Ideal theory in semigroups based on intersectional soft sets.  
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The notions of intsoft semigroups and intsoft 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 intsoft products, characterizations of intsoft semigroups and intsoft left (resp., right) ideals are discussed. We prove that the soft intersection of intsoft left (resp., right) ideals (resp., intsoft semigroups) is also intsoft left (resp., right) ideals (resp., intsoft semigroups). The concept of intsoft quasiideals is also introduced, and characterization of a regular semigroup is discussed. 
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Seok Zun Song; Hee Sik Kim; Young Bae Jun 
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Type: Journal Article Date: 20140623 
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Title: TheScientificWorldJournal Volume: 2014 ISSN: 1537744X ISO Abbreviation: ScientificWorldJournal Publication Date: 2014 
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Created Date: 20140807 Completed Date:  Revised Date:  
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Languages: eng Pagination: 136424 Citation Subset: IM 
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Journal Information Journal ID (nlmta): ScientificWorldJournal Journal ID (isoabbrev): ScientificWorldJournal Journal ID (publisherid): TSWJ ISSN: 23566140 ISSN: 1537744X Publisher: Hindawi Publishing Corporation 
Article Information Download PDF Copyright © 2014 Seok Zun Song et al. openaccess: Received Day: 18 Month: 3 Year: 2014 Accepted Day: 22 Month: 5 Year: 2014 Print publication date: Year: 2014 Electronic publication date: Day: 23 Month: 6 Year: 2014 Volume: 2014Elocation ID: 136424 PubMed Id: 25101310 ID: 4094860 DOI: 10.1155/2014/136424 
Ideal Theory in Semigroups Based on Intersectional Soft Sets  
Seok Zun Song^{1}  
Hee Sik Kim^{2}*  http://orcid.org/0000000153215919 
Young Bae Jun^{3}  
^{1}Department of Mathematics, Jeju National University, Jeju 690756, Republic of Korea 

^{2}Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133791, Republic of Korea 

^{3}Department of Mathematics Education, Gyeongsang National University, Jinju 660701, Republic of Korea 

Correspondence: *Hee Sik Kim: heekim@hanyang.ac.kr [other] Academic Editor: Feng Feng 
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 FPsoft 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 [^{5}–^{8}]. 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 [^{12}–^{26}]). 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 intsoft semigroups and intsoft 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 intsoft products, we give characterizations of intsoft semigroups and intsoft left (resp., right) ideals. We show that the soft intersection of intsoft left (resp., right) ideals (resp., intsoft semigroups) is also intsoft left (resp., right) ideals (resp., intsoft semigroups). We also introduce the concept of intsoft quasiideals and discuss a characterization of a regular semigroup by using the notion of intsoft quasiideals.
Let S be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows:
(1)
AB={ab∈S ∣ a∈A,b∈B}. 
A semigroup S is said to be regular if for every x ∈ S there exists a ∈ S such that xax = x.
A nonempty subset A of S is called
 a subsemigroup of S if AA⊆A, that is, ab ∈ A for all a, b ∈ A,
 a left (resp., right) ideal of S if SA⊆A (resp., SA⊆A), that is, xa ∈ A (resp., ax ∈ A) for all x ∈ S and a ∈ A,
 a twosided ideal of S if it is both a left and a right ideal of S,
 a quasiideal of S if AS∩SA⊆A.
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
(2)
(α,A):={(x,α(x)):x∈E,α(x)∈P(U)}, 
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
(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 γinclusive set of (α, A), denoted by i_{A}(α; γ), is defined to be the set
(4)
iA(α;γ):={x∈A ∣ γ⊆α(x)}. 
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, intsoft semigroup) over U if it satisfies
(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, intsoft left (resp., right) ideal) over U if it satisfies
(6)
(∀x,y∈S) (α(xy)⊇α(y)(resp.,α(xy)⊇α(x))). 
If a soft set (α, S) over U is both an intsoft left ideal and an intsoft right ideal over U, we say that (α, S) is an intersectional soft twosided ideal (briefly, intsoft twosided ideal) over U.
Example 5 .
Let S = {a, b, c, d} be a semigroup with the following Cayley table:
Let (α, S) be a soft set over U defined as follows:(8)
α:S⟶P(U), x⟼{γ1,if x=a,γ2,if x=b,γ4,if x=c,γ3,if x=d, 
Obviously, every intsoft left (resp., right) ideal over U is an intsoft 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:

(1)
Let (α, S) be a soft set over U defined as follows:
[Formula ID: eq10]where γ_{1}, γ_{2}, γ_{3}, γ_{4}, and γ_{5} are subsets of U with γ_{1}⊋γ_{2}⊋γ_{3}⊋γ_{4}⊋γ_{5}. Then (α, S) is an intsoft semigroup over U. But it is not an intsoft left ideal over U since α(3 · 5) = α(3) = γ_{4}⊉γ_{3} = α(5).
(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, 
(2)
Let (β, S) be a soft set over U defined as follows:
[Formula ID: eq11]where γ_{1}, γ_{2}, γ_{3}, and γ_{4} are subsets of U with γ_{1}⊋γ_{2}⊋γ_{3}⊋γ_{4}. Then (β, S) is an intsoft semigroup over U. But it is not an intsoft right ideal over U since β(3 · 4) = β(2) = γ_{3}⊉γ_{2} = β(3).
(11)β:S⟶P(U), x⟼{γ1if x∈{0,1},γ3if x=2,γ2if x=3,γ4if x∈{4,5},
For a nonempty subset A of S, define a map χ_{A} as follows:
(12)
χA:S⟶P(U), x⟼{U,if x∈ A,∅,otherwise. 
Theorem 7 .
For any nonempty subset A of S, the following are equivalent.
 A is a left (resp., right) ideal of S.
 The characteristic soft set (χ_{A}, S) over U is an intsoft left (resp., right) ideal over U.
Proof
Assume that A is a left ideal of S. For any x, y ∈ S, if y ∉ A then χ_{A}(xy)⊇∅ = χ_{A}(y). If y ∈ A, then xy ∈ A since A is a left ideal of S. Hence χ_{A}(xy) = U = χ_{A}(y). Therefore (χ_{A}, S) is an intsoft left ideal over U. Similarly, (χ_{A}, S) is an intsoft right ideal over U when A is a right ideal of S.
Conversely suppose that (χ_{A}, S) is an intsoft left ideal over U. Let x ∈ S and y ∈ A. Then χ_{A}(y) = U, and so χ_{A}(xy)⊇χ_{A}(y) = U; that is, χ_{A}(xy) = U. Thus xy ∈ A and therefore A is a left ideal of S. Similarly, we can show that if (χ_{A}, S) is an intsoft 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.
 A is a twosided ideal of S.
 The characteristic soft set (χ_{A}, S) over U is an intsoft twosided ideal over U.
Theorem 9 .
A soft set (α, S) over U is an intsoft semigroup over U if and only if the nonempty γinclusive set of (α, S) is a subsemigroup of S for all γ⊆U.
Proof
Assume that (α, S) over U is an intsoft semigroup over U. Let γ⊆U be such that i_{S}(α; γ) ≠ ∅. Let x, y ∈ i_{S}(α; γ). Then α(x)⊇γ and α(y)⊇γ. It follows from (5) that
(13)
α(xy)⊇α(x)∩α(y)⊇γ, 
Conversely, suppose that the nonempty γinclusive set of (α, S) is a subsemigroup of S for all γ⊆U. Let x, y ∈ S be such that α(x) = γ_{x} and α(y) = γ_{y}. Taking γ = γ_{x}∩γ_{y} implies that x, y ∈ i_{S}(α; γ). Hence xy ∈ i_{S}(α; γ), and so α(xy)⊇γ = γ_{x}∩γ_{y} = α(x)∩α(y). Therefore (α, S) is an intsoft semigroup over U.
Theorem 10 .
A soft set (α, S) over U is an intsoft left (resp., right) ideal over U if and only if the nonempty γinclusive set of (α, S) is a left (resp., right) ideal of S for all γ⊆U.
Proof
It is the same as the proof of Theorem 9.
Corollary 11 .
A soft set (α, S) over U is an intsoft twosided ideal over U if and only if the nonempty γinclusive set of (α, S) is a twosided ideal of S for all γ⊆U.
For any soft sets (α, S) and (β, S) over U, we define
(14)
(α,S)⊆~(β,S) if α(x)⊆β(x) ∀x∈S. 
(15)
(α∪~β)(x)=α(x)∪β(x) ∀x∈S. 
(16)
(α ∩~ β)(x)=α(x)∩β(x) ∀x∈S. 
(17)
(α ∘~ β)(x) ={⋃x=yz{α(y)∩β(z)},if ∃y,z∈S such that x=yz,∅,otherwise. 
Proposition 12 .
Let (α_{1}, S), (α_{2}, S), (β_{1}, S), and (β_{2}, S) be soft sets over U. If
(18)
(α1,S)⊆~(β1,S), (α2,S) ⊆~ (β2,S), 
Proof
Let x ∈ S. If x is not expressed as x = yz for y, z ∈ S, then clearly
(19)
(α1 ∘~ α2,S) ⊆~ (β1 ∘~ β2,S). 
(20)
(α1∘~ α2)(x)=⋃x=yz{α1(y)∩α2(z)}⊆⋃x=yz{β1(y)∩β2(z)}=(β1∘~ β2)(x). 
Lemma 13 .
Let (χ_{A}, S) and (χ_{B}, S) be characteristic soft sets over U where A and B are nonempty subsets of S. Then the following properties hold:
 χA ∩~ χB=χA∩B,
 χA ∘~ χB=χAB.
Proof
(1) Let x ∈ S. If x ∈ A∩B, then x ∈ A and x ∈ B. Thus we have
(21)
(χA ∩~ χB)(x)=χA(x)∩χB(x)=U=χA∩B(x). 
(22)
(χA ∩~ χB)(x)=χA(x)∩χB(x)=∅=χA∩B(x). 
(2) For any x ∈ S, suppose x ∈ AB. Then there exist a ∈ A and b ∈ B such that x = ab. Thus we have
(23)
(χA ∘~ χB)(x)=⋃x=yz{χA(y)∩χB(z)}⊇χA(a)∩χB(b)=U, 
(24)
(χA ∘~ χB)(x)=⋃x=yz{χA(y)∩χB(z)}=∅=χAB(x). 
(25)
(χA ∘~ χB)(x)=∅=χAB(x). 
Theorem 14 .
A soft set (α, S) over U is an intsoft semigroup over U if and only if (α ∘~ α,S) ⊆~ (α,S).
Proof
Assume that (α ∘~ α,S) ⊆~ (α,S) and let x, y ∈ S. Then
(26)
α(xy)⊇(α ∘~ α)(xy)⊃α(x)∩α(y), 
Conversely, suppose that (α, S) is an intsoft semigroup over U. Then α(x)⊇α(y)∩α(z) for all x ∈ S with x = yz. Thus
(27)
α(x)⊇⋃x=yz{α(y)∩α(z)}=(α ∘~ α)(x) 
Theorem 15 .
For the identity soft set (χ_{S}, S) and a soft set (β, S) over U, the following are equivalent:
 (β, S) is an intsoft left ideal over U,
 (χS ∘~ β,S) ⊆~ (β,S).
Proof
Suppose that (β, S) is an intsoft left ideal over U. Let x ∈ S. If x = yz for some y, z ∈ S, then
(28)
(χS ∘~ β)(x)=⋃x=yz{χS(y)∩β(z)}⊆⋃x=yz{U∩β(yz)}=β(x). 
Conversely, assume that (χS ∘~ β,S) ⊆~ (β,S). For any x, y ∈ S, we have
(29)
β(xy)⊇(χS ∘~ β)(xy)⊇χS(x)∩β(y)=U∩β(y)=β(y). 
Similarly, we have the following theorem.
Theorem 16 .
For the identity soft set (χ_{S}, S) over U and a soft set (β, S) over U, the following assertions are equivalent:
 (β, S) is an intsoft right ideal over U,
 (β ∘~ χS,S) ⊆~ (β,S).
Corollary 17 .
For the identity soft set (χ_{S}, S) over U and a soft set (β, S) over U, the following assertions are equivalent:
 (β, S) is an intsoft twosided ideal over U,
 (χS ∘~ β,S) ⊆~ (β,S) and (β ∘~ χS,S) ⊆~ (β,S).
Theorem 18 .
If (α, S) and (β, S) are intsoft semigroups over U, then so is the soft intersection (α ∩~ β,S).
Proof
Let x, y ∈ S. Then
(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). 
By similar manner, we can prove the following theorem.
Theorem 19 .
If (α, S) and (β, S) are intsoft left ideals (resp., intsoft right ideals) over U, then so is the soft intersection (α ∩~ β,S).
Corollary 20 .
If (α, S) and (β, S) are intsoft twosided 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 intsoft left ideal over U, then so is the intsoft product (α ∘~ β,S).
Proof
Let x, y ∈ S. If y = ab for some a, b ∈ S, then xy = x(ab) = (xa)b and
(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). 
Similarly, we have the following theorem.
Theorem 22 .
Let (α, S) and (β, S) be soft sets over U. If (β, S) is an intsoft right ideal over U, then so is the intsoft product (α ∘~ β,S).
Corollary 23 .
The intsoft product of two intsoft twosided ideals over U is an intsoft twosided ideal over U.
Let (α, S) be a soft set over U. For a subset γ of U with i_{S}(α; γ) ≠ ∅, define a soft set (α*, S) over U by
(32)
α∗:S⟶P(U), x⟼{α(x),if x∈iS(α;γ),δ,otherwise, 
Theorem 24 .
If (α, S) is an intsoft semigroup over U, then so is (α*, S).
Proof
Let x, y ∈ S. If x, y ∈ i_{S}(α; γ), then xy ∈ i_{S}(α; γ) since i_{S}(α; γ) is a subsemigroup of S by Theorem 9. Hence we have
(33)
α∗(xy)=α(xy)⊇α(x)∩α(y)=α∗(x)∩α∗(y). 
(34)
α∗(xy)⊇δ=α∗(x)∩α∗(y). 
By similar manner, we can prove the following theorem.
Theorem 25 .
If (α, S) is an intsoft left ideal (resp., intsoft right ideal) over U, then so is (α*, S).
Corollary 26 .
If (α, S) is an intsoft twosided ideal over U, then so is (α*, S).
Theorem 27 .
If (α, S) is an intsoft right ideal over U and (β, S) is an intsoft left ideal over U, then (α ∘~ β,S) ⊆~ (α ∩~ β,S).
Proof
Let x ∈ S. If x is not expressible as x = ab for a, b ∈ S, then (α ∘~ β)(x)=∅⊆(α ∩~ β)(x). Assume that there exist a, b ∈ S such that x = ab. Then
(35)
(α ∘~ β)(x)=⋃x=ab{α(a)∩β(b)}⊆⋃x=ab{α(ab)∩β(ab)}=α(x)∩β(x)=(α ∩~ β)(x). 
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 intsoft right ideal over U, then (α ∩~ β,S) ⊆~ (α ∘~ β,S) for every soft set (β, S) over U.
Proof
Let x ∈ S. Then there exists a ∈ S such that xax = x since S is regular. Thus
(36)
(α ∘~ β)(x)=⋃x=yz{α(y)∩β(z)}. 
(37)
(α ∩~ β)(x)=α(x)∩β(x)⊆α(xa)∩β(x) 
(38)
α(xa)∩β(x)⊆⋃x=yz{α(y)∩β(z)}=(α ∘~ β)(x). 
In a similar way we prove the following.
Theorem 29 .
Let S be a regular semigroup. If (β, S) is an intsoft 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 intsoft right ideal (α, S) and intsoft left ideal (β, S) over U.
Proof
Assume that S is a regular semigroup and let (α, S) and (β, S) be an intsoft right ideal and an intsoft 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 intsoft quasiideal over U if
(39)
(α ∘~ χS,S)∩~(χS ∘~ α,S)⊆~(α,S). 
Obviously, every intsoft left (resp., right) ideal is an intsoft quasiideal 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:
Let (α, S) be a soft set over U defined as follows:(41)
α:S⟶P(U), x⟼{γ,if x∈{0,a},∅,if x∈{b,c}, 
Theorem 33 .
Let G be a nonempty subset of S. Then G is a quasiideal of S if and only if the characteristic soft set (χ_{G}, S) is an intsoft quasiideal over U.
Proof
We first assume that G is a quasiideal of S. Let a be any element of S. If a ∈ G, then
(42)
((χG ∘~ χS)∩~(χS ∘~ χG))(a)⊆~ U=χG(a). 
(43)
((χG ∘~ χS)∩~(χS ∘~ χG))(a)=U. 
(44)
⋃a=xy{χG(x)∩χS(y)}=(χG ∘~ χS)(a)=U, ⋃a=xy{χS(x)∩χG(y)}=(χS ∘~ χG)(a)=U. 
Conversely, suppose that (χ_{G}, S) is an intsoft quasiideal over U. Let a be any element of GS∩SG. Then bx = a = yc for some b, c ∈ G and x, y ∈ S. It follows from (39) that
(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 
Theorem 34 .
For a semigroup S, the following are equivalent:
 S is regular,
 (α,S)=(α ∘~ χS ∘~ α,S) for every intsoft quasiideal (α, S) over U.
Proof
Assume that S is regular and let a ∈ S. Then a = axa for some x ∈ S. Hence
(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), 
(47)
(α ∘~ χS ∘~ α,S)⊆~(α ∘~ χS,S)∩~(χS ∘~ α,S)⊆~(α,S). 
Conversely, suppose that (2) is valid and let A be a quasiideal of S. Then ASA⊆AS∩SA⊆A and (χ_{A}, S) is an intsoft quasiideal over U. For any a ∈ A, we have
(48)
⋃a=yz{(χA ∘~ χS)(y)∩χA(z)} =((χA ∘~ χS) ∘~ χA)(a)=χA(a)=U. 
(49)
U=(χA ∘~ χS)(b)=⋃b=pq{χA(p)∩χS(q)}, 
The authors are grateful to the referee for valuable suggestions and help.
The authors declare that there is no conflict of interests regarding the publication of this paper.
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