (x) =
f S (x)
⊇ α
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and ( fS ◦ S)(a) =
{
{ f S (n) ◦ S(m)}
a=nm
⊇
S(r ) f S (y) ∩
=
f S (y)
⊇
α
Since f S is an S I -quasi-ideal of S, we have f S (a) ⊇ ( S ◦ f S )(a) ∩ ( f S ◦ S)(a) ⊇ α, thus a ∈ U( f S ; α). This shows that U( f S ; α) is a quasi-ideal of S. Definition 20. Let f S be an S I -quasi-ideal of S over U . Then, the quasiideals U( f S ; α) are called upper α-quasi-ideals of f S . Proposition 20. ( [19], p.85) A semigroup S is a group if and only if it contains no proper quasi-ideal. Theorem 13. Let S be a semigroup without zero. Then, the following conditions are equivalent: 1. 2.
S is a group. Every S I -quasi-ideal of S is a constant function.
Proof. First assume that (1) holds. Let f S be any S I -quasi-ideal of S and let a and b be any element of S. Then, since S is a group, there exist elements x and y of S such that a = bx = yb. Since f S is an S I -quasi-ideal of S, we have f S (a)
⊇ (( f S ◦ S) ∩( S ◦ f S ))(a) = ( fS ◦ S)(a) ∩ ( S ◦ f S )(a) S( p) ∩ f S (q)} = { f S ( p) ∩ S(q)} ∩ { a= pq
a= pq
S(x)) ∩ ( S(y) ∩ f S (b)) ⊇ ( f S (b) ∩ =
f S (b)
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It can be seen in a similar way that f S (b) ⊇ f S (a). Thus, f S (a) = f S (b) for every pair of elements a and b of S. This implies that f S is a constant function. Thus, (1) implies (2). Conversely, assume that (2) holds. If S is not a group, then it follows by Proposition 20 that S contains a proper quasi-ideal Q of S. Since Q is not empty and S Q is a constant function by assumption, we have S Q (x) = U for all x ∈ S. This implies that S ⊆ Q and so S = Q, which contradicts the condition (2). Thus, S is a group and so (2) implies (1). Proposition 21. Let f S be any S I -quasi-ideal of a commutative semigroup S and a be any element of A. Then, f S (a n ) ⊆ f S (a n+1 ) for every positive integer n. Proof. For any positive integer n, we have ( fS ◦ S)(a n+1 )
=
( f S (x) ∩ S(y))
a n+1 =x y
⊇
f S (a n ) ∩ S(a)
=
f S (a n ).
Similarly, ( S ◦ f S )(a n+1 ) ⊇ f S (a n ). Thus, since f S is an S I -quasi-ideal of S f S (a n+1 )
⊇ (( f S ◦ S) ∩( S ◦ f S ))(a n+1 ) = ( fS ◦ S)(a n+1 ) ∩ ( S ◦ f S ))(a n+1 )
This completes the proof.
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⊇
f S (a n ) ∩ f S (a n )
=
f S (a n )
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5 SOFT INTERSECTION GENERALIZED BI-IDEALS OF SEMIGROUPS In this section, we define soft intersection generalized bi-ideals and study their properties as regards soft set operations and soft intersection product. Definition 21. A soft set over U is called a soft intersection generalized biideal of S over U if f S (x yz) ⊇ f S (x) ∩ f S (z) for all x, y, z ∈ S. For the sake of brevity, soft intersection generalized bi-ideal is abbreviated by S I -generalized bi-ideal in what follows. It is clear that every S I -bi-ideal of S is an S I -generalized bi-ideal of S, but the converse of this statement does not hold in general. This is shown by the following example: Example 7. Consider the semigroup S in Example 1. Define the soft set f S over U = Z5 such that f S (a) = {0, 2, 4}, f S (b) = {0}, f S (c) = {0, 2}, f S (d) = {0}. Then, one can easily show that f S is an S I -generalized bi-ideal of S over U . However since f S (cc) = f S (b) f S (c) ∩ f S (c) f S is not an S I -bi-ideal of S. However the following theorem shows that the converse of this holds for a regular semigroup. Proposition 22. Every S I -generalized bi-ideal of a regular semigroup is an S I -bi-ideal of S. Proof. Let f S be an S I -generalized bi-ideal of S and let a and b be any element of S. Then, since S is regular, there exists an element x ∈ S such that b = bxb. Thus, we have f S (ab) = f S (a(bxb)) = f S (a(bx)b) ⊇ f S (a) ∩ f S (b). This implies that f S is an S I -semigroup of S, and so f S is an S I -bi-ideal of S.
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It is known that a nonempty subset A of S is a generalized bi-ideal of S if and only if AS A ⊆ A. It is natural to extend this property to S I -generalized bi-ideal with the following: Theorem 14. Let f S be a soft set over U . Then, f S is an S I -generalized bi-ideal of S over U if and only if fS S ◦ fS⊆ fS ◦ Proof. First assume that f S is an S I -generalized bi-ideal of S over U . Let s ∈ fS, S ◦ f S )(s) = ∅, then it is clear that f S ◦ S ◦ fS⊆ S. In the case, when ( f S ◦ Otherwise, there exist elements x, y, p, q ∈ S such that s = x y and x = pq Then, since f S is an S I -generalized bi-ideal of S over U , we have: f S (s) = f S (x y) = f S (( pq)y) ⊇ f S ( p) ∩ f S (y) Thus, we have ( fS ◦ S ◦ f S )(s) = [( f S ◦ S) ◦ f S ](s) [( f S ◦ S)(x) ∩ f S (y)] = s=x y
=
[(
s=x y
=
( f S ( p) ∩ S(q)) ∩ f S (y)]
x= pq
[(
s=x y
=
( f S ( p) ∩ U ) ∩ f S (y)]
x= pq
( f S ( p) ∩ f S (y))
s= pqy
⊆
f S ( pqy)
s= pqy
=
f S (x y)
=
f S (s)
f S . Here, note that if x = pq, then ( f S ◦ Hence, f S ◦ S ◦ fS⊆ S)(x) = ∅, and so, ( f S ◦ S ◦ f S )(s) = ∅ ⊆ f S (s).
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f S . Let x, y, z ∈ S and s = x yz. Conversely, assume that f S ◦ S ◦ fS⊆ f S , we have S ◦ fS⊆ Then, since f S ◦ f S (x yz)
=
f S (s)
S ◦ f S )(s) ⊇ ( fS ◦ S) ◦ f S ](s) = [( f S ◦ [( f S ◦ S)(m) ∩ f S (n)] = s=mn
S)(x y) ∩ f S (z) ⊇ ( fS ◦ ( f S ( p) ∩ S(q)] ∩ f S (z) = [ x y= pq
⊇ (( f S (x) ∩ S(y)) ∩ f S (z) = (( f S (x) ∩ U ) ∩ f S (z) =
f S (x) ∩ f S (z)
Thus, f S is an S I -generalized bi-ideal of S over U . This completes the proof. Theorem 15. Let X be a nonempty subset of a semigroup S. Then, X is a generalized bi-ideal of S if and only if S X is an S I -generalized bi-ideal of S over U . Proof. Assume that X is a generalized bi-ideal of S, that is, X S X ⊆ X . Then, we have: S X (since X S X ⊆ X ) SX ◦ S ◦ S X = S X ◦ SS ◦ S X = S X S X ⊆ This means that S X is a generalized bi-ideal of S. Conversely, let S X be an S I -generalized bi-ideal of S over U and x ∈ X S X . Thus; S X (x) ⊇ (S X ◦ S ◦ S X )(x) = (S X ◦ S S ◦ S X )(x) = S X S X (x) = U and so x ∈ X . Thus, X S X ⊆ X and X is a generalized bi-ideal of S. It is known that every left (right, two sided) ideal of a semigroup S is a bi-ideal of S. Moreover, we have the following: Theorem 16. Every S I -left (right, two sided) ideal of a semigroup S over U is an S I -generalized bi-ideal of S over U .
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Proof. Let f S be an S I -left (right, two sided) ideal of S over U and x, y, z ∈ S. Then, f S (x yz) ⊇ f S ((x y)z) ⊇ f S (z) ⊇ f S (x) ∩ f S (z) Thus, f S is an S I -generalized bi-ideal of S. Theorem 17. Let f S be any soft subset of a semigroup S and gS be any S I bi-ideal of S over U . Then, the soft intersection products f S ◦ g S and g S ◦ f S are S I -generalized bi-ideals of S over U . Proof. We show the proof for f S ◦ g S . ( f S ◦ gS ) ◦ S ◦ ( f S ◦ gS ) = ⊆ ⊆
f S ◦ (g S ◦ ( S ◦ f S ) ◦ gS ) S ◦ gS ) f S ◦ (g S ◦ f S ◦ gS
Thus, it follows that f S ◦ g S is an S I -generalized bi-ideal of S over U . It can be seen in a similar way that g S ◦ f S is an S I -generalized bi-ideal of S over U . This completes the proof. Proposition 23. Let f S and f T be S I -generalized bi-ideals over U . Then, f S ∧ f T is an S I -generalized bi-ideal of S × T over U . Proposition 24. If f S and f T are S I -generalized bi-ideals of S over U , then so is f S × f T of S × T over U × U . Proposition 25. If f S and h S are two S I -generalized bi-ideals of S over U , then so is f S ∩h S of S over U . Proposition 26. Let f S be a soft set over U and α be a subset of U such that α ∈ I m( f S ). If f S is an S I -generalized bi-ideal of S over U , then U( f S ; α) is a generalized bi-ideal of S. Definition 22. If f S is an S I -generalized bi-ideal of S over U , then generalized bi-ideals U( f S ; α) are called upper α generalized bi-ideals of f S . Proposition 27. Let f S be a soft set over U , U( f S ; α) be upper α generalized bi-ideals of f S for each α ⊆ U and I m( f S ) be an ordered set by inclusion. Then, f S is an S I -generalized bi-ideal of S over U .
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Proposition 28. Let f S and f T be soft sets over U and be a semigroup isomorphism from S to T . If f S is an S I -generalized bi-ideal of S over U , then so is ( f S ) of T over U . Proposition 29. Let f S and f T be soft sets over U and be a semigroup homomorphism from S to T . If f T is an S I -generalized bi-ideal of T over U , then so is −1 ( f T ) of S over U . 6 REGULAR SEMIGROUPS In this section, we characterize a regular semigroup in terms of S I -interior ideals, S I -quasi-ideals and S I -generalized-bi-ideals. Proposition 30. [26] For a semigroup S, the following conditions are equivalent: 1. 2. 3.
S is regular. R L = R ∩ L for every right ideal R and left ideal L of S. AS A = A for every quasi-ideal A of S.
Theorem 18. For a semigroup S, the following conditions are equivalent: 1. 2. 3. 4.
S is regular. fS = fS ◦ S ◦ f S for every S I -generalized bi-ideal f S of S over U . S ◦ f S for every S I -bi-ideal f S of S over U . fS = fS ◦ S ◦ f S for every S I -quasi-ideal f S of S over U . fS = fS ◦
Proof. First assume that (1) holds. Let f S be any S I -generalized bi-ideal f S of S over U and s be any element of S. Then, since S is regular, there exists an element x ∈ S such that s = sxs. Thus, we have; ( fS ◦ S ◦ f S )(s) = [( f S ◦ S) ◦ f S ](s) [( f S ◦ S)(a) ∩ f S (b)] = s=ab
⊇ ( fS ◦ S)(sx) ∩ f S (s) {( f S (m) ∩ S(n)} ∩ f S (s) = sx=mn
S(x)) ∩ f S (s) ⊇ ( f S (s) ∩ = ( f S (s) ∩ U ) ∩ f S (s) =
f S (s)
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f S . Since f S is an S I -generalized bi-ideal of S, and so, we have f S ◦ S ◦ fS⊇ f S . Thus, f S ◦ S ◦ fS⊆ S ◦ f S = f S which means that (1) implies (2). fS ◦ (2) implies (3) and (3) implies (4) is obvious. Assume that (4) holds. In order to show that S is regular, we need to illustrate that AS A = A for every quasi-ideal A of S. Let A be any quasi-ideal of S. Then, since AS A ⊆ A(SS) ∩ (SS)A ⊆ AS ∩ S A ⊆ A, AS A ⊆ A. Therefore, it is enough to show that A ⊆ AS A. Let a ∈ A. Then, by Theorem 15, the soft characteristic function S A of A is an S I -quasi-ideal of S. Thus, we have; S ◦ S A )(a) = (S A )(a) = U (S AS A )(a) = (S A ◦ S S ◦ S A )(a) = (S A ◦ which means that a ∈ AS A. Thus, A ⊆ AS A and so A = AS A. It follows by Proposition 9 that S is regular, so (4) implies (1). Theorem 19. For a semigroup S the following conditions are equivalent: 1. 2. 3. 4. 5. 6. 7.
S is regular. ∩g S = f S ◦ g S ◦ f S for every S I -quasi-ideal f S of S and S I -ideal g S f S of S over U . f S ∩g S = f S ◦ g S ◦ f S for every S I -quasi-ideal f S of S and S I -interior ideal g S of S over U . f S ∩g S = f S ◦ g S ◦ f S for every S I -bi-ideal f S of S and S I -ideal g S of S over U . ∩g S = f S ◦ g S ◦ f S for every S I -bi-ideal f S of S and S I -interior f S ideal g S of S over U . f S ∩g S = f S ◦ g S ◦ f S for every S I -generalized bi-ideal f S of S and S I ideal g S of S over U . f S ∩g S = f S ◦ g S ◦ f S for every S I -generalized bi-ideal f S of S and S I interior ideal g S of S over U .
Proof. First assume that (1) holds. Let f S be any S I -generalized bi-ideal and g S be any S I -interior ideal of S over U . Then, fS ◦ fS S ◦ fS⊆ f S ◦ gS ◦ f S ⊆ and gS S⊆ f S ◦ gS ◦ f S ⊆ S ◦ gS ◦
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f S f S ◦ g S ◦ f S holds, let s be so f S ◦ g S ◦ f S ⊆ ∩g S . To show that f S ∩g S ⊆ any element of S. Since S is regular, there exists an element x in S such that s = sxs (s = sx(sxs)) Since g S is an S I -interior ideal of S, we have ( f S ◦ g S ◦ f S )(s) =
[ f S ◦ (g S ◦ f S )](s) [ f S (m) ∩ (g S ◦ f S )(n)] = s=mn
=
f S (s) ∩ (g S ◦ f S )(xsxs) [g S (y) ∩ f S (z)]} f S (s) ∩ {
=
f S (s) ∩ (g S (xsx) ∩ f S (s))
⊇
f S (s) ∩ g S (s) ∩ f S (s)
⊇
f S (s) ∩ g S (s) ( f S ∩g S )(s)
⊇
xsxs=yz
=
f S ◦ g S ◦ f S . Thus we obtain that f S so we have f S ∩g S ⊆ ∩g S = f S ◦ g S ◦ f S , hence (1) implies (7). It is clear that (7) implies (5), (5) implies (3), and that (3) implies (2). Also, (7) implies (6), (6) implies (4) and (4) implies (2) is obvious. Assume that (2) holds. In order to show that S is regular, it is enough to S ◦ f S for all S I -quasi-ideal f S of S over U by Theorem show that f S = f S ◦ 18. Since S is an S I -ideal of S, we have f S = f S S ◦ fS. ∩ S = fS ◦ Thus, S is regular and (2) implies (1). This completes the proof. Theorem 20. For a semigroup S the following conditions are equivalent: 1. 2. 3. 4.
S is regular. f S ∩g S ⊆ f S ◦ g S for every S I -quasi-ideal f S of S and S I -left ideal g S of S over U . f S ∩g S ⊆ f S ◦ g S for every S I -bi-ideal f S of S and S I -left ideal g S of S over U . ∩g S ⊆ f S ◦ g S for every S I -generalized bi-ideal f S of S and S I -left f S ideal g S of S over U .
Proof. First assume that (1) holds. Let f S be any S I -generalized bi-ideal and g S be any S I -left ideal of S over U . Let s be any element of S. Then, since S
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is regular, there exists an element x in S such that s = sxs. Thus, we have
( f S ◦ g S )(s) =
( f S (a) ∩ g S (b))
s=ab
⊇
f S (s) ∩ g S (xs)
⊇
( f S (s) ∩ g S (s)) ( f S ∩g S )(s)
=
f S Thus, f S ◦ g S ⊇ ∩g S . Hence, we obtain that (1) implies (4). It is clear that (4) implies (3), (3) implies (2). Assume that (2) holds. f S ◦ g S always holds for every S I -right ideal of S is an ∩g S ⊇ Since f S S I -quasi-ideal of S, we have f S ∩g S = f S ◦ g S for every S I -right ideal f S and S I -left ideal g S of S. Thus, it follows by Theorem 4 that S is regular and (2) implies (1). Theorem 21. For a semigroup S the following conditions are equivalent: 1. 2. 3. 4.
S is regular. h S ◦ f S ◦ g S for every S I -right ideal h S , every S I -quasih S ∩ f S ∩g S ⊆ ideal f S and every S I -left ideal g S of S. h S ◦ f S ◦ g S for every S I -right ideal h S , every S I -bi-ideal h S ∩ f S ∩g S ⊆ f S and every S I -left ideal g S of S. h S ◦ f S ◦ g S for every S I -right ideal h S , every S I h S ∩ f S ∩g S ⊆ generalized bi-ideal f S and every S I left-ideal g S of S.
Proof. Assume that (1) holds. Let h S , f S and g S be any S I -right ideal, S I generalized bi-ideal and S I -left ideal of S, respectively. Let a be any element of S. Since S is regular, there exists an element x in S such that a = axa. Hence, we have: (h S ◦ f S ◦ g S )(a) =
[h S ◦ ( f S ◦ g S )](a) [h S (y) ∩ ( f S ◦ g S )(z)] = a=yz
⊇ =
h S (ax) ∩ ( f S ◦ g S )(a) [ f S ( p) ∩ g S (q)]} h S (ax) ∩ {
⊇
h S (a) ∩ ( f S (a) ∩ g S (xa))
⊇
h S (a) ∩ ( f S (a) ∩ g S (a)) (h S ∩ f S ∩g S )(a)
a= pq
=
h S so we have h S ◦ f S ◦ g S ⊇ ∩ f S ∩g S . Thus, (1) implies (2).
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It is clear that (4) implies (3), (3) implies (2). Assume that (2) holds. Let h S and g S be any S I -right ideal and S I -left ideal of S, respectively. It is obvious that h S h S ◦ gS ⊆ ∩g S . Since S itself is an S I -quasi-ideal of S, by assumption we have: h S ◦ h S ◦ g S . h S S ◦ gS ⊆ ∩g S = h S ∩ S ∩g S ⊆ h S ◦ g S for every S I -right ideal h S and S I -left ideal It follows that h S ∩g S ⊆ g S of S. It follows by Theorem 4 that S is regular. Hence, (2) implies (1). This completes the proof. Proposition 31. [18] Let S be a regular semigroup. Then, every S I -left (right) ideal of S is idempotent. Proposition 32. Let S be a regular semigroup and f S be an S I -quasi-ideal of S. Then, ∩( f S ◦ S) = f S . ( S ◦ f S ) f S . Thus, Proof. Let f S be any S I -quasi-ideal of S. Then, ( S ◦ f S ) ∩( f S ◦ S)⊆ ( ∩( f S ◦ S). One can easily show that it suffices to show that f S ⊆ S ◦ f S ) f S ∪( S ◦ f S ) is an S I -left ideal of S. In fact, S ◦ ( f S S ◦ f S ) ∪( S ◦ ( S ◦ f S )) = ( S ◦ f S ) ∪(( S ◦ S) ◦ f S ) = ∪( S ◦ f S )) = ( (S ◦ f S ) ∪(S ◦ f S ) = S ◦ f S ⊆ f S ∪(S ◦ f S ). Since S is regular, every S I -left (right) ideal of S is idempotent by Proposition 31. Thus, we have f S ∪( S ◦ fS) fS⊆
=
∪( S ◦ f S )] ◦ [ f S ∪( S ◦ f S )] [ f S
=
∪( S ◦ f S )] ◦ f S } ∪( S ◦ f S )] ◦ ( ∪{[( f S S ◦ f S )} {[( f S
=
{(( f S ◦ f S ) ∪(( S ◦ f S ) ◦ f S )} ∪{(( f S ◦ ( S ◦ f S )) ∪(( S ◦ f S ) ◦ ( S ◦ f S )}
=
∪( S ◦ ( f S ◦ f S ))} ∪{(( f S ◦ ( S ◦ f S )) ∪(( S ◦ f S )2 } {(( f S ◦ f S )
⊆
fS) (( S ◦ f S ) ∪( S ◦ f S )) ∪(( S ◦ ( S ◦ f S )) ∪( S ◦ f S )2 )(since f S ◦ f S ⊆
⊆
∪( S ◦ f S ) ∪( S ◦ f S ) ∪( S ◦ f S )2 (since( S ◦ ( S ◦ f S ))⊆ S ◦ fS) ( S ◦ f S )
⊆
S ◦ fS
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fS ◦ that is to say f S ⊆ S. Thus, S ◦ f S . Similarly, one can show that f S ⊆ ( ∩( f S ◦ S) and so, fS⊆ S ◦ f S ) ( S ◦ f S ) ∩( f S ◦ S) = f S . Theorem 22. Let f S be a soft set and S be a regular semigroup. Then, the following conditions are equivalent: 1. 2.
f S is an S I -quasi-ideal of S. f S may be presented in the form f S = g S ◦ h S , where g S is an S I -right ideal and h S is an S I -left ideal of S.
Proof. Assume that (1) holds. Since S is regular, it follows by Theorem 18 that f S = f S ◦ S ◦ f S , where f S is an S I -quasi-ideal of S. Thus, fS = fS ◦ S ◦ f S = f S ◦ ( S ◦ S) ◦ f S = ( f S ◦ S) ◦ ( S ◦ fS) Since f S ◦ S is an S I -right ideal of S and S ◦ f S is an S I -right ideal of S, (1) implies (2). Conversely, assume that f S = g S ◦ h S , where g S is an S I -right ideal and h S is an S I -left ideal of S. Then, by Proposition 16, g S ◦ h S is an S I -quasiideal of S. Proposition 33. Let S be a regular semigroup and f S be an S I -quasi-ideal of S. Then, ( f S )2 = ( f S )3 . Proof. Let S be a regular semigroup and f S be an S I -quasi-ideal of S. Then, by Corollary 1, ( f S )2 is an S I -quasi-ideal of S and by Theorem 18, ( f S )2 = ( f S )2 ◦ S ◦ ( f S )2 = f S ◦ f S ◦ S ◦ fS ◦ fS = fS ◦ ( fS ◦ S ◦ f S ) ◦ f S = f S ◦ f S ◦ f S = ( f S )3 Example 8. Consider the semigroup S = {0, x, 1} defined by the following table: . 0 x 1
0 0 0 0
x 0 x x
1 0 x 1
Note that S is a regular semigroup. Let f S be a soft set over S such that f S (0) = {0, x, 1}, f S (x) = {0, x}, f S (1) = {x}. Then, one can easily show
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that f S is an S I -quasi-ideal of S over U . Moreover ( f S ◦ f S )(0) = {0, x, 1}, ( f S ◦ f S )(x) = {0, x}, ( f S ◦ f S )(1) = {0, x, 1} and so ( f S )3 (0) = (( f S )2 (0) ∩ f S (0)) ∪ (( f S )2 (0) ∩ f S (x)) ∪ (( f S )2 (0) ∩ f S (1))∪ (( f S )2 (x) ∩ f S (0)) ∪ (( f S )2 (1) ∩ f S (0)) = {0, x, 1} = ( f S )2 (0) ( f S )3 (x) = (( f S )2 (x) ∩ f S (x)) ∪ (( f S )2 (x) ∩ f S (1)) ∪ (( f S )2 (1) ∩ f S (x)) = {0, x} = ( f S )2 (x) ( f S )3 (1) = (( f S )2 (1) ∩ f S (1)) = {x} = ( f S )2 (1). Thus, ( f S )3 = ( f S )2 . 7 INTRA-REGULAR SEMIGROUPS In this section, we characterize an intra-regular semigroup in terms of S I interior ideals, S I -quasi-ideals and S I -generalized-bi-ideals. A semigroup S is called intra-regular if for every element a of S there exist elements x and y in S such that a = xa 2 y Proposition 34. For a soft set f S of an intra-regular semigroup S, the following conditions are equivalent: 1. 2.
f S is an S I -ideal of S. f S is an S I -interior ideal of S.
Proof. (1) implies (2) is clear. Assume that (2) holds. Let a and b be any elements of S. Then, since S is intra-regular, there exist elements x, y, u and v in S such that a = xa 2 y and b = ub2 v. Since f S is an S I -interior ideal of S, we have f S (ab) = f S ((xa 2 y)b) = f S ((xa)a(yb)) ⊇ f S (a) and f S (ab) = f S (a(ub2 v)) = f S ((au)b(bv)) ⊇ f S (b)
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Hence, f S is an S I -ideal of S and (2) implies (1). Theorem 23. For a semigroup S, the following conditions are equivalent: 1. 2. 3.
S is intra-regular. Every S I -interior ideal of S is soft semiprime. f S (a) = f S (a 2 ) for all S I -interior ideal of S and for all a ∈ S.
Proof. First assume that (1) holds. Let f S be any S I -interior ideal of S and a any element of S. Since S is intra-regular, there exist elements x and y in S such that a = xa 2 y. Thus, f S (a) = f S (xa 2 y) ⊇ f S (xa 2 ) ⊇ f S (a 2 ) = f S (a(xa 2 y)) = f S ((ax)a(ay)) ⊇ f S (a)
so, we have f S (a) = f S (a 2 ). Hence, (1) implies (3). The equivalence of (2) and (3) follows from Proposition 15. Assume that (3) holds. It is known that I [a 2 ] is an interior-ideal of S. Thus, the soft characteristic function S I [a 2 ] is an S I -interior ideal of S. Since a 2 ∈ I [a 2 ], we have; S I [a 2 ] (a) = S I [a 2 ] (a 2 ) = U Thus, a ∈ I [a 2 ] = {a 2 } ∪ {a 4 } ∪ Sa 2 S ⊆ Sa 2 S. Here, S is intra-regular. Thus, (3) implies (1). This completes the proof. Theorem 24. Let S be an intra-regular semigroup. Then, for every S I interior ideal f S of S, f S (ab) = f S (ba) for all a, b ∈ S. Proof. Let f S be an S I -ideal of an intra-regular semigroup S. Then, by Theorem 23, we have; f S (ab) = f S ((ab)2 ) = f S (a(ba)b) ⊇ f S (ba) = f S ((ba)2 ) = f S (b(ab)a) ⊇ f S (ab)
so, we have f S (ab) = f S (ba). This completes the proof. Theorem 25. A semigroup S is simple if and only if it is intra-regular and archimedean.
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Proof. First assume that S is simple. Let f S be any S I -interior ideal of S. Then, it follows by Theorem 9 that f S is a constant function. Hence, any element a of S, f S (a) = f S (a 2 ). Thus, it follows by Theorem 23 that S is intra-regular. Since S is simple, there exist elements x, y ∈ S such that a 1 = a = xby ( [21], Lemma I,3.9). Therefore, S is archimedean. Conversely, assume that S is intra-regular and archimedean. Let f S be any S I -interior ideal of S. Since S intra-regular, f S is soft semiprime by Theorem 23. Since S is archimedean, f S is a constant function by Proposition 5. Thus, it follows by Theorem 9 that S is simple. Theorem 26. [27] A semigroup S is regular and intra-regular if and only if every quasi-ideal of S is idempotent. Theorem 27. For a semigroup S, the following conditions are equivalent: 1. 2.
S is both regular and intra-regular. f S ◦ f S = f S for every S I -quasi-ideal f S of S. (That is, every S I -quasiideal of S is idempotent). 3. f S ◦ f S = f S for every S I -bi-ideal f S of S. (That is, every S I -bi-ideal of S is idempotent). ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -quasi-ideals f S and g S of S. ∩g S ⊆ 4. f S ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -quasi-ideal f S and S I -bi5. f S ∩g S ⊆ ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -quasi-ideal f S and for every 6. f S ∩g S ⊆ S I -generalized bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -bi-ideal f S and for every S I 7. f S ∩g S ⊆ quasi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -bi-ideals f S and g S of S. 8. f S ∩g S ⊆ ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -bi-ideal f S and for every S I 9. f S ∩g S ⊆ generalized bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -generalized-bi-ideal f S and 10. f S ∩g S ⊆ for every S I -quasi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -generalized-bi-ideal f S and 11. f S ∩g S ⊆ for every S I -bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -generalized bi-ideals f S and 12. f S ∩g S ⊆ g S of S. Proof. First assume that (1) holds. In order to show that (12) holds, let f S and g S be S I -generalized bi-ideals of S and a ∈ S. Since S is intra-regular, there exist elements y and z in S such that a = ya 2 z for every element a of S. Thus, a = axa = (axa)xa = ax(ya 2 z)xa = (ax ya)(azxa)
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Then, we have ( f S ◦ g S )(a) =
( f S (b) ∩ g S (c))
a=bc
⊇
f S (a(x y)a) ∩ g S (a(zx)a)
f S (a) ∩ g S (a) = ( f S ∩g S )(a) ⊇
f S ◦ g S . This shows that (1) implies (12). and so we have f S ∩g S ⊆ It is obvious that (12) implies (11), (11) implies (10), (10) implies (4), (4) implies (2) and (12) implies (9), (9) implies (8), (8) implies (7), (7) implies (4), (12) implies (6), (6) implies (5), (5) implies (4) and (8) implies (3) and (3) implies (2). Assume that (2) holds. Let Q be quasi-ideal of S and a be any element of Q. Then, Q Q ⊆ Q always holds. We show that Q ⊆ Q Q. Since Q is a quasi-ideal of S, the soft characteristic function S Q is an S I -quasi-ideal of S. So we have; (S Q Q )(a) = (S Q ◦ S Q )(a) = S Q (a) = U which means that a ∈ Q Q. Thus, Q ⊆ Q Q and so Q = Q Q = Q 2 . It follows that Q is both regular and intra-regular, so (2) implies (1) by Theorem 36. Theorem 28. [18] For a semigroup S the following conditions are equivalent: 1. 2.
S is intra-regular. g S ◦ f S for every S I -right ideal f S of S and S I -left ideal g S of gS ∩ fS⊆ S over U .
Theorem 29. For a semigroup S the following conditions are equivalent: 1. 2. 3. 4. 5. 6. 7.
S is both regular and intra-regular. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -right ideal ∩g S ⊆ f S S I -left ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -right ideal f S ∩g S ⊆ S I -quasi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -right ideal f S ∩g S ⊆ S I -bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -right ideal f S ∩g S ⊆ S I -generalized bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -left ideal f S f S ∩g S ⊆ quasi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -left ideal f S f S ∩g S ⊆ bi-ideal g S of S.
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( f S ◦ g S ) f S ∩(g S ◦ f S ) for every S I -left ideal f S and for every S I ∩g S ⊆ generalized bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -quasi-ideals f S and g S of S. f S ∩g S ⊆ ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -quasi-ideal f S and for every f S ∩g S ⊆ S I -bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -quasi-ideal f S and for every f S ∩g S ⊆ S I -generalized bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -bi-ideals f S and g S of S. f S ∩g S ⊆ ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -bi-ideal f S and for every S I f S ∩g S ⊆ generalized bi-ideal g S of S. ( f S ◦ g S ) ∩(g S ◦ f S ) for every S I -generalized bi-ideals f S and f S ∩g S ⊆ g S of S.
Proof. Assume that (1) holds. Let f S and g S be any S I -generalized bi-ideals f S ◦ g S . Moreover, we of S. Then, it follows by Theorem 28 that f S ∩g S ⊆ have gS ◦ f S . f S ∩g S = g S ∩ fS⊆ ( f S ◦ g S ) Thus, we have f S ∩(g S ◦ f S ) and so (1) implies (14). It is obvi∩g S ⊆ ous that (14) implies (13), (13) implies (12), (12) implies (9), (9) implies (6) and (6) implies (2) and (14) implies (11), (11) implies (10), (10) implies (9) and (14) implies (8), (8) implies (7), (7) implies (6) and (14) implies (5), (5) implies (4), (4) implies (3) and (3) implies (2). Assume that (2) holds. Let f S and g S be any S I -right ideal and S I -left ideal of S, respectively. Then, gS ◦ f S ( f S ◦ g S ) ∩(g S ◦ f S )⊆ ∩g S ⊆ f S It follows by Theorem 28 that S is intra-regular. On the other hand, f S ◦ gS . ( f S ◦ g S ) ∩(g S ◦ f S )⊆ f S ∩g S ⊆ f S Since f S ◦ g S ⊆ ∩g S always holds, we have f S ◦ g S = f S ∩g S . Thus, it follows by Theorem 4 that S is regular. Thus, (2) implies (1). Theorem 30. For a semigroup S the following conditions are equivalent: 1. 2.
S is both regular and intra-regular. f S ◦ g S ◦ f S for every S I -quasi-ideal f S and for every S I -left f S ∩g S ⊆ ideal g S of S. f S ◦ g S ◦ f S for every S I -quasi-ideal f S and for every S I -right 3. f S ∩g S ⊆ ideal g S of S. f S ◦ g S ◦ f S for every S I -quasi-ideals f S and g S of S. 4. f S ∩g S ⊆
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f S ◦ g S ◦ f S for every S I -quasi-ideal f S and for every S I -bif S ∩g S ⊆ ideal g S of S. f S ◦ g S ◦ f S for every S I -quasi-ideal f S and for every S I f S ∩g S ⊆ generalized bi-ideal g S of S. f S ◦ g S ◦ f S for every S I -bi-ideal f S and for every S I -left ideal f S ∩g S ⊆ g S of S. f S ◦ g S ◦ f S for every S I -bi-ideal f S and for every S I -right f S ∩g S ⊆ ideal g S of S. f S ◦ g S ◦ f S for every S I -bi-ideal f S and for every S I -quasif S ∩g S ⊆ ideal g S of S. f S ◦ g S ◦ f S for every S I -bi-ideals f S and g S of S. f S ∩g S ⊆ f S ◦ g S ◦ f S for every S I -bi-ideal f S and for every S I f S ∩g S ⊆ generalized bi-ideal g S of S. f S ◦ g S ◦ f S for every S I -generalized bi-ideal f S and for every f S ∩g S ⊆ S I -left ideal g S of S. f S ◦ g S ◦ f S for every S I -generalized bi-ideal f S and for every f S ∩g S ⊆ S I -right ideal g S of S. f S ◦ g S ◦ f S for every S I -generalized bi-ideal f S and for every f S ∩g S ⊆ S I -quasi-ideal g S of S. f S ◦ g S ◦ f S for every S I -generalized bi-ideal f S and for every f S ∩g S ⊆ S I -bi-ideal g S of S. f S ◦ g S ◦ f S for every S I -generalized bi-ideals f S and g S of S. f S ∩g S ⊆
Proof. Assume that (1) holds. Let f S and g S be any S I -generalized bi-ideals of S and a ∈ S. Since S is regular, there exists element x in S such that a = axa. Since S is intra-regular, there exist elements y, z in S such that a = ya 2 z. Thus, we have a = axa = (axa)x(axa) = (ax(yaaz))x((yaaz)xa) = (ax ya)(azx ya)(azxa)
Therefore, we have ( f S ◦ g S ◦ f S )(a) =
[ f S ◦ (g S ◦ f S )](a) [ f S ( p) ∩ (g S ◦ h S )(q)] = a= pq
⊇ =
f S (ax ya) ∩ (g S ◦ h S )((azx yaazxa) [g S (u) ∩ h S (v)]} f S (a) ∩ { azx yaazxa=uv
⊇
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f S (a) ∩ (g S (azx ya) ∩ h S (azxa))
⊇
f S (a) ∩ (g S (a) ∩ f S (a))
=
( f S ∩g S )(a)
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f S ◦ g S ◦ f S . Thus, (1) implies (16). so we have f S ∩g S ⊆ It is obvious that (11) implies (6), (6) implies (5), (5) implies (4), (4) implies (3) and (16) implies (15), (15) implies (14), (14) implies (13), (13) implies (8) and (8) implies (3). Also, (14) implies (12), (12) implies (7) and (7) implies (2) and (16) implies (11), (11) implies (10), (10) implies (9) and (9) implies (8) is obvious. S Assume that (3) holds. Let f S be any S I -quasi-ideal of S. Then, since itself is an S I -right ideal of S, we have f S = f S S ◦ fS ∩ S = fS ◦ It follows by Theorem 20 that S is regular. Let f S and g S be any S I -left ideal and S I -right ideal of S, respectively. Then, since f S is an S I -quasi-ideal of S, we have f S ◦ gS f S ◦ gS ◦ f S ⊆ f S ◦ (g S ◦ f S S)⊆ ∩g S ⊆ It follows by Theorem 28 that S is intra-regular. Thus, (3) implies (1). One can similarly prove that (2) implies (1). Theorem 31. For a semigroup S, the following conditions are equivalent: 1. 2. 3. 4.
S is both regular and intra-regular. f S ◦ g S ◦ h S for every S I -quasi-ideal f S , every S I -right f S ∩g S ∩h S ⊆ ideal g S and every S I -left ideal h S of S. f S ◦ g S ◦ h S for every S I -bi-ideal f S , S I -right ideal g S and f S ∩g S ∩h S ⊆ every S I -left ideal h S of S. f S ◦ g S ◦ h S for every S I -generalized bi-ideal f S , S I -right f S ∩g S ∩h S ⊆ ideal g S and every S I -left ideal h S of S.
Proof. First assume that (1) holds. In order to show that (4) holds, let f S be any S I -generalized bi-ideal, g S be any S I -left ideal and h S be any S I -right ideal of S and a be any element in S. Since S is regular, there exists element x in S such that a = axa. Since S is intra-regular, there exist elements y, z in S such that a = ya 2 z. Thus, we have a
= axa = (axa)x(axa) = (ax(yaaz))x((yaaz)xa) = (ax ya)(azx ya)(azxa)
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Therefore, we have ( f S ◦ g S ◦ h S )(a) = [ f S ◦ (g S ◦ h S )](a) [ f S ( p) ∩ (g S ◦ h S )(q)] = a= pq
=
f S (ax ya) ∩ (g S ◦ h S )(azx yaazxa) [g S (u) ∩ h S (v)]} f S (a) ∩ {
⊇
f S (a) ∩ (g S (azx ya) ∩ h S (azxa))
⊇
azx yaazxa=uv
f S (a) ∩ g S (a) ∩ h S (a) = ( f S ∩g S ∩h S )(a)
⊇
f S ◦ g S ◦ h S . Thus, (1) implies (4). so we have f S ∩g S ∩h S ⊆ It is clear that (4) implies (3) and (3) implies (2). Assume that (2) holds. Let f S and g S be S I -left and S I -right ideal of S, S, itself is an S I -left ideal respectively. Since f S is an S I -quasi-ideal of S and of S, by assumption we have f S ◦ gS f S ◦ gS ◦ f S S = f S ◦ (g S ◦ S)⊆ ∩g S = f S ∩g S ∩ S⊆ Thus, it follows from Theorem 28 that S is intra-regular. Moreover, since S, itself is an S I -right ideal of S, by g S is an S I -quasi-ideal of S and since assumption we have gS ◦ f S gS S ◦ f S = (g S ◦ S) ◦ f S ⊆ ∩ f S = gS ∩ S ∩ f S = gS ◦ gS Since g S ◦ f S ⊆ ∩ f S always holds, gS ◦ f S = gS ∩ fS. It follows from Theorem 4 that S is regular. Thus, (2) implies (1).
8 COMPLETELY REGULAR SEMIGROUPS In this section, we characterize a completely regular semigroups in terms of S I -ideals. An element a of S is called a completely regular if there exists an element x ∈ S such that a = axa and ax = xa
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A semigroup S is called completely regular if every element of S is completely regular. A semigroup is called left (right) regular if for each element a of S, there exists an element x ∈ S such that a = xa 2 (a = a 2 x). Proposition 35. [28] For a semigroup S the following conditions are equivalent: 1. S is completely regular. 2. S is left and right regular, that is, a ∈ Sa 2 and a ∈ a 2 S for all a ∈ S. 3. a ∈ a 2 Sa 2 for all a ∈ S. Theorem 32. For a semigroup S the following conditions are equivalent: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
S is completely regular. Every quasi-ideal of S is semiprime. Every bi-ideal of S is semiprime. Every generalized bi-ideal of S is semiprime. Every S I -quasi-ideal of S is soft semiprime. Every S I -bi-ideal of S is soft semiprime. Every S I -generalized bi-ideal of S is soft semiprime. f S (a) = f S (a 2 ) for every S I -quasi-ideal f S of S and for all a ∈ S. f S (a) = f S (a 2 ) for every S I -bi-ideal f S of S and for all a ∈ S. f S (a) = f S (a 2 ) for every S I -generalized bi-ideal f S of S and for all a ∈ S.
Proof. First assume that (1) holds. Let f S be any S I -generalized bi-ideal of S. Since S is completely regular, there exists an element x ∈ S such that a = a 2 xa 2 . Thus, we have f S (a) = f S (a 2 xa 2 ) ⊇ f S (a 2 ) ∩ f S (a 2 ) = f S (a 2 ) = f S (aa) = f S (a(a 2 xa 2 ) = f S (a(a 2 xa)a) ⊇ f S (a) ∩ f S (a) = f S (a)
and so, f S (a) = f S (a 2 ). Thus (1) implies (10). It is clear that (10) implies (9), (9) implies (8), (8) implies (5) and (10) implies (7), (7) implies (6), (6) implies (5) and that (10) implies (4), (4) implies (3) and (3) implies (2). Assume that (5) holds. Let Q be any quasi-ideal of S and a 2 ∈ Q. Since the soft characteristic function S Q of Q is an S I -quasi-ideal of S, it is soft semiprime by hypothesis. Thus, S Q (a) ⊇ S Q (a 2 ) = U
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Hence, a ∈ Q and so Q is semiprime. Thus (5) implies (2). Finally assume that (2) holds. Let a be any element of S. Then, since the principal ideal Q[a 2 ] generated by a 2 is quasi-ideal and so by assumption semiprime and since a 2 ∈ Q[a 2 ], S Q[a 2 ] (a) = S Q[a 2 ] (a 2 ) = U implying that a ∈ Q[a 2 ] = {a 2 } ∪ {a 4 } ∪ (a 2 S ∩ Sa 2 ). Hence, S is completely regular. Thus (2) implies (1).
9 QUASI-REGULAR SEMIGROUPS In this section, we study a semigroup whose S I -left (right, two-sided) ideals are all idempotent. A semigroup S is called left (right) quasi-regular if every left (right) ideal of S is idempotent, and is called quasi-regular if every left ideal and right ideal of S is idempotent ( [29]). It is easy to prove that S is left (right) quasi-regular if and only if a ∈ SaSa (a ∈ aSaS), this implies that there exist elements x, y ∈ S such that a = xaya (a = axay). Theorem 33. [18] A semigroup S is left (right) quasi-regular if and only if every S I -left (right) ideal is idempotent. Theorem 34. Let S be a semigroup. If f S = ( f S ◦ S)2 ∩( S ◦ f S )2 for every S I -quasi-ideal f S of S, then S is quasi-regular. Proof. Let f S be an S I -quasi-ideal of S. Then, since f S is S I -right ideal of S, we have fS ( f S ◦ fS ◦ fS⊆ fS ◦ fS = ( fS ◦ S)2 S)2 ⊆ S⊆ ∩( S ◦ f S )2 ⊆ and so f S = ( f S )2 . It follows that S is right quasi-regular by Theorem 33. One can similarly show that S is left quasi-regular. This completes proof. Theorem 35. For a semigroup S the following conditions are equivalent: 1. 2.
S is both intra-regular and left quasi-regular. gS ∩h S ∩ f S = g S ◦ h S ◦ f S for every S I -quasi-ideal f S , for every S I -left ideal g S and every S I -right ideal h S of S.
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43
gS ∩h S ∩ f S = g S ◦ h S ◦ f S for every S I -bi-ideal f S , for every S I -left ideal g S and every S I -right ideal h S of S. gS ∩h S ∩ f S = g S ◦ h S ◦ f S for every S I -generalized bi-ideal f S , for every S I -left ideal g S and every S I -right ideal h S of S.
Proof. Assume that (1) holds. Let f S be any S I -generalized bi-ideal, g S be any S I -left ideal and h S be any S I -right ideal of S. Let a be any element of S. Since S is intra-regular, there exist elements x, y ∈ S such that a = xa 2 y. Since S is left quasi-regular, there exist elements u, v ∈ S such that a = uava. Hence a = uava = u(xaay)va = ((ux)a)((a(yv)a) Thus, (g S ◦ h S ◦ f S )(a) = [g S ◦ (h S ◦ f S )](a) [g S ((ux)a)) ∩ (h S ◦ f S )(a(yv)a))] = a=((ux)a)((a(yv)a)
⊇ ⊇
g S ((ux)a)) ∩ (h S ◦ f S )(a(yv)a)) h S (m) ∩ f S (n)) g S (a) ∩ (
⊇
g S (a) ∩ (h S (a(yv)) ∩ f S (a))
(a(yv))a=mn)
⊇
g S (a) ∩ h S (a) ∩ f S (a) = (g S ∩h S ∩ f S )(a) and so g S ◦ h S ◦ f S ⊇ g S ∩h S ∩ f S . Thus, (1) implies (4). It is clear that (4) implies (3) and (3) implies (2). Assume that (2) holds. Let g S be any S I -left ideal and f S be any S I -right ideal of S. Then, since S itself is an S I -right S I -left ideal g S is an S I -quasi-ideal of S, and since ideal of S, we have gS gS ◦ S ◦ gS = gS ◦ gS ⊆ S⊆ ∩ S ∩g S = g S ◦ gS = gS Hence g S = g S ◦ g S . Thus, by Theorem 33, S is left quasi-regular. S itself Now, since S I -right ideal f S is an S I -quasi-ideal of S, and since is an S I -right ideal of S, we have: gS ◦ f S S ◦ fS⊆ ∩ f S = gS ∩ S ∩ f S = gS ◦ gS
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Thus, by Theorem 28, S is intra-regular. Hence (2) implies (1). This completes the proof.
10 WEAKLY REGULAR SEMIGROUPS In this section, we characterize a weakly regular semigroup in terms of certain S I -ideals. A semigroup S is called weakly-regular if for every x ∈ S, x ∈ (x S)2 Theorem 36. [18] For a monoid S, the following conditions are equivalent: 1. 2.
S is weakly regular. f S ◦ g S for every S I -right ideal f S of S and for every S I -ideal ∩g S ⊆ f S g S of S.
Theorem 37. For a monoid S, the following conditions are equivalent: 1. 2. 3.
S is weakly regular. f S ◦ g S ◦ h S for every S I -bi-ideal f S of S, for every S I f S ∩g S ∩h S ⊆ ideal g S of S and for every S I -right ideal h S of S. f S ◦ g S ◦ h S for every S I -quasi-ideal f S of S, for every S I f S ∩g S ∩h S ⊆ ideal g S of S and for every S I -right ideal h S of S.
Proof. First assume that (1) holds. Let x ∈ S. Then, x ∈ (x S)2 . Thus, x = xsxt for some s, t ∈ S. Hence, ( f S ◦ g S ◦ h S )(x) =
[ f S ◦ (g S ◦ h S )](x) [ f S (x) ∩ (g S ◦ h S )(sxt)] = x=xsxt
⊇
f S (x) ∩ {
(g S ( p) ∩ h S (v))}
sxt= pv
⊇
f S (x) ∩ g S (sxs) ∩ h S (xt 2 )
⊇
f S (x) ∩ g S (x) ∩ h S (x) ( f S ∩g S ∩h S )(x)
=
since sxt = s(xsxt)t = (sxs)(xt 2 ). Thus, (1) implies (2). Since every S I -quasi-ideal of S is an S I -bi-ideal of S, (2) implies (3) is obvious.
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Now, assume that (3) holds. Let f S be an S I -right ideal of S, g S be an S. Then, we have S I -ideal of S and let h S = f S ∩g S ∩h S = f S ∩g S ∩ S = f S ∩g S and f S ◦ gS f S ◦ gS ◦ h S = f S ◦ gS ◦ S = f S ◦ (g S ◦ S)⊆ f S ◦ gS ◦ h S ⊆ f S ◦ g S that is, f S f S ◦ gS Then, f S ∩g S = f S ∩g S ∩h S ⊆ ∩g S ⊆ for every S I -right ideal f S of S and S I -ideal g S of S. Thus, S is weakly regular by Theorem 36. Hence (2) implies (1). This completes the proof. Theorem 38. For a monoid S, the following conditions are equivalent: 1. 2. 3.
S is weakly regular. f S ◦ g S for every S I -bi-ideal f S of S and for every S I -ideal g S f S ∩g S ⊆ of S. f S ◦ g S for every S I -quasi-ideal f S of S and for every S I -ideal f S ∩g S ⊆ g S of S.
Proof. Similar to the the proof of Theorem 37.
11 CONCLUSION It is known that the formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics because they are the abstract algebraic underpinning of “memoryless” systems: timedependent systems that start from scratch at each iteration. In applied mathematics, semigroups are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time. The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata. In probability theory, semigroups are associated with Markov processes. Throughout this paper, we have dealt with semigroup theory from a different aspect of study via the soft sets. We have introduced and studied the concepts of soft intersection interior ideals, quasi-ideals and generalized bi-ideals of semigroups. Furthermore, we have characterized regular, intraregular, completely regular, weakly regular and quasi-regular semigroups by the properties of these ideals. Based on these results, some further work can
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be done on the properties of soft intersection semigroups, which may be useful to characterize the semigroups which have a wide-range application fields in mathematics.
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