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Ideal theory in semigroups based on intersectional soft sets.
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The notions of int-soft semigroups and int-soft left (resp., right) ideals are introduced, and several properties are investigated. Using these notions and the notion of inclusive set, characterizations of subsemigroups and left (resp., right) ideals are considered. Using the notion of int-soft products, characterizations of int-soft semigroups and int-soft left (resp., right) ideals are discussed. We prove that the soft intersection of int-soft left (resp., right) ideals (resp., int-soft semigroups) is also int-soft left (resp., right) ideals (resp., int-soft semigroups). The concept of int-soft quasi-ideals is also introduced, and characterization of a regular semigroup is discussed.
Authors:
Seok Zun Song; Hee Sik Kim; Young Bae Jun
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Type:  Journal Article     Date:  2014-06-23
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Title:  TheScientificWorldJournal     Volume:  2014     ISSN:  1537-744X     ISO Abbreviation:  ScientificWorldJournal     Publication Date:  2014  
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Journal ID (nlm-ta): ScientificWorldJournal
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Received Day: 18 Month: 3 Year: 2014
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DOI: 10.1155/2014/136424

Ideal Theory in Semigroups Based on Intersectional Soft Sets
Seok Zun Song1
Hee Sik Kim2* http://orcid.org/0000-0001-5321-5919
Young Bae Jun3
1Department of Mathematics, Jeju National University, Jeju 690-756, Republic of Korea
2Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea
3Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Republic of Korea
Correspondence: *Hee Sik Kim: heekim@hanyang.ac.kr
[other] Academic Editor: Feng Feng

1. Introduction

Molodtsov [1] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties. He pointed out several directions for the applications of soft set theory. At present, works on soft set theory are progressing rapidly. Maji et al. [2] described the application of soft set theory to a decision making problem. Maji et al. [3] also studied several operations on the theory of soft sets.

Çağman and Enginoğlu [4] introduced fuzzy parameterized (FP) soft sets and their related properties. They proposed a decision making method based on FP-soft set theory and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Decision making based on soft sets was further developed in [58]. It is worth noting that soft sets are closely related to many other soft computing models such as rough sets and fuzzy sets. Feng and Li [9] initiated soft approximation spaces and soft rough sets, which extended Pawlak's rough sets. Moreover, Feng [10] considered the application of soft rough approximations in multicriteria group decision making problems. Aktaş and Çağman [11] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. After that, many algebraic properties of soft sets were studied by several researchers (see [1226]). Recently, Feng et al. [27] investigated the relationships among five different types of soft subsets. They also explored free soft algebras associated with soft product operations, showing that soft sets have some nonclassical algebraic properties. In this paper, we introduce the notion of int-soft semigroups and int-soft left (resp., right) ideals. Using these notions, we provide characterizations of subsemigroups and left (resp., right) ideals. Using the notion of inclusive set, we also establish characterizations of subsemigroups and left (resp., right) ideals. Using the notion of int-soft products, we give characterizations of int-soft semigroups and int-soft left (resp., right) ideals. We show that the soft intersection of int-soft left (resp., right) ideals (resp., int-soft semigroups) is also int-soft left (resp., right) ideals (resp., int-soft semigroups). We also introduce the concept of int-soft quasi-ideals and discuss a characterization of a regular semigroup by using the notion of int-soft quasi-ideals.


2. Preliminaries

Let S be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows:

[Formula ID: eq1]
(1) 
    AB={ab∈S ∣ a∈A,b∈B}.

A semigroup S is said to be regular if for every xS there exists aS such that xax = x.

A nonempty subset A of S is called

  1. a subsemigroup of S if AAA, that is, abA for all a, bA,
  2. a left (resp., right) ideal of S if SAA (resp., SAA), that is, xaA (resp., axA) for all xS and aA,
  3. a two-sided ideal of S if it is both a left and a right ideal of S,
  4. a quasi-ideal of S if ASSAA.

A soft set theory was introduced by Molodtsov [1], and Çağman and Enginoğlu [5] provided new definitions and various results on soft set theory.

In what follows, let U be an initial universe set and let E be a set of parameters. Let P(U) denote the power set of U and A, B, C,…⊆E.

Definition 1 (see [1, 5]).

A soft set (α, A) over U is defined to be the set of ordered pairs

[Formula ID: eq2]
(2) 
(α,A):={(x,α(x)):x∈E,α(x)∈P(U)},
where α : EP(U) such that α(x) = if xA.

The function α is called approximate function of the soft set (α, A). The subscript A in the notation α indicates that α is the approximate function of (α, A).

Definition 2 (see [28]).

Assume that E has a binary operation ↪. For any nonempty subset A of E, a soft set (α, A) over U is said to be intersectional over U if it satisfies

[Formula ID: EEq2.1]
(3) 
(x↪y∈A⟹α(x)∩α(y)⊆α(x↪y))                  (∀x,y∈A).

For a soft set (α, A) over U and a subset γ of U, the γ-inclusive set of (α, A), denoted by iA(α; γ), is defined to be the set

[Formula ID: eq4]
(4) 
    iA(α;γ):={x∈A ∣ γ⊆α(x)}.


3. Intersectional Soft Ideals

In what follows, we take E = S, as a set of parameters, which is a semigroup unless otherwise specified.

Definition 3 .

A soft set (α, S) over U is called an intersectional soft semigroup (briefly, int-soft semigroup) over U if it satisfies

[Formula ID: EEq3.1]
(5) 
    (∀x,y∈S) (α(x)∩α(y)⊆α(xy)).

Definition 4 .

A soft set (α, S) over U is called an intersectional soft left (resp., right) ideal (briefly, int-soft left (resp., right) ideal) over U if it satisfies

[Formula ID: EEq3.2]
(6) 
    (∀x,y∈S) (α(xy)⊇α(y)(resp.,α(xy)⊇α(x))).

If a soft set (α, S) over U is both an int-soft left ideal and an int-soft right ideal over U, we say that (α, S) is an intersectional soft two-sided ideal (briefly, int-soft two-sided ideal) over U.

Example 5 .

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

[Formula ID: eq7]
(7) 
Let (α, S) be a soft set over U defined as follows:
[Formula ID: eq8]
(8) 
α:S⟶P(U), x⟼{γ1,if  x=a,γ2,if  x=b,γ4,if  x=c,γ3,if  x=d,
where γ1, γ2, γ3, and γ4 are subsets of U with γ1γ2γ3γ4. Then (α, S) is an int-soft two-sided ideal over U.

Obviously, every int-soft left (resp., right) ideal over U is an int-soft semigroup over U. But the converse is not true as seen in the following example.

Example 6 .

Let S = {0,1, 2,3, 4,5} be a semigroup with the following Cayley table:

[Formula ID: eq9]
(9) 
  • (1)  Let (α, S) be a soft set over U defined as follows:
    [Formula ID: eq10]
    (10) 
    α:S⟶P(U), x⟼{γ1,if  x=0,γ2,if  x=1,γ5,if  x∈{2,4},γ4,if  x=3,γ3,if  x=5,
    where γ1, γ2, γ3, γ4, and γ5 are subsets of U with γ1γ2γ3γ4γ5. Then (α, S) is an int-soft semigroup over U. But it is not an int-soft left ideal over U since α(3 · 5) = α(3) = γ4γ3 = α(5).
  • (2)  Let (β, S) be a soft set over U defined as follows:
    [Formula ID: eq11]
    (11) 
    β:S⟶P(U), x⟼{γ1if  x∈{0,1},γ3if  x=2,γ2if  x=3,γ4if  x∈{4,5},
    where γ1, γ2, γ3, and γ4 are subsets of U with γ1γ2γ3γ4. Then (β, S) is an int-soft semigroup over U. But it is not an int-soft right ideal over U since β(3 · 4) = β(2) = γ3γ2 = β(3).

For a nonempty subset A of S, define a map χA as follows:

[Formula ID: eq12]
(12) 
χA:S⟶P(U), x⟼{U,if  x∈  A,∅,otherwise.
Then (χA, S) is a soft set over U, which is called the characteristic soft set. The soft set (χS, S) is called the identity soft set over U.

Theorem 7 .

For any nonempty subset A of  S, the following are equivalent.

  1. A is a left (resp., right) ideal of  S.
  2. The characteristic soft set (χA, S) over  U is an int-soft left (resp., right) ideal over  U.

Proof

Assume that A is a left ideal of S. For any x, yS, if yA then χA(xy)⊇ = χA(y). If yA, then xyA since A is a left ideal of S. Hence χA(xy) = U = χA(y). Therefore (χA, S) is an int-soft left ideal over U. Similarly, (χA, S) is an int-soft right ideal over U when A is a right ideal of S.

Conversely suppose that (χA, S) is an int-soft left ideal over U. Let xS and yA. Then χA(y) = U, and so χA(xy)⊇χA(y) = U; that is, χA(xy) = U. Thus xyA and therefore A is a left ideal of S. Similarly, we can show that if (χA, S) is an int-soft right ideal over U, then A is a right ideal of S.

Corollary 8 .

For any nonempty subset A of  S, the following are equivalent.

  1. A is a two-sided ideal of  S.
  2. The characteristic soft set (χA, S) over  U  is an int-soft two-sided ideal over  U.

Theorem 9 .

A soft set (α, S) over  U is an int-soft semigroup over U if and only if the nonempty γ-inclusive set of (α, S) is a subsemigroup of  S for all γU.

Proof

Assume that (α, S) over U is an int-soft semigroup over U. Let γU be such that iS(α; γ) ≠ . Let x, yiS(α; γ). Then α(x)⊇γ and α(y)⊇γ. It follows from (5) that

[Formula ID: eq13]
(13) 
    α(xy)⊇α(x)∩α(y)⊇γ,
so that xyiS(α; γ). Thus iS(α; γ) is a subsemigroup of S.

Conversely, suppose that the nonempty γ-inclusive set of (α, S) is a subsemigroup of S for all γU. Let x, yS be such that α(x) = γx and α(y) = γy. Taking γ = γxγy implies that x, yiS(α; γ). Hence xyiS(α; γ), and so α(xy)⊇γ = γxγy = α(x)∩α(y). Therefore (α, S) is an int-soft semigroup over U.

Theorem 10 .

A soft set (α, S) over U is an int-soft left (resp., right) ideal over U if and only if the nonempty γ-inclusive set of  (α, S) is a left (resp., right) ideal of  S for all γU.

Proof

It is the same as the proof of Theorem 9.

Corollary 11 .

A soft set (α, S) over  U is an int-soft two-sided ideal over U if and only if the nonempty γ-inclusive set of  (α, S) is a two-sided ideal of  S for all γU.

For any soft sets (α, S) and (β, S) over U, we define

[Formula ID: eq14]
(14) 
(α,S)⊆~(β,S) if  α(x)⊆β(x) ∀x∈S.
The soft union of (α, S) and (β, S) is defined to be the soft set (α  ∪~  β,S) over U in which α  ∪~  β is defined by
[Formula ID: eq15]
(15) 
    (α∪~β)(x)=α(x)∪β(x) ∀x∈S.
The soft intersection of (α, S) and (β, S) is defined to be the soft set (α  ∩~  β,S) over U in which α  ∩~  β is defined by
[Formula ID: eq16]
(16) 
    (α  ∩~  β)(x)=α(x)∩β(x) ∀x∈S.
The int-soft product of (α, S) and (β, S) is defined to be the soft set (α  ∘~  β,S) over U in which α  ∘~  β is a mapping from S to P(U) given by
[Formula ID: eq17]
(17) 
(α  ∘~  β)(x) ={⋃x=yz{α(y)∩β(z)},if  ∃y,z∈S  such  that  x=yz,∅,otherwise.  

Proposition 12 .

Let  (α1, S), (α2, S), (β1, S), and  (β2, S) be soft sets over U. If

[Formula ID: eq18]
(18) 
(α1,S)⊆~(β1,S),  (α2,S)  ⊆~  (β2,S),
then (α1  ∘~  α2,S)  ⊆~  (β1∘~β2,S).

Proof

Let xS. If x is not expressed as x = yz for y, zS, then clearly

[Formula ID: eq19]
(19) 
(α1 ∘~ α2,S)  ⊆~  (β1  ∘~  β2,S).
Suppose that there exist y, zS such that x = yz. Then
[Formula ID: eq20]
(20) 
(α1∘~  α2)(x)=⋃x=yz{α1(y)∩α2(z)}⊆⋃x=yz{β1(y)∩β2(z)}=(β1∘~  β2)(x).
Therefore (α1∘~  α2,S)  ⊆~  (β1∘~  β2,S).

Lemma 13 .

Let (χA, S) and (χB, S) be characteristic soft sets over  U where A and B are nonempty subsets of  S. Then the following properties hold:

  1. χA  ∩~  χB=χA∩B,
  2. χA  ∘~  χB=χAB.

Proof

(1) Let xS. If xAB, then xA and xB. Thus we have

[Formula ID: eq21]
(21) 
(χA  ∩~  χB)(x)=χA(x)∩χB(x)=U=χA∩B(x).
If xAB, then xA or xB. Hence we have
[Formula ID: eq22]
(22) 
(χA  ∩~  χB)(x)=χA(x)∩χB(x)=∅=χA∩B(x).
Therefore χA  ∩~  χB=χA∩B.

(2) For any xS, suppose xAB. Then there exist aA and bB such that x = ab. Thus we have

[Formula ID: eq23]
(23) 
(χA  ∘~  χB)(x)=⋃x=yz{χA(y)∩χB(z)}⊇χA(a)∩χB(b)=U,
and so (χA  ∘~  χB)(x)=U. Since xAB, we get χAB(x) = U. Suppose xAB. Then xab for all aA and bB. If x = yz for some y, zS, then yA or zB. Hence
[Formula ID: eq24]
(24) 
(χA  ∘~  χB)(x)=⋃x=yz{χA(y)∩χB(z)}=∅=χAB(x).
If xyz for all x, yS, then
[Formula ID: eq25]
(25) 
(χA  ∘~  χB)(x)=∅=χAB(x).
In any case, we have χA  ∘~  χB=χAB.

Theorem 14 .

A soft set (α, S) over U is an int-soft semigroup over U if and only if (α  ∘~  α,S)  ⊆~  (α,S).

Proof

Assume that (α  ∘~  α,S)  ⊆~  (α,S) and let x, yS. Then

[Formula ID: eq26]
(26) 
    α(xy)⊇(α  ∘~  α)(xy)⊃α(x)∩α(y),
and so (α, S) is an int-soft semigroup over U.

Conversely, suppose that (α, S) is an int-soft semigroup over U. Then α(x)⊇α(y)∩α(z) for all xS with x = yz. Thus

[Formula ID: eq27]
(27) 
    α(x)⊇⋃x=yz{α(y)∩α(z)}=(α  ∘~  α)(x)
for all xS. Hence (α  ∘~  α,S)  ⊆~  (α,S).

Theorem 15 .

For the identity soft set (χS, S) and a soft set (β, S) over  U, the following are equivalent:

  1. (β, S) is an int-soft left ideal over U,
  2. (χS  ∘~  β,S)  ⊆~  (β,S).

Proof

Suppose that (β, S) is an int-soft left ideal over U. Let xS. If x = yz for some y, zS, then

[Formula ID: eq28]
(28) 
(χS  ∘~  β)(x)=⋃x=yz{χS(y)∩β(z)}⊆⋃x=yz{U∩β(yz)}=β(x).
Otherwise, we have (χS  ∘~  β)(x)=∅⊆β(x). Therefore (χS  ∘~  β,S)  ⊆~  (β,S).

Conversely, assume that (χS  ∘~  β,S)  ⊆~  (β,S). For any x, yS, we have

[Formula ID: eq29]
(29) 
β(xy)⊇(χS  ∘~  β)(xy)⊇χS(x)∩β(y)=U∩β(y)=β(y).
Hence (β, S) is an int-soft left ideal over U.

Similarly, we have the following theorem.

Theorem 16 .

For the identity soft set (χS, S) over U and a soft set (β, S) over U, the following assertions are equivalent:

  1. (β, S) is an int-soft right ideal over U,
  2. (β  ∘~  χS,S)  ⊆~  (β,S).

Corollary 17 .

For the identity soft set (χS, S) over U and a soft set (β, S) over U, the following assertions are equivalent:

  1. (β, S) is an int-soft two-sided ideal over U,
  2. (χS  ∘~  β,S)  ⊆~  (β,S) and (β  ∘~  χS,S)  ⊆~  (β,S).

Theorem 18 .

If (α, S) and (β, S) are int-soft semigroups over  U, then so is the soft intersection (α  ∩~  β,S).

Proof

Let x, yS. Then

[Formula ID: eq30]
(30) 
(α  ∩~  β)(xy)=α(xy)∩β(xy)⊇(α(x)∩α(y))∩(β(x)∩β(y))    =(α(x)∩β(x))∩(α(y)∩β(y))  =(α  ∩~  β)(x)∩(α  ∩~  β)(y).
Thus (α  ∩~  β,S) is an int-soft semigroup over U.

By similar manner, we can prove the following theorem.

Theorem 19 .

If  (α, S) and (β, S) are int-soft left ideals (resp., int-soft right ideals) over U, then so is the soft intersection (α  ∩~  β,S).

Corollary 20 .

If  (α, S) and (β, S) are int-soft two-sided ideals over U, then so is the soft intersection (α  ∩~  β,S).

Theorem 21 .

Let  (α, S) and (β, S) be soft sets over U. If (α, S) is an int-soft left ideal over U, then so is the int-soft product (α  ∘~  β,S).

Proof

Let  x, yS. If y = ab for some a, bS, then xy = x(ab) = (xa)b and

[Formula ID: eq31]
(31) 
(α  ∘~  β)(y)=⋃y=ab{α(a)∩β(b)}⊆⋃xy=(xa)b{α(xa)∩β(b)}⊆⋃xy=cb{α(c)∩β(b)}=(α  ∘~  β)(xy).
If y is not expressible as y = ab for a, bS, then (α  ∘~  β)(y)=∅⊆(α  ∘~  β)(xy). Thus (α  ∘~  β)(y)⊆(α  ∘~  β)(xy) for all x, yS, and so (α ∘~ β,S) is an int-soft left ideal over U.

Similarly, we have the following theorem.

Theorem 22 .

Let (α, S) and  (β, S) be soft sets over U. If (β, S) is an int-soft right ideal over U, then so is the int-soft product (α  ∘~  β,S).

Corollary 23 .

The int-soft product of two int-soft two-sided ideals over U is an int-soft two-sided ideal over U.

Let (α, S) be a soft set over U. For a subset γ of U with iS(α; γ) ≠ , define a soft set (α*, S) over U by

[Formula ID: eq32]
(32) 
α∗:S⟶P(U), x⟼{α(x),if  x∈iS(α;γ),δ,otherwise,
where δ is a subset of U with δα(x).

Theorem 24 .

If  (α, S) is an int-soft semigroup over U, then so is (α*, S).

Proof

Let x, yS. If  x, yiS(α; γ), then xyiS(α; γ) since iS(α; γ) is a subsemigroup of S by Theorem 9. Hence we have

[Formula ID: eq33]
(33) 
    α∗(xy)=α(xy)⊇α(x)∩α(y)=α∗(x)∩α∗(y).
If  xiS(α; γ) or yiS(α; γ), then α*(x) = δ or α*(y) = δ. Thus
[Formula ID: eq34]
(34) 
    α∗(xy)⊇δ=α∗(x)∩α∗(y).
Therefore (α*, S) is an int-soft semigroup over U.

By similar manner, we can prove the following theorem.

Theorem 25 .

If  (α, S) is an int-soft left ideal (resp., int-soft right ideal) over U, then so is (α*, S).

Corollary 26 .

If  (α, S) is an int-soft two-sided ideal over U, then so is (α*, S).

Theorem 27 .

If  (α, S) is an int-soft right ideal over U and (β, S) is an int-soft left ideal over U, then (α  ∘~  β,S)  ⊆~  (α  ∩~  β,S).

Proof

Let xS. If x is not expressible as x = ab for a, bS, then (α  ∘~  β)(x)=∅⊆(α  ∩~  β)(x). Assume that there exist a, bS such that x = ab. Then

[Formula ID: eq35]
(35) 
(α  ∘~  β)(x)=⋃x=ab{α(a)∩β(b)}⊆⋃x=ab{α(ab)∩β(ab)}=α(x)∩β(x)=(α  ∩~  β)(x).
In any case, we have (α ∘~ β,S)  ⊆~  (α  ∩~  β,S).

If we strengthen the condition of the semigroup S, then we can induce the reverse inclusion in Theorem 27 as follows.

Theorem 28 .

Let S be a regular semigroup. If  (α, S) is an int-soft right ideal over  U, then (α  ∩~  β,S)  ⊆~  (α  ∘~  β,S) for every soft set (β, S) over  U.

Proof

Let xS. Then there exists aS such that xax = x since S is regular. Thus

[Formula ID: eq36]
(36) 
    (α  ∘~  β)(x)=⋃x=yz{α(y)∩β(z)}.
On the other hand, we have
[Formula ID: eq37]
(37) 
    (α  ∩~  β)(x)=α(x)∩β(x)⊆α(xa)∩β(x)
since (α, S) is an int-soft right ideal over U. Since xax = x, we obtain
[Formula ID: eq38]
(38) 
    α(xa)∩β(x)⊆⋃x=yz{α(y)∩β(z)}=(α  ∘~  β)(x).
Therefore (α  ∩~  β)(x)⊆(α  ∘~  β)(x), and so (α  ∩~  β,S)  ⊆~  (α  ∘~  β,S).

In a similar way we prove the following.

Theorem 29 .

Let S be a regular semigroup. If  (β, S) is an int-soft left ideal over  U, then (α  ∩~  β,S)  ⊆~  (α  ∘~  β,S) for every soft set (α, S) over  U.

Theorem 30 .

If a semigroup S is regular, then (α  ∩~  β,S)=(α  ∘~  β,S) for every int-soft right ideal (α, S) and int-soft left ideal (β, S) over  U.

Proof

Assume that S is a regular semigroup and let (α, S) and (β, S) be an int-soft right ideal and an int-soft left ideal, respectively, over U. By Theorem 28, we have (α  ∩~  β,S)  ⊆~  (α ∘~ β,S). Since (α ∘~ β,S)  ⊆~  (α  ∩~  β,S) by Theorem 27, we have (α  ∩~  β,S)=(α ∘~ β,S).

Definition 31 .

A soft set (α, S) over U is called an int-soft quasi-ideal over U if

[Formula ID: EEq3.3]
(39) 
    (α  ∘~  χS,S)∩~(χS  ∘~  α,S)⊆~(α,S).

Obviously, every int-soft left (resp., right) ideal is an int-soft quasi-ideal over U, but the converse does not hold in general.

In fact, we have the following example.

Example 32 .

Let S = {0, a, b, c} be a semigroup with the following Cayley table:

[Formula ID: eq40]
(40) 
Let (α, S) be a soft set over U defined as follows:
[Formula ID: eq41]
(41) 
α:S⟶P(U), x⟼{γ,if  x∈{0,a},∅,if  x∈{b,c},
where γ is a subset of U. Then (α, S) is an int-soft quasi-ideal over U and is not an int-soft left (resp., right) ideal over U.

Theorem 33 .

Let G be a nonempty subset of  S. Then G is a quasi-ideal of S if and only if the characteristic soft set (χG, S) is an int-soft quasi-ideal over  U.

Proof

We first assume that G is a quasi-ideal of S. Let a be any element of S. If aG, then

[Formula ID: eq42]
(42) 
    ((χG  ∘~  χS)∩~(χS  ∘~  χG))(a)⊆~  U=χG(a).
If aG, then χG(a) = . On the other hand, assume that
[Formula ID: eq43]
(43) 
    ((χG  ∘~  χS)∩~(χS  ∘~  χG))(a)=U.
Then
[Formula ID: eq44]
(44) 
    ⋃a=xy{χG(x)∩χS(y)}=(χG  ∘~  χS)(a)=U,    ⋃a=xy{χS(x)∩χG(y)}=(χS  ∘~  χG)(a)=U.
This implies that there exist elements b, c, d, and e of S with a = bc = de such that χG(b) = U and χG(e) = U. Hence a = bc = deGSSGG, which contradicts that aG. Thus we have (χG  ∘~  χS,S)  ∩~  (χS  ∘~  χG,S)  ⊆~  (χG,S) and so (χG, S) is an int-soft quasi-ideal over U.

Conversely, suppose that (χG, S) is an int-soft quasi-ideal over U. Let a be any element of GSSG. Then bx = a = yc for some b, cG and x, yS. It follows from (39) that

[Formula ID: eq45]
(45) 
χG(a)⊇((χG  ∘~  χS)∩~(χS  ∘~  χG))(a)=(χG  ∘~  χS)(a)∩(χS  ∘~  χG)(a)=(⋃a=uv{χG(u)∩χS(v)})∩(⋃a=uv{χS(u)∩χG(v)})=(⋃a=uv{χG(u)})∩(⋃a=uv{χG(v)})=U
and so aG. Thus GSSGG, and hence G is a quasi-ideal of S.

Theorem 34 .

For a semigroup S, the following are equivalent:

  1. S is regular,
  2. (α,S)=(α  ∘~  χS  ∘~  α,S) for every int-soft quasi-ideal (α, S) over  U.

Proof

Assume that S is regular and let aS. Then a = axa for some xS. Hence

[Formula ID: eq46]
(46) 
(α  ∘~  χS  ∘~  α)(a)=⋃a=uv{(α  ∘~  χS)(u)∩α(v)}⊇(α  ∘~  χS)(ax)∩α(a)=(⋃ax=cd{α(c)∩χS(d)})∩α(a)=(⋃ax=cd{α(c)})∩α(a)=α(a),
and so (α,S)  ⊆~  (α  ∘~  χS  ∘~  α,S). On the other hand, since (α, S) is an int-soft quasi-ideal over U,
[Formula ID: eq47]
(47) 
    (α  ∘~  χS  ∘~  α,S)⊆~(α  ∘~  χS,S)∩~(χS  ∘~  α,S)⊆~(α,S).
Hence (α,S)=(α  ∘~  χS  ∘~  α,S).

Conversely, suppose that (2) is valid and let A be a quasi-ideal of S. Then ASAASSAA and (χA, S) is an int-soft quasi-ideal over U. For any aA, we have

[Formula ID: eq48]
(48) 
⋃a=yz{(χA  ∘~  χS)(y)∩χA(z)}  =((χA  ∘~  χS)  ∘~  χA)(a)=χA(a)=U.
This implies that there exist b, cS such that a=bc,(χA  ∘~  χS)(b)=U and χA(c) = U. Then
[Formula ID: eq49]
(49) 
    U=(χA  ∘~  χS)(b)=⋃b=pq{χA(p)∩χS(q)},
and so b = st,  χA(s) = U = χS(t) for some s, tS. It follows that c, sA and tS so that a = bc = (st)cASA. Hence AASA, and thus A = ASA. Therefore S is regular.


Acknowledgment

The authors are grateful to the referee for valuable suggestions and help.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


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