# Concave Soft Sets, Critical Soft Points, and Union-Soft Ideals of Ordered Semigroups

Authors: Young Bae Jun; Seok Zun Song; G. Muhiuddin
Journal: The Scientific World Journal
 (2014)

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## Abstract

The notions of union-soft semigroups, union-softl-ideals, and union-soft r-ideals are introduced, and related properties are investigated. Characterizations of a union-soft semigroup, a union-soft l-ideal, and a union-soft r-ideal are provided. The concepts of union-soft products and union-soft semiprime soft sets are introduced, and their properties related to union-soft l-ideals and union-soft r-ideals are investigated. Using the notions of union-soft l-ideals and union-soft r-ideals, conditions for an ordered semigroup to be regular are considered. The concepts of concave soft sets and critical soft points are introduced, and their properties are discussed.


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## Body

## 1. Introduction

The uncertainty which has appeared in economics, engineering, environmental science, medical science and social science, and so forth is too complicated to be captured within a traditional mathematical framework. Molodtsov’s soft set theory [1] is a kind of new mathematical model for coping with uncertainty from a parameterization point of view. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields. At present, works on the soft set theory with algebraic applications are progressing rapidly (see [2–5]). Mainly, Kehayopulu et al. studied ordered semigroups (see [6–10]). Feng et al. discussed soft relations in semigroups (see [11]) and explored decomposition of fuzzy soft sets with finite value spaces (see [12]). Also, Feng and Li [13] considered soft product operations. Jun et al. [14] applied the concept of soft set theory to ordered semigroups. They applied the notion of soft sets by Molodtsov to ordered semigroups and introduced the notions of (trivial, whole) soft ordered semigroups, soft ordered subsemigroups, soft l-ideals, soft r-ideals, and l-idealistic and r-idealistic soft ordered semigroups. They investigated various related properties.The aim of this paper is to lay a foundation for providing a soft algebraic tool (in ordered semigroups) in considering many problems that contain uncertainties. We introduce the notions of union-soft semigroups, union-softl-ideals, and union-soft r-ideals and investigate their properties. We consider characterizations of a union-soft semigroup, a union-soft l-ideal, and a union-soft r-ideal. We introduce the concepts of union-soft products and union-soft semiprime soft sets and investigate their properties related to union-soft l-ideals and union-soft r-ideals. Using the notions of union-soft l-ideals and union-soft r-ideals, we provide conditions for an ordered semigroup to be regular. We also introduce the concepts of concave soft sets and critical soft points and discuss their properties.


## 2. Preliminaries

### 2.1. Basic Results on Ordered Semigroups

Anordered semigroup (or,po-semigroup) is an ordered set (S,≤) which is a semigroup such that
(1)(∀a,b,x∈S)(a≤b⟹xa≤xb,ax≤bx).ForA⊆S, we denote
(2)(A]:={t∈S∣t≤hforsomeh∈A}.
For any a∈S, denote by R(a) (resp., L(a)) the right (resp., left) ideal of S generated by a. Note that R(a)=(a∪aS] and L(a)=(a∪Sa].An ordered semigroupS is said to be(i)
left (resp.,right)regular if it satisfies
(3)(∀a∈S)(∃x∈S)(a≤xa2)(resp.a≤a2x);(ii)
regular if it satisfies
(4)(∀a∈S)(∃x∈S)(a≤axa);(iii)
intraregular if it satisfies
(5)(∀a∈S)(∃x,y∈S)(a≤xa2y).Lemma 1 (see [9]).
An ordered semigroupS is regular if and only if
(6)(∀a∈S)(R(a)∩L(a)⊆(R(a)L(a)]).A nonempty subsetI of an ordered semigroup S is called aleft (resp.,right)ideal of S if
(7)SI⊆I(resp.IS⊆I).(∀a,b∈S)(a∈I,b≤a⟹b∈I).


### 2.2. Basic Results on Soft Sets

A soft set theory is introduced by Molodtsov [1], and Çağman and Enginoğlu [15] provided new definitions and various results on soft set theory.In what follows, letU be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U and I,J,…⊆E.Definition 2 (see [1, 15]).
A soft set(F,I) over U is defined to be the set of ordered pairs
(8)(F,I):={(x,F(x)):x∈E,F(x)∈P(U)},
where F:E→P(U) such that F(x)=∅ if x∉I.For a soft set(F,I) over U and a subset δ of U, the δ-exclusive set of (F,I), denoted by eI(F;δ), is defined to be the set
(9)eI(F;δ):={x∈I∣δ⊇F(x)}.For any soft sets(F,E) and (G,E) over U, we define
(10)(F,E)⊆~(G,E)ifF(x)⊆G(x)∀x∈E.
Thesoft union, denoted by (F,E)∪~(G,E), of (F,E) and (G,E) is defined to be the soft set (F∪~G,E) over U in which F∪~G is defined by
(11)(F∪~G)(x)=F(x)∪G(x)∀x∈E.
Thesoft intersection, denoted by (F,E)∩~(G,E), of (F,E) and (G,E) is defined to be the soft set (F∩~G,E) over U in which F∩~G is defined by
(12)(F∩~G)(x)=F(x)∩G(x)∀x∈E.


## 2.1. Basic Results on Ordered Semigroups

Anordered semigroup (or,po-semigroup) is an ordered set (S,≤) which is a semigroup such that
(1)(∀a,b,x∈S)(a≤b⟹xa≤xb,ax≤bx).ForA⊆S, we denote
(2)(A]:={t∈S∣t≤hforsomeh∈A}.
For any a∈S, denote by R(a) (resp., L(a)) the right (resp., left) ideal of S generated by a. Note that R(a)=(a∪aS] and L(a)=(a∪Sa].An ordered semigroupS is said to be(i)
left (resp.,right)regular if it satisfies
(3)(∀a∈S)(∃x∈S)(a≤xa2)(resp.a≤a2x);(ii)
regular if it satisfies
(4)(∀a∈S)(∃x∈S)(a≤axa);(iii)
intraregular if it satisfies
(5)(∀a∈S)(∃x,y∈S)(a≤xa2y).Lemma 1 (see [9]).
An ordered semigroupS is regular if and only if
(6)(∀a∈S)(R(a)∩L(a)⊆(R(a)L(a)]).A nonempty subsetI of an ordered semigroup S is called aleft (resp.,right)ideal of S if
(7)SI⊆I(resp.IS⊆I).(∀a,b∈S)(a∈I,b≤a⟹b∈I).


## 2.2. Basic Results on Soft Sets

A soft set theory is introduced by Molodtsov [1], and Çağman and Enginoğlu [15] provided new definitions and various results on soft set theory.In what follows, letU be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U and I,J,…⊆E.Definition 2 (see [1, 15]).
A soft set(F,I) over U is defined to be the set of ordered pairs
(8)(F,I):={(x,F(x)):x∈E,F(x)∈P(U)},
where F:E→P(U) such that F(x)=∅ if x∉I.For a soft set(F,I) over U and a subset δ of U, the δ-exclusive set of (F,I), denoted by eI(F;δ), is defined to be the set
(9)eI(F;δ):={x∈I∣δ⊇F(x)}.For any soft sets(F,E) and (G,E) over U, we define
(10)(F,E)⊆~(G,E)ifF(x)⊆G(x)∀x∈E.
Thesoft union, denoted by (F,E)∪~(G,E), of (F,E) and (G,E) is defined to be the soft set (F∪~G,E) over U in which F∪~G is defined by
(11)(F∪~G)(x)=F(x)∪G(x)∀x∈E.
Thesoft intersection, denoted by (F,E)∩~(G,E), of (F,E) and (G,E) is defined to be the soft set (F∩~G,E) over U in which F∩~G is defined by
(12)(F∩~G)(x)=F(x)∩G(x)∀x∈E.


## 3. Union-Soft Ideals

In what follows, we takeE=S as a set of parameters, which is an ordered semigroup unless otherwise specified.Definition 3.
A soft set(F,S) over U is called a union-soft semigroup over U if it satisfies
(13)(∀x,y∈S)(F(xy)⊆F(x)∪F(y)).Theorem 4.
A soft set(F,S) over U is a union-soft semigroup over U if and only if the nonempty δ-exclusive set of (F,S) is a subsemigroup of S for all δ⊆U.Proof.
Assume that(F,S) over U is a union-soft semigroup over U. Let δ⊆U be such that eS(F;δ)≠∅. Let x,y∈eS(F;δ). Then F(x)⊆δ and F(y)⊆δ. It follows from (13) that
(14)F(xy)⊆F(x)∪F(y)⊆δ,
so that xy∈eS(F;δ). Thus eS(F;δ) is a subsemigroup of S.
Conversely, suppose that the nonemptyδ-exclusive set of (F,S) is a subsemigroup of S for all δ⊆U. Let x,y∈S be such that F(x)=δx and F(y)=δy. Taking δ=δx∪δy implies that x,y∈eS(F;δ). Hence xy∈eS(F;δ), and so F(xy)⊆δ=δx∪δy=F(x)∪F(y). Therefore (F,S) is a union-soft semigroup over U.Definition 5.
For a left idealA of S, a soft set (F,A) over U is called aunion-softl-ideal over U related to A if it satisfies
(15)(∀x∈S)(∀a∈A)(F(xa)⊆F(a)),(16)(∀x,y∈A)(x≤y⇒F(x)⊆F(y)).Definition 6.
For a right idealA of S, a soft set (F,S) over U is called aunion-softr-ideal over U related to A if it satisfies (16) and
(17)(∀x∈S)(∀a∈A)(F(ax)⊆F(a)).A union-softl-ideal (resp., union-soft r-ideal) over U related to A=S is called a union-soft l-ideal (resp., union-soft r-ideal) over U.If a soft set(F,S) over U is both a union-soft l-ideal and a union-soft r-ideal over U, we say that (F,S) is a union-soft ideal over U.Example 7.
LetS={a,b,c,d,e,f} be an ordered semigroup with the following Cayley table and order (see [10]):
(18)·abcdefaaaadaababbdbbcabcdeedaaddddeabcdeefabcdef≤:={(a,a),(b,b),(c,c),(d,d),(e,e),(f,e),(f,f)}.(1)
Let(F,S) be a soft set over U={1,2,3,4,5,6,7,8,9,10} in which F is given as follows:
(19)F:S⟶P(U),x⟼{{1,3,5},ifx∈{a,d},{1,2,3,5,6},ifx=b,{1,2,3,5,6,8},ifx∈{c,e,f}.
Routine calculations show that (F,S) is a union-soft ideal over U.(2)
Let(G,S) be a soft set over U={1,2,3,4,5,6,7,8,9,10} in which G is given as follows:
(20)G:S⟶P(U),x⟼{{1,3},ifx=a,{2,8},ifx=d,{1,2,3,8},ifx=b,{1,2,3,8,9},ifx=c,{1,2,3,5,8},ifx∈{e,f}.Then (G,S) is a union-soft l-ideal over U. But it is not a union-soft r-ideal over U since G(ce)=G(e)={1,2,3,5,8}⊈{1,2,3,8,9}=G(c).Example 8.
LetS={a,b,c,d,e,f} be an ordered semigroup with Cayley table and Hasse diagramas shown in Figure 1. In Figure 1, (F,S) is a soft set over U={1,2,3,4,5,6,7,8,9,10} in which F is given as follows:
(21)F:S⟶P(U),x⟼{{3,5},ifx=a,{1,3,5,6,9},ifx∈{b,d},{1,3,5,6},ifx∈{e,f},{2,4,6,8},ifx=c.
Routine calculations show that (F,S) is a union-soft r-ideal over U. But it is not a union-soft l-ideal over U since F(ce)=F(c)={2,4,6,8}⊈{1,3,5,6}=F(e).Figure 1For a nonempty subsetI of S, thecharacteristic soft set is defined to be the soft set (χI,S) over U in which χI is given as follows:
(22)χI:S⟶P(U),x⟼{U,ifx∈I,∅,otherwise.
The soft set (US,S), where US(x)=U for all x∈S, is called theidentity soft set over U. For the characteristic soft set (χI,S) over U, the soft set (χIc,S) over U is given as follows:
(23)χIc:S⟶P(U),x⟼{∅,ifx∈I,U,otherwise.Theorem 9.
For any nonempty subsetI of S, the following are equivalent. (1)
I is a left (resp., right) ideal of S.(2)
The soft set(χIc,S) over U is a union-soft l-ideal (resp., union-soft r-ideal) over U.Proof.
Assume thatI is a left ideal of S. Let x,y∈S be such that x≤y. If y∉I, then χIc(y)=U and so χIc(x)⊆U=χIc(y). If y∈I, then χIc(y)=∅. Since x≤y and I is a left ideal of S, we have x∈I and thus χIc(x)=∅=χIc(y). For any x,y∈S, if y∉I then χIc(xy)⊆U=χIc(y). If y∈I, then xy∈I since I is a left ideal of S. Hence χIc(xy)=∅=χIc(y). Therefore (χIc,S) is a union-soft l-ideal over U. Similarly, (χIc,S) is a union-soft r-ideal over U when I is a right ideal of S.
Conversely suppose that(χIc,S) is a union-soft l-ideal over U. Let x∈S and y∈I. Then χIc(y)=∅, and so χIc(xy)⊆χIc(y)=∅; that is, χIc(xy)=∅. Thus xy∈I. Let x∈S and y∈I be such that x≤y. Then χIc(x)⊆χIc(y)=∅, and thus x∈I. Therefore I is a left ideal of S. Similarly, we can show that if (χIc,S) is a union-soft r-ideal over U, then I is a right ideal of S.Corollary 10.
For any nonempty subsetI of S, the following are equivalent. (1)
I is an ideal of S.(2)
The soft set(χIc,S) over U is a union-soft ideal over U.Theorem 11.
If a soft set(F,S) over U is a union-soft l-ideal (resp., union-soft r-ideal) over U, then the nonempty δ-exclusive set of (F,S) is a left (resp., right) ideal of S for all δ⊆U.Proof.
Assume that(F,S) is a union-soft l-ideal over U. Let δ⊆U be such that eS(F;δ)≠U. Let x∈S and y∈eS(F;δ). Then F(y)⊆δ, and thus F(xy)⊆F(y)⊆δ by (15). Hence xy∈eS(F;δ). Let x∈eS(F;δ) and y∈S be such that y≤x. Using (16), we have F(y)⊆F(x)⊆δ. Thus y∈eS(F;δ). Therefore eS(F;δ) is a left ideal of S. The right case can be seen in a similar way.Corollary 12.
If a soft set(F,S) over U is a union-soft l-ideal (resp., union-soft r-ideal) over U, then the set
(24)Ia:={b∈S∣F(b)⊆F(a)}
is a left (resp., right) ideal of S for every a∈S.Theorem 13.
Let(F,S) be a soft set over U in which (Im⁡(F),⊆) is a chain. If the nonempty δ-exclusive set of (F,S) is a left (resp., right) ideal of S for all δ⊆U, then (F,S) is a union-soft l-ideal (resp., union-soft r-ideal) over U.Proof.
Letx,y∈S be such that x≤y. If F(x)⊋F(y), then y∈eS(F;τ) and x∉eS(F;τ) by taking τ=F(y). This is a contradiction, and so F(x)⊆F(y). Assume that F(xy)⊋F(y) for some x,y∈S. Then y∈eS(F;τ) and xy∉eS(F;τ) by taking τ=F(y). This is a contradiction, and thus F(xy)⊆F(y) for all x,y∈S. Therefore (F,S) is a union-soft l-ideal over U. The right case can be seen in a similar way.Question. In Theorem 13, can we delete the condition “(Im⁡(F),⊆) is a chain”?For any soft sets(F,S) and (G,S) over U, the union-soft product, denoted by (F,S)⊙~(G,S), of (F,S) and (G,S) is defined to be the soft set (F⊙~G,S) over U in which F⊙~G is a mapping from S to P(U) given by
(25)(F⊙~G)(x)={⋂(y,z)∈Ax{F(y)∪G(z)},ifAx≠∅,U,ifAx=∅,
where Ax={(y,z)∈S×S∣x≤yz}.Proposition 14.
Let(χIc,S) and (χJc,S) be soft sets over U where I and J are nonempty subsets of S. Then the following properties hold: (1)
(χIc,S)∪~(χJc,S)=(χI∪Jc,S).(2)
(χIc,S)⊙~(χJc,S)=(χ(IJ]c,S).Proof.
(1) Letx∈S. If x∈I∪J, then x∈I or x∈J. Thus we have
(26)(χIc∪~χJc)(x)=χIc(x)∪χJc(x)=∅=χI∪Jc(x).
If x∉I∪J, then x∉I and x∉J. Hence we have
(27)(χIc∪~χJc)(x)=χIc(x)∪χJc(x)=U=χI∪Jc(x).
Therefore (χIc,S)∪~(χJc,S)=(χI∪Jc,S).
(2) For anyx∈S, suppose x∈(IJ]. Then x≤ab for some a∈I and b∈J, and so (a,b)∈Ax. Thus we have
(28)(χIc⊙~χJc)(x)=⋂(y,z)∈Ax{χIc(y)∪χJc(z)}⊆χIc(a)∪χJc(b)=∅,
and so (χIc⊙~χJc)(x)=∅. Since x∈(IJ], we get χ(IJ]c(x)=∅. Suppose x∉(IJ]. Then χ(IJ]c(x)=U. If Ax=∅, then (χIc⊙~χJc)(x)=U and (χIc⊙~χJc)(x)=χ(IJ]c(x). Assume that Ax≠∅. Then
(29)(χIc⊙~χJc)(x)=⋂(y,z)∈Ax{χIc(y)∪χJc(z)}.
We now prove that χIc(y)∪χJc(z)=U for all (y,z)∈Ax. Let (y,z)∈Ax. Then x≤yz. If y∈I and z∈J, then yz∈IJ and so x∈(IJ]. This is impossible. Thus we have y∉I or z∉J. If y∉I, then χIc(y)=U and so χIc(y)∪χJc(z)=U. Similarly, if z∉J then χIc(y)∪χJc(z)=U. In any case, we have (χIc,S)⊙~(χJc,S)=(χ(IJ]c,S).Proposition 15.
Let(F1,S), (F2,S), (G1,S), and (G2,S) be soft sets over U. If (F1,S)⊆~(G1,S) and (F2,S)⊆~(G2,S), then
(30)(F1,S)⊙~(F2,S)⊆~(G1,S)⊙~(G2,S).Proof.
For anyx∈S, if Ax=∅ then clearly (F1,S)⊙~(F2,S)⊆~(G1,S)⊙~(G2,S). Assume that Ax≠∅. Then
(31)(F1⊙~F2)(x)=⋂(y,z)∈Ax{F1(y)∪F2(z)}⊆⋂(y,z)∈Ax{G1(y)∪G2(z)}=(G1⊙~G2)(x),
and so (F1,S)⊙~(F2,S)⊆~(G1,S)⊙~(G2,S).Proposition 16.
Let(F,S) and (G,S) be a union-soft r-ideal and a union-soft l-ideal over U, respectively. Then
(32)(F,S)∪~(G,S)⊆~(F,S)⊙~(G,S).Proof.
Letx∈S. If Ax=∅, then (F⊙~G)(x)=U⊇(F∪~G)(x). Suppose that Ax≠∅. Then (F⊙~G)(x)=⋂(y,z)∈Ax{F(y)∪G(z)}. Let y,z∈S be such that (y,z)∈Ax. Then x≤yz. Since (F,S) is a union-soft r-ideal over U, it follows that F(x)⊆F(yz)⊆F(y). Since (G,S) is a union-soft l-ideal over U, we have G(x)⊆G(yz)⊆G(z). Hence F(x)∪G(x)⊆F(y)∪G(z) for all (y,z)∈Ax, and so
(33)(F⊙~G)(x)=⋂(y,z)∈Ax{F(y)∪G(z)}⊇F(x)∪G(x)=(F∪~G)(x).
Therefore (F,S)⊙~(G,S)⊇~(F,S)∪~(G,S).Proposition 17.
Let(F,S) and (G,S) be soft sets over U. If S is regular and (F,S) is a union-soft r-ideal over U, then
(34)(F,S)∪~(G,S)⊇~(F,S)⊙~(G,S).Proof.
Let(F,S) be a union-soft r-ideal over U. Let a∈S. Then there exists x∈S such that a≤(ax)a since S is regular. Thus (ax,a)∈Aa; that is, Aa≠∅, and so
(35)(F⊙~G)(a)=⋂(y,z)∈Aa{F(y)∪G(z)}.
On the other hand, since (F,S) is a union-soft r-ideal over U, we have
(36)F(ax)⊆F(a).
Hence (F∪~G)(a)=F(a)∪G(a)⊇F(ax)∪G(a). Since (ax,a)∈Aa, it follows that
(37)(F∪~G)(a)⊇F(ax)∪G(a)⊇⋂(y,z)∈Aa{F(y)∪G(z)}=(F⊙~G)(a).
Therefore (F,S)∪~(G,S)⊇~(F,S)⊙~(G,S).In a similar way we prove the following.Proposition 18.
Let(F,S) and (G,S) be soft sets over U. If S is regular and (G,S) is a union-soft l-ideal over U, then the soft inclusion (34) is valid.Corollary 19.
Let(F,S) and (G,S) be a union-soft r-ideal and a union-soft l-ideal over U, respectively. If S is regular, then
(38)(F,S)⊙~(G,S)=(F,S)∪~(G,S).We now provide a characterization of a regular ordered semigroup.Theorem 20.
An ordered semigroupS is regular if and only if the soft inclusion (34) is valid for every union-soft r-ideal (F,S) and every union-soft l-ideal (G,S) over U.Proof.
Assume thatS is regular. Proposition 17 (or Proposition 18) implies that the soft inclusion (34) is valid.
Conversely, assume that the soft inclusion (34) is valid for every union-soft r-ideal (F,S) and every union-soft l-ideal (G,S) over U. Let a∈S and b∈R(a)∩L(a). Since R(a) (resp., L(a)) is a right (resp., left) ideal of S, it follows from Theorem 9 that (χR(a)c,S) (resp., (χL(a)c,S)) is a union-soft r-ideal (resp., union-soft l-ideal) over U. Using (34), we have
(39)(χR(a)c,S)∪~(χL(a)c,S)⊇~(χR(a)c,S)⊙~(χL(a)c,S).
Since b∈R(a)∩L(a), we have χR(a)c(b)=∅=χL(a)c(b). Hence
(40)∅=χR(a)c(b)∪χL(a)c(b)=(χR(a)c∪~χL(a)c)(b)⊇(χR(a)c⊙~χL(a)c)(b),
and so (χR(a)c⊙~χL(a)c)(b)=∅. Therefore Ab≠∅. Let y,z∈S be such that (y,z)∈Ab. Suppose that y∉R(a) or z∉L(a). Then χR(a)c(y)∪χL(a)c(z)=U, and so
(41)(χR(a)c⊙~χL(a)c)(b)=⋂(y,z)∈Ab{χR(a)c(y)∪χL(a)c(z)}=U.
This is a contradiction, and thus y∈R(a) and z∈L(a). Therefore b≤yz∈R(a)L(a), which implies that b∈(R(a)L(a)]. Hence
(42)R(a)∩L(a)⊆(R(a)L(a)].
Using Lemma 1, we know that S is regular.Theorem 21.
For a union-softl-ideal (G,S) over U, the following assertion is valid
(43)(∅S,S)⊙~(G,S)⊇~(G,S),
where ∅S is an empty soft set over U; that is, ∅S(x)=∅ for all x∈S.Proof.
Suppose that(G,S) is a union-soft l-ideal over U. Let x∈S. If Ax≠∅, then
(44)(∅S⊙~G)(x)=⋂(y,z)∈Ax{∅S(y)∪G(z)}⊇⋂(y,z)∈Ax{∅∪G(yz)}⊇G(x).
Assume that Ax=∅. Then (∅S⊙~G)(x)=U⊇G(x). This completes the proof.Theorem 22.
Let(∅S,S) be the empty soft set over U and (G,S) be a soft set over U. If (G,S) satisfies the conditions (43) and (16), then (G,S) is a union-soft l-ideal over U.Proof.
For anyx,y∈S, we have
(45)G(xy)⊆(∅S⊙~G)(xy)=⋂(x,y)∈Axy{∅S(x)∪G(y)}⊆∅S(x)∪G(y)=∅∪G(y)=G(y).
Hence (G,S) is a union-soft l-ideal over U.Similarly, we have the following theorem.Theorem 23.
For the empty soft set(∅S,S) over U and a soft set (G,S) over U, the following assertions are equivalent: (1)
(G,S) is a union-soft r-ideal over U.(2)
(G,S) satisfies the conditions (16) and
(46)(G,S)⊙~(∅S,S)⊇~(G,S).Corollary 24.
For the empty soft set(∅S,S) over U and a soft set (G,S) over U, the following assertions are equivalent: (1)
(G,S) is a union-soft ideal over U.(2)
(G,S) satisfies the conditions (16) and
(47)(∅S,S)⊙~(G,S)⊇~(G,S),(G,S)⊙~(∅S,S)⊇~(G,S).Lemma 25.
If(F,S) (resp., (G,S)) is a union-soft r-ideal (resp., union-soft l-ideal) over U, then (1)
(F,S)⊙~(∅S,S)⊇~(F,S) (resp., (∅S,S)⊙~(G,S)⊇~(G,S)).(2)
(F,S)⊙~(F,S)⊇~(F,S) (resp., (G,S)⊙~(G,S)⊇~(G,S)).Proof.
(1) Assume that(F,S) is a union-soft r-ideal over U and let x∈S. If Ax=∅, then (F⊙~∅S)(x)=U⊇F(x). Assume that Ax≠∅ and let y,z∈S be such that (y,z)∈Ax. Then x≤yz, and so F(x)⊆F(yz)⊆F(y) by (16) and (17). Hence F(y)∪∅S(z)=F(y)∪∅=F(y)⊇F(x) for all (y,z)∈Ax. Therefore
(48)(F⊙~∅S)(x)=⋂(y,z)∈Ax{F(y)∪∅S(z)}⊇F(x).
Consequently, (F,S)⊙~(∅S,S)⊇~(F,S). Similarly, (∅S,S)⊙~(G,S)⊇~(G,S) for each union-soft l-ideal (G,S) over U.
(2) Note that(F,S)⊇~(∅S,S) and (F,S)⊆~(F,S). Using Proposition 15 and (1), we have (F,S)⊙~(F,S)⊇~(F,S)⊙~(∅S,S)⊇~(F,S). In a similar way, we obtain
(49)(G,S)⊙~(G,S)⊇~(G,S)
for each union-soft l-ideal (G,S) over U.Proposition 26.
LetS be a regular ordered semigroup. If (F,S) (resp., (G,S)) is a union-soft r-ideal (resp., union-soft l-ideal) over U, then
(50)(F,S)⊇~(F,S)⊙~(F,S)(resp.(G,S)⊇~(G,S)⊙~(G,S)).Proof.
Let(F,S) be a union-soft r-ideal over U and let a∈S. Then there exists x∈S such that a≤axa. Hence Aa≠∅ since (ax,a)∈Aa. Thus
(51)(F⊙~F)(a)=⋂(y,z)∈Aa{F(y)∪F(z)}⊆F(ax)∪F(a)=F(a)
since F(a)⊆F((ax)a)⊆F(ax)⊆F(a). Therefore (F,S)⊇~(F,S)⊙~(F,S). Similarly we have (G,S)⊇~(G,S)⊙~(G,S) for each union-soft l-ideal (G,S) over U.We say that a soft set(F,S) over U issoft idempotent if (F,S)⊙~(F,S)=(F,S).By Lemma25(2) and Proposition 26 we have the following result.Proposition 27.
IfS is a regular ordered semigroup, then every union-soft r-ideal (resp., union-soft l-ideal) over U is soft idempotent.Definition 28.
A soft set(F,S) over U is said to be union-soft semiprime if it satisfies
(52)(∀x∈S)(F(x)⊆F(x2)).Theorem 29.
IfS is left regular, then every union-soft l-ideal is a union-soft semiprime.Proof.
Let(F,S) be a union-soft l-ideal over U and let a∈S. Then a≤xa2 for some x∈S since S is left regular. It follows from (16) and (15) that
(53)F(a)⊆F(xa2)⊆F(a2).
Hence (F,S) is union-soft semiprime.In a similar way, we have the following theorem.Theorem 30.
IfS is right regular, then every union-soft r-ideal is union-soft semiprime.Theorem 31.
IfS is intraregular, then every union-soft ideal is union-soft semiprime.Proof.
Let(F,S) be a union-soft ideal over U and let a∈S. Then a≤xa2y for some x,y∈S since S is intraregular. It follows from (16), (15), and (17) that
(54)F(a)⊆F(xa2y)⊆F(a2y)⊆F(a2).
Hence (F,S) is union-soft semiprime.Corollary 32.
IfS is intraregular, then every union-soft ideal (F,S) over U satisfies the following equality:
(55)(∀x,y∈S)(F(xy)=F(yx)).Proof.
Using Theorem31, we have
(56)F(xy)⊆F((xy)2)=F(x(yx)y)⊆F(yx),F(yx)⊆F((yx)2)=F(y(xy)x)⊆F(xy)
for all x,y∈S. This completes the proof.


## 4. Concave Soft Sets and Critical Soft Points

For any soft set(F,S) over U, consider a soft set ([[F]],S) over U where
(57)[[F]]:S⟶P(U),x⟼⋂x≤yF(y).
Since x≤x for all x∈S, we have
(58)[[F]](x)=⋂x≤yF(y)⊆F(x)
for all x∈S. Hence (F,S)⊇~([[F]],S).A soft set(F,S) over U is said to be concave if ([[F]],S)⊇~(F,S), and hence (F,S)=([[F]],S).Theorem 33.
For a soft set(F,S) over U, the following are equivalent: (1)
(F,S) is concave.(2)
(∀x,y∈S)  (x≤y⇒F(x)⊆F(y)).Proof.
Assume that(F,S) is concave. Let x,y∈S be such that x≤y. Then
(59)F(x)=[[F]](x)=⋂x≤wF(w)⊆F(y).
Conversely, if (2) is valid, then[[F]](x)=⋂x≤yF(y)⊇F(x) for all x∈S. Hence ([[F]],S)⊇~(F,S); that is, (F,S) is concave.Proposition 34.
For any soft sets(F,S), (G,S), and (H,S) over U, we have (1)
if(F,S)⊆~(G,S), then ([[F]],S)⊆~([[G]],S).(2)
([[F]],S)⊙~([[G]],S)⊇~([[F⊙~G]],S).(3)
([[F]],S) is concave.Proof.
(1) If(F,S)⊆~(G,S), then F(x)⊆G(x) for all x∈S. Thus
(60)[[F]](x)=⋂x≤yF(y)⊆⋂x≤yG(y)=[[G]](x)
for all x∈S. Therefore ([[F]],S)⊆~([[G]],S).
(2) Letx∈S. If Ax=∅, then ([[F]]⊙~[[G]])(x)=U⊇[[F⊙~G]](x). If Ax≠∅, then x≤yz for some y,z∈S. Thus
(61)([[F]]⊙~[[G]])(x)=⋂‍(y,z)∈Ax{[[F]](y)∪[[G]](z)}=⋂(y,z)∈Ax{(⋂y≤sF(s))∪(⋂z≤tG(t))}=⋂(y,z)∈Ax{⋂y≤s,z≤t{F(s)∪G(t)}}⊇⋂(y,z)∈Ax{⋂yz≤st{F(s)∪G(t)}}=⋂x≤yz(F⊙~G)(yz)=[[F⊙~G]](x).
Therefore ([[F]],S)⊙~([[G]],S)⊇~([[F⊙~G]],S).
(3) Letx,y∈S be such that x≤y. Then
(62)[[F]](y)=⋂y≤zF(z)⊇⋂x≤zF(z)=[[F]](x).
It follows from Theorem 33 that ([[F]],S) is concave.Let(F,S) be a soft set over U. For any a∈S and any proper subset λ of U, acritical soft point, denoted by ((a]λ,S), over U is defined to be a soft set over U where
(63)(a]λ:S⟶P(U),x⟼{λ,ifx∈(a],U,otherwise.Proposition 35.
For any proper subsetsλ and δ of U, if ((a]λ,S) and ((b]δ,S) are critical soft points over U, then ((a]λ⊙~(b]δ,S)=((ab]λ∪δ,S).Proof.
Letx∈S. If x∈(ab], then Ax≠∅ and so
(64)((a]λ⊙~(b]δ)(x)=⋂(y,z)∈Ax{(a]λ(y)∪(b]δ(z)}⊆(a]λ(a)∪(b]δ(b)=λ∪δ.
Note that (a]λ(y)∪(b]δ(z)⊇λ∪δ for all y,z∈S. Hence ((a]λ⊙~(b]δ)(x)⊇λ∪δ. It follows that
(65)((a]λ⊙~(b]δ)(x)=λ∪δ=(ab]λ∪δ(x).
For x∉(ab], assume that ((a]λ⊙~(b]δ)(x)≠U. Then
(66)((a]λ⊙~(b]δ)(x)=⋂(y,z)∈Ax{(a]λ(y)∪(b]δ(z)}≠U,
and so (a]λ(y0)∪(b]δ(z0)≠U for some y0,z0∈S with x≤y0z0. Thus (a]λ(y0)=λ and (b]δ(z0)=δ; that is, y0∈(a] and z0∈(b]. It follows that y0z0⊆(a](b]⊆(ab] and that x∈(ab]. This is a contradiction. Therefore ((a]λ⊙~(b]δ)(x)=U=(ab]λ∪δ(x). Consequently, we know that
(67)((a]λ⊙~(b]δ)(x)=(ab]λ∪δ(x)
for all x∈S; that is, ((a]λ⊙~(b]δ,S)=((ab]λ∪δ,S).Corollary 36.
For any proper subsetsλ and δ of U, if ((a]λ,S) and ((b]δ,S) are critical soft points over U, then
(68)((a]λ⊙~(b]δ,S)=((b]δ⊙~(a]λ,S)⟺ab=ba.Proof.
It is straightforward.Proposition 37.
If(F,S) is a concave soft set over U, then
(69)(F,S)=(⋂(F,S)⊆~((a]λ,S)~(a]λ,S)
for any a∈S and a proper subset λ of U.Proof.
Let((a]λ,S) be a critical soft point over U such that (F,S)⊆~((a]λ,S). Then (a]λ(x)⊇F(x) for all x∈S, and so
(70)(⋂(F,S)⊆~((a]λ,S)~(a]λ)(x)=⋂(F,S)⊆~((a]λ,S)(a]λ(x)⊇⋂(F,S)⊆~((a]λ,S)F(x)=F(x)
for all x∈S. Hence
(71)(⋂(F,S)⊆~((a]λ,S)~(a]λ,S)⊇~(F,S).
On the other hand, let F(x)=λ for x∈S. Then (F,S)⊆~(xλ,S). In fact, if y∉(x] then xλ(y)=U⊇F(y). If y∈(x], then y≤x and xλ(y)=λ. Since (F,S) is concave, it follows from Theorem 33 that F(y)⊆F(x)=λ=xλ(y). Therefore (xλ,S)⊇~(F,S), and so
(72)F(x)=λ=xλ(x)⊇(⋂(F,S)⊆~((a]λ,S)(a]λ)(x).
Hence
(73)(F,S)⊇~(⋂(F,S)⊆~((a]λ,S)~(a]λ,S).Theorem 38.
For any soft sets(F,S), (G,S), and (H,S) over U, the following items are valid: (1)
if(F,S) is a union-soft ideal over U, then it is concave; that is, (F,S)=([[F]],S);(2)
if(F,S) and (G,S) are union-soft l-ideals (resp., union-soft r-ideals) over U, then so are (F⊙~G,S) and (F∩~G,S).Proof.
(1) Since every union-soft ideal(F,S) over U satisfies the condition (16), it follows from Theorem 33.
(2) Assume that(F,S) and (G,S) are union-soft l-ideals over U. For any x,y∈S with x≤y, we have
(74)(F⊙~G)(x)=⋂(a,b)∈Ax{F(a)∪G(b)}⊆⋂(a,b)∈Ay{F(a)∪G(b)}=(F⊙~G)(y).
Theorem 21 implies that
(75)(∅S,S)⊙~((F,S)⊙~(G,S))=((∅S,S)⊙~(F,S))⊙~(G,S)⊇~(F,S)⊙~(G,S).
It follows from Theorem 22 that (F⊙~G,S) is a union-soft l-ideal over U.
It is easy to verify that(∅S,S)⊙~((F,S)∩~(G,S))⊇~(F,S)∩~(G,S). Let x,y∈S be such that x≤y. Then F(x)⊆F(y) and G(x)⊆G(y). Hence
(76)(F∩~G)(x)=F(x)∩G(x)⊆F(y)∩G(y)=(F∩~G)(y).
Therefore (F∩~G,S) is a union-soft l-ideal over U by Theorem 22. Similarly, one can prove that (F⊙~G,S) and (F∩~G,S) are union-soft r-ideals over U when (F,S) and (G,S) are union-soft r-ideals over U.Theorem 39.
For any critical soft point((a]λ,S) over U, let (Fl((a]λ),S) be a soft set over U in which Fl((a]λ) is given as follows:
(77)Fl((a]λ):S⟶P(U),x⟼{λ,ifx∈L(a),U,otherwise.
Then (Fl((a]λ),S) is the greatest union-soft l-ideal over U which is contained in the critical soft point ((a]λ,S).Proof.
Letx,y∈S. If Fl((a]λ)(y)=U, then it is clear that Fl((a]λ)(xy)⊆Fl((a]λ)(y). If Fl((a]λ)(y)≠U, then Fl((a]λ)(y)=λ and y∈(S1a]. Thus y≤ba for some b∈S1, and so xy≤(xb)a. Hence xy∈L(a), and thus Fl((a]λ)(xy)=λ⊆Fl((a]λ)(y). Assume that x≤y. If y∉L(a) then Fl((a]λ)(y)=U⊇Fl((a]λ)(x). If y∈L(a) then x∈L(a) since x≤y. Thus Fl((a]λ)(x)=λ⊆Fl((a]λ)(y). Consequently, (Fl((a]λ),S) is a union-soft l-ideal over U. For each x∈S, if x∈(a] then x∈L(a) and so (a]λ(x)=λ=Fl((a]λ)(x). If x∉(a] then (a]λ(x)=U⊇Fl((a]λ)(x). Therefore ((a]λ,S)⊇~(Fl((a]λ),S). Let (G,S) be a union-soft l-ideal over U such that ((a]λ,S)⊇~(G,S). If x∈L(a), then there exists b∈S1 such that x≤ba. Hence
(78)Fl((a]λ)(x)=λ=(a]λ(a)⊇G(a)⊇G(ba)⊇G(x).
If x∉L(a), then G(x)⊆U=Fl((a]λ)(x). Therefore (G,S)⊆~(Fl((a]λ),S). This completes the proof.Similarly, we have the following theorem.Theorem 40.
For any critical soft point((a]λ,S) over U, let (Fr((a]λ),S) be a soft set over U in which Fr((a]λ) is given as follows:
(79)Fr((a]λ):S⟶P(U),x⟼{λ,ifx∈R(a),U,otherwise.
Then (Fr((a]λ),S) is the greatest union-soft r-ideal over U which is contained in the critical soft point ((a]λ,S).Theorem 41.
Let((a]λ,S) be a critical soft points over U. Then (∅S,S)⊙~((a]λ,S)⊙~(∅S,S) is a union-soft ideal over U, and
(80)(∅S⊙~(a]λ⊙~∅S)(x)={λ,ifx∈(SaS],U,ifx∉(SaS],
for all x∈S.Proof.
Letx∈S. If x∈(SaS], then there exist y,z∈S such that x≤yaz. Hence
(81)(∅S⊙~(a]λ⊙~∅S)(x)=⋂x≤x1x2x3{∅S(x1)∪(a]λ(x2)∪∅S(x3)}⊆∅S(y)∪(a]λ(a)∪∅S(z)=∅∪λ∪∅=λ.
On the other hand, ∅S(x1)∪(a]λ(x2)∪∅S(x3)=(a]λ(x2)⊇λ for all x1,x2,x3∈S, and so
(82)(∅S⊙~(a]λ⊙~∅S)(x)=⋂x≤x1x2x3{∅S(x1)∪(a]λ(x2)∪∅S(x3)}⊇λ
for all x∈S. It follows that (∅S⊙~(a]λ⊙~∅S)(x)=λ for all x∈(SaS]. Assume that x∉(SaS]. If there exist x1,x2,x3∈S such that x≤x1x2x3, then
(83)(∅S⊙~(a]λ⊙~∅S)(x)=⋂x≤x1x2x3{∅S(x1)∪(a]λ(x2)∪∅S(x3)}=⋂x≤x1x2x3(a]λ(x2).
If (∅S⊙~(a]λ⊙~∅S)(x)≠U, then there exist a,b,c∈S such that x≤abc and b∈(a]. Thus x∈(SaS] which leads a contradiction. Therefore (∅S⊙~(a]λ⊙~∅S)(x)=U. If there does not exist x1,x2,x3∈S such that x≤x1x2x3, then (∅S⊙~(a]λ⊙~∅S)(x)=U. Now, it is easy to verify that
(84)(∅S,S)⊙~(∅S⊙~(a]λ⊙~∅S,S)⊇~(∅S⊙~(a]λ⊙~∅S,S),(∅S⊙~(a]λ⊙~∅S,S)⊙~(∅S,S)⊇~(∅S⊙~(a]λ⊙~∅S,S).
Let x,y∈S be such that x≤y. Obviously, (∅S⊙~(a]λ⊙~∅S)(x)⊇λ for all x∈S. If x∈(SaS], then (∅S⊙~(a]λ⊙~∅S)(x)=λ⊆(∅S⊙~(a]λ⊙~∅S)(y). If x∉(SaS], then y∉(SaS] and so (∅S⊙~(a]λ⊙~∅S)(x)=U=(∅S⊙~(a]λ⊙~∅S)(y). Therefore (∅S,S)⊙~((a]λ,S)⊙~(∅S,S) is a union-soft ideal over U by Theorems 22 and 23.Similarly, we have the following theorems.Theorem 42.
Let((a]λ,S) be a critical soft points over U. Then (∅S,S)⊙~((a]λ,S) is a union-soft l-ideal over U, and
(85)(∅S⊙~(a]λ)(x)={λ,ifx∈(Sa],U,ifx∉(Sa],
for all x∈S.Theorem 43.
Let((a]λ,S) be critical soft points over U. Then ((a]λ,S)⊙~(∅S,S) is a union-soft r-ideal over U, and
(86)((a]λ⊙~∅S)(x)={λ,ifx∈(aS],U,ifx∉(aS],
for all x∈S.Proposition 44.
For any critical soft points((a]λ,S) and ((b]δ,S) over U, we have
(87)(∅S,S)⊙~((a]λ,S)⊙~(∅S,S)⊆~((b]δ,S)⟺b∈(SaS],iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiδ⊆λ.Proof.
If(∅S,S)⊙~((a]λ,S)⊙~(∅S,S)⊆~((b]δ,S), then (∅S⊙~(a]λ⊙~∅S)(b)⊆(b]δ(b)=δ≠U. Hence
(88)(∅S⊙~(a]λ⊙~∅S)(b)=λ⊇δ,b∈(SaS]
by Theorem 41.
Conversely, assume thatb∈(SaS] and δ⊆λ. For any x∈S, if x∈(b] then x∈(b]⊆((SaS]]=(SaS]. It follows from Theorem 41 that (∅S⊙~(a]λ⊙~∅S)(x)=λ⊇δ=(b]δ(x). If x∉(b], then (b]δ(x)=U⊇(∅S⊙~(a]λ⊙~∅S)(x). Therefore
(89)(∅S,S)⊙~((a]λ,S)⊙~(∅S,S)⊆~((b]δ,S).
This completes the proof.For any subsetD of S and a proper subset λ of U, let (FD,S) and ((λF)D,S) be soft sets over U given as follows:
(90)FD:S⟶P(U),x⟼{∅,ifx∈D,U,otherwise,(λF)D:S⟶P(U),x⟼{λ,ifx∈D,U,otherwise,
respectively. Obviously, if D=(a] then (λF)D=(a]λ.Proposition 45.
For any nonempty subsetsD and E of S and any proper subset λ of U, we have the following assertions. (1)
((λF)D⊙~(λF)E,S)=((λF)(DE],S).(2)
((λF)D∪~(λF)E,S)=((λF)D∩E,S).(3)
((λF)D∩~(λF)E,S)=((λF)D∪E,S).(4)
(91)((λF)(D],S)=(⋂a∈D~(a]λ,S).Proof.
(1) Ifx∈(DE] then F(DE](x)=∅ and x≤ab for some a∈D and b∈E. Hence (a,b)∈Ax; that is, Ax≠∅, and thus
(92)((λF)D⊙~(λF)E)(x)=⋂(y,z)∈Ax{(λF)D(y)∪(λF)E(z)}⊆(λF)D(a)∪(λF)E(b)=λ∪λ=λ.
Since (λF)D(y)⊇λ and (λF)E(z)⊇λ for all y,z∈S, we get ((λF)D⊙~(λF)E)(x)⊇λ. Therefore ((λF)D⊙~(λF)E)(x)=λ=(λF)(DE](x).
Ifx∉(DE] then (λF)(DE](x)=U. For the case Ax=∅, we have
(93)((λF)D⊙~(λF)E)(x)=U=(λF)(DE](x).
The case Ax≠∅ implies that x≤yz for all (y,z)∈Ax. If y∈D and z∈E, then yz∈DE and so x∈(DE]. This is impossible, and thus y∉D or z∉E. If y∉D, then (λF)D(y)=U and thus (λF)D(y)∪(λF)E(z)=U. Similarly, if z∉E then (λF)D(y)∪(λF)E(z)=U. Therefore
(94)((λF)D⊙~(λF)E)(x)=⋂(y,z)∈Ax{(λF)D(y)∪(λF)E(z)}=U.
Consequently, (1) is true.
(2) Letx∈S. If x∈D∩E, then x∈D and x∈E, and so
(95)(λF)D∩E(x)=λ=λ∪λ=(λF)D(x)∪(λF)E(x)=((λF)D∪~(λF)E)(x).
Assume that x∉D∩E. Then (λF)D∩E(x)=U. If x∉D, then (λF)D(x)=U and so
(96)((λF)D∪~(λF)E)(x)=(λF)D(x)∪(λF)E(x)=U=(λF)D∩E(x).
Similarly, if x∉E then ((λF)D∪~(λF)E)(x)=(λF)D∩E(x). Therefore
(97)((λF)D∪~(λF)E,S)=((λF)D∩E,S).
(3) Letx∈S. If x∈D∪E, then x∈D or x∈E, which implies that (λF)D(x)=λ or (λF)E(x)=λ. Hence
(98)(λF)D∪E(x)=λ=λ∩λ=(λF)D(x)∩(λF)E(x)=((λF)D∩~(λF)E)(x).
Suppose that x∉D∪E. Then x∉D and x∉E. It follows that
(99)((λF)D∩~(λF)E)(x)=(λF)D(x)∩(λF)E(x)=U=(λF)D∪E(x).
Therefore ((λF)D∩~(λF)E,S)=((λF)D∪E,S).
(4) Letx∈S. If x∈(D], then x≤b for some b∈D. Hence
(100)(⋂a∈D~(a]λ)(x)=⋂a∈D(a]λ(x)⊆bλ(x)=λ.
Note that (a]λ(x)⊇λ for any critical soft point (a]λ over U. Thus
(101)(⋂a∈D~(a]λ)(x)=⋂‍a∈D(a]λ(x)⊇λ.
Conditions (100) and (101) induce
(102)(⋂a∈D~(a]λ)(x)=λ=(λF)(D](x).
If x∉(D], then(i)
(λF)(D](x)=U and(ii)
x∉(a] for all a∈D, and so (a]λ(x)=U for all a∈D.It follows that
(103)(⋂a∈D~(a]λ)(x)=⋂a∈D(a]λ(x)=U=(λF)(D](x).
Therefore
(104)((λF)(D],S)=(⋂a∈D~(a]λ,S).Theorem 46.
IfD is a left ideal of S, then ((λF)D,S) is a union-soft l-ideal over U.Proof.
Suppose thatD is a left ideal of S. Let x,y∈S. If y∈D, then xy∈D and so (λF)D(xy)=λ=(λF)D(y). If y∉D, then (λF)D(y)=U⊇(λF)D(xy). Assume that x≤y. If y∈D, then x∈D and thus (λF)D(x)=λ=(λF)D(y). If y∉D, then (λF)D(y)=U⊇(λF)D(x). Therefore ((λF)D,S) is a union-soft l-ideal over U.In the same way, we can verify the following result.Theorem 47.
IfD is a right ideal of S, then ((λF)D,S) is a union-soft r-ideal over U.



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*Source: 467968-2014-10-19.xml*