On Uni-Soft Ideals and Uni-Soft Interior Ideals of Ordered Semigroups Raees Khan

(1)

and Biological Sciences www.textroad.com

On Uni-Soft Ideals and Uni-Soft Interior Ideals of Ordered Semigroups

Raees Khan^*1, Asghar Khan², Imarn Khan¹, M. Uzair Khan¹, Zia Ur Rahman³, Shahid Ali³

1Department of Mathematics &Statistics, Bacha Khan University, Charsadda, KPK, Pakistan

2Department of Mathematics, Abdul Wali Khan University, Mardan, KPK, Pakistan

3Department of Computer Science, Bacha Khan University, Charsadda, KPK, Pakistan

Received: January 18, 2017 Accepted: April 9, 2017

ABSTRACT

In this paper, we define and investigate some properties of uni-soft ideals and uni-soft interior ideals of ordered semigroups. Furthermore, we prove that in regular ordered semigroups the uni-soft ideals and the uni-soft interior ideals coincide. Finally, we introduce the concept of uni-soft simple ordered semigroups and show that in a simple ordered semigroup every uni-soft interior ideal is a constant function. Using the notion of uni-soft left (right) ideals, characterizations of a left (right) simple semigroup are provided.

KEYWORDS: Ordered semigroups, Soft sets, uni-soft interior ideal, simple ordered semigroup, regular (intra- regular) ordered semigroup, characteristic soft set.

1. INTRODUCTION

“

Molodstov [1] pointed out that the important existing theories viz. Probability theory, Fuzzy set theory [2], Intuitionistic Fuzzy set theory [3], rough set theory [4] etc, which can be considered as mathematical tools for dealing with uncertainties, have their own difficulties. He further pointed out that the reason for these difficulties is the inadequacy of parameterization tools of the theory. In

1999

he initiated the novel concept of soft set theory to deal with the uncertainty associated with the data, whereas on the other hand it has the ability to represent the data in a useful manner.”In soft set theory, the problem of setting the membership function does not arises, which make the theory easily applied to many different fields including game theory, Riemann integration, operation research, perron integration, theory of measurement and so on. At present work, soft set theory and its applications are progressing rapidly. After Molodstov work, some operations and applications of soft sets were studied by many researchers including Ali et al. [5], Aktas and Cağman [6], Chen et al. [7], and Maji et al. [8]. “Recently, Neog and Sut [9, 10] have studied the notion of fuzzy soft union, fuzzy soft intersection, complement of a fuzzy set and several other properties of fuzzy soft sets along with examples and proofs of certain results. Jun et al., [11] 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 sub semigroups, soft

r

-ideals, soft

l

-ideals, and

r

-idealistic and

l

- idealistic soft ordered semigroups. They investigated various related properties by using these notions. In [18, 19, 20] Khan et al., characterized different classes of ordered semigroups by using uni-soft quasi-ideals and uni- soft ideals. In this paper the concept of a uni-soft interior ideal is introduced and it is given that in regular and intra regular ordered semigroups, the concepts of uni-soft ideals and uni-soft interior ideals are coincide.”We also prove that in a simple ordered semigroup every uni-soft interior ideal is a constant function. Using the notion of uni-soft left (right) ideals, characterizations of a left (right) simple semigroup are provided. It is shown that if

A

is an interior ideal of

S

if and only if

( ) ε , δ −

characteristic soft set

( χ

A⁽^ε^,^δ⁾

, ^S )

over

U

is a uni-soft interior ideal over

U .

2. Basic definitions and preliminaries

“

By an ordered semigroup

( or po - semigroup )

we mean a structure

( S , ⋅, ≤ )

in which

( ) ^{S ⋅} ^,

is a semigroup,

( ^{S ≤} ^, )

is a poset and for all

a , b , x ∈ S ,

the condition

a ≤ b ⇒ xa ≤ xb

and

ax ≤ bx

is satsfied.”

For subsets

A

and

B

of an ordered semigroup

S ,

we denote

}.

,

| {

: ab a A b B

AB = ∈ ∈

(2)

“If

A ⊆ S

, we denote

( A ] : = { t ∈ S | t ≤ h

for some

h ∈ A }.

For

a ∈ S ,

we write

(a ]

instead of

⁽ { } ^a ^]

^.

For subsets

A

and

B

of an ordered semigroup

S

, we have

A ⊆ ( ] A

. If

A ⊆ B

, then

( A ⊆ ] ( B ]

,

]

( ] ](

( A B ⊆ AB

,

(( A = ]] ( A ]

and

(( A ]( B ]] ⊆ ( AB ].

Let

( S , ⋅ , ≤ )

be an ordered semigroup. A non-empty subset of

S

is called a subsemigroup of

S

if

A ⊆

²

A

.”

A non-empty subset of

S

is called a right (resp., left) ideal of

S

if:

(1)

AS ⊆ A

(resp.,

SA ⊆ A

) and (2) if

a ∈ A

and

S ∋ b ≤ a

, then

b ∈ A

.

If

A

is both a right and a left ideal of

S

, then it is called an ideal of

S

. A non-empty subset of

S

is called ainteriorideal of

S

if:

(1)

SAS ⊆ A

,

(2) if

a ∈ A

and

S ∋ b ≤ a

, then

b ∈ A

.

An ordered semigroup

S

is said to be regular if for every

x ∈ S ,

there exist

a ∈ S

such that

a ≤ axa .

. An ordered semigroup

S

is called intra-regular if for every

a ∈ S

, there exist

x , y ∈ S

such that

a ≤ xa

²

y

. An ordered semigroup

S

is called semi simple if for every

a ∈ S

, there exist

x , y , z ∈ S

such that

xayaz

a ≤

.

An ordered semigroup

S

is said to be left (resp., right) simple if it contains no proper left

( resp., right )

ideal.

An ordered semigroup is said to be simple if it contains no proper two-sided ideal.

From now on,

U

be an initial universe set,

E

be a set of parameters,

P (U )

denotes the power set of

U

and

E

C B

A , , ... ⊆

.

A soft set theory is introduced by Çağman [6] provided new definitions and various results on soft set theory.

“A pair

( f

_A

, E )

is called a soft set over

U

if and only if

f

_A is a mapping of

E

into the set of all subsets of the set

U

.

e, g

{ ( x , f ( x )) | x E , f ( x ) P ( U ) } ,

f

_A

=

_A

∈

_A

∈

where

f

_A

: E → P ( U )

such that

f

_A

(x ) = φ

if

x ∉ A

. The function

f

_A is also called anapproximation function of the soft

( f

_A

, E )

. It is clear that a soft set is a parameterized family of subsets of

U

. Note that the set of all soft sets over

U

will be denoted

S (U )

and we take

E = S .

_”

For a soft

( f

_A

, S )

over

U

and

δ ⊆ U

. The

δ −

exclusive set of

( f

_A

, S )

, denoted by

e

_A

( f

_A

; δ )

, is defined as

{ ⁽ ⁾ } ^.

)

;

( f δ = x ∈ A f x ⊆ δ

e

_A _A _A

For any soft sets

( f

_A

, S )

and

( f

_B

, S )

over

U

, we define

) ,

~ ( ) ,

( f

_A

S ⊆ f

_B

S

if

f

_A

( x ) ⊆ f

_B

( x ), ∀ x ∈ S

The soft union of

( f

_A

, S )

and

( f

_B

, S ) ,

denoted by

( f

_A

, S ) ∪ ~ ( f

_B

, S ) = ( f

_A_∪^~_B

, S )

is defined by

. ),

( ) ( )

~ )(

( f

_A

∪ f

_B

x = f

_A

x ∪ f

_B

x ∀ x ∈ S

The soft intersection of

( f

_A

, S )

and

( f

_B

, S ) ,

denoted by

( f

_A

, S ) ∩ ~ ( f

_B

, S ) = ( f

_A_∩^~_B

, S )

is defined by

. ),

( ) ( )

~ )(

( f

_A

∩ f

_B

x = f

_A

x ∩ f

_B

x ∀ x ∈ S

“For a non-empty subset

A

of

S

, the characteristic soft set

( χ

_A

, S )

over

U

is a soft set defined as follows:

 



∈

→ ∈

.

\ if

, if ),

(

: x S A

A x x U

U P

A

S φ

χ ^a

Let

( f

_S

, S )

and

( g

_S

, S )

be uni-soft set over

U

. The uni-soft product of

( f

_S

, S )

and

( g

_S

, S )

is defined to be the soft

( f

_S

◊ g

_S

, S )

over

U

in which

f ◊

_S

g

_S is a mapping from

S

to

P ( ) U

given by

( )( ) { ( ) ( ) }

 

 ∪ ≤ ∈

=

◊

^≤

otherwise.

, , some for if

U

S z y yz

x z g y x f

g

f

_S _S ^x

I

^yz ^S ^S

”

(3)

“Definition 1.(see [11]) Let

S

be an ordered semigroup. A soft set

( f

_S

, S )

of

S

over

U

is called a uni-soft semigroup of

S

over

U

if:

. , all for ) ( ) ( )

( xy f x f y x y S

f

_S

⊆

_S

∪

_S

∈

Definition 2.(see [11]) Let

( S , ⋅ , ≤ )

( f

_S

, S )

of

S

over

U

is called a uni-soft left (resp., right) ideal of

S

over

U

if:

(1)

x ≤ y ⇒ f

_S

( x ) ⊆ f

_S

( y ),

(2)

f

_S

( xy ) ⊆ f

_S

( y ) (

resp.,

f

_S

( xy ) ⊆ f

_S

( x ))

for all

x , y ∈ S .

If

( f

_S

, S )

is both a uni-soft left ideal and a uni-soft right ideal over

U

, then

( f

_S

, S )

is called a uni-soft two-sided ideal over

U .

”

3. Uni-soft ideals and uni-soft interior ideals of ordered semigroups

In this section, we first introduce the concepts of uni-soft interior ideals of ordered semigroup. And And then show that every uni-soft ideal is a uni-soft interior ideal.

“Definition3.Let

( S , ⋅ , ≤ )

be an ordered semigroup. A soft

( f

_S

, S )

over

U

is called uni-soft interior ideal over

U

if:

(1)

x ≤ y ⇒ f

_S

( x ) ⊆ f

_S

( y ),

(2)

f

_S

( xay ) ⊆ f

_S

( a )

for all

a , x , y ∈ S .

”

“Example 1.[19] Let

S = { a , b , c , d }

be an ordered semigroup with the following multiplication table and order relation which are given in the following:

b b a a d

a b a a c

a a a a b

a a a a a

d c b

⋅ a

)}.

, ( ), , ( ), , ( ), , ( ), , ( ), , {(

: = a a b b c c d d a b a d

≤

Let

( f

_S

, S )

be a soft set over

U = Z

defined by

6N if , : ( ), 3Z if { , }, 3N if .

S

x a

f S P U x x b d

x c

 =

→   ∈

 =

 a

Then

f

_S

( xyz ) = f

_S

( a ) = 6 N ⊆ f

_S

( y )

for every

x , y , z ∈ S

. Therefore,

( f

_S

, S )

is a uni-soft inteior ideal over

U

.”

“Example 2.Consider the ordered semigroup in Example

1

. Let

( f

_S

, S )

be a soft set over

{ x y x y e xy yx } { e x y yx }

D

U =

₂

= < , > :

²

=

²

= , = = , , ,

defined by

{ }

{ } { }

 { }



 



=

∈

=

→

. if ,

, , if , ,

, if )

( ),

( :

c x x

e

d b x y x e

a x e

x f x U P S

f

_S

a

_S

Then

f

_S

( xyz ) = f

_S

( a ) = { } e ⊆ f

_S

( y )

for every

x , y , z ∈ S

. Therefore,

( f

_S

, S )

U

.”

“Lemma 1.Let

( S , ⋅, ≤ )

be an ordered semigroup. Then every uni-soft two-sided ideal over

U

U .

_”

Proof. Let

( ^f

_S

^, ^S )

be an uni soft two-sided ideal over

U

and

x , y , a ∈ S

. Then,

(4)

).

( ) ( ) ) ((

)

( xay f xa y f xa f a

f

_S

=

_S

⊆

_S

⊆

_S

Let

x , y ∈ S

be such that

x ≤ y

. Then

f

_S

( x ) ⊆ f

_S

( y )

, because

( ^f

S

, ^S )

is a uni-soft two-sided ideal over

.

U

Thus

( f

_S

, S )

U .

“As it is well-known that every uni-soft two-sided ideal is a uni-soft interior ideal, but the converse is not true in general.

The uni-soft interior ideal

( f

_S

, S )

over

U = Z

in Example 1, is not a uni-soft left ideal over

U

, since

) ( N 3 Z 3 ) ( )

( dc f b f c

f

_S

=

_S

= ⊄ =

_S and hence it is not a uni-soft two-sided ideal over

U = Z .

”

“Example 1. Consider the ordered semigroup in Example

1

. Define a soft set

( f

_S

, S )

over

{ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 }

=

U

such that

{ }

 { }



 



=

∈

=

→

. if 2 , 1

}, , { if 3 , 2 , 1

, if 2 ) ( ),

( :

c x

d b x

a x x

f x U P S

f

_S

a

_S

Then

f

_S

( xyz ) = f

_S

( a ) = { } 2 ⊆ f

_S

( y )

for every

x , y , z ∈ S

. Therefore,

( f

_S

, S )

U

_.

Consider the uni-soft interior ideal

( f

_S

, S )

over

^U ⁼ { ¹ ^, ² ^, ³ ^, ⁴ ^, ⁵ ^, ⁶ ^, ⁷ ^, ⁸ ^, ⁹ ^, ¹⁰ }

in Example 2, is not a uni- soft left ideal over

U

, since

f

_S

( dc ) = f

_S

( b ) = { 1 , 2 , 3 , } { } ⊄ 1 , 2 = f

_S

( c )

and hence it is not a uni-soft two- sided ideal over

U .

In the following result we provide a condition for a uni-soft interior ideal over

U

to be a uni-soft two-sided ideal over

U

.”

“Theorem 1.Let

( S , ⋅, ≤ )

be a regular ordered semigroup. Then every uni-soft interior ideal over

U

U .

”

“Proof. Let

( f S

_S

, )

U

and let

x , y ∈ S .

Since

S

is a regular, then there exist

S

b

a , ∈

such that

x ≤ xax

and

y ≤ yby .

we have”

( ) ⁾ ⁽ ( ) ⁾ ⁽ ^),

( )

( xy f xax y f xa xy f x

f

_S

⊆

_S

=

_S

⊆

_S

and

^f

S

⁽ ^xy ⁾ ⊆ ^f

S

⁽ ^x ( ^yby ) ⁾ = ^f

S

⁽ ^xy ( ) ^by ⁾ ⊆ ^f

S

⁽ ^y ^).

Now let

x , y ∈ S

be such that

x ≤ y

. Then

f

_S

( x ) ⊆ f

_S

( y )

, because

( ^f

S

, ^S )

is a uni-soft interior ideal of

S

over

U .

Therefore

( ^f

S

, ^S )

U .

By Lemma

1

and Theorem

1

we have the following:

“Remark 1.In regular ordered semigroups, the concepts of uni-soft two-sided ideals and uni-soft interior ideals coincide.”

“Theorem 2.Let

( S , ⋅, ≤ )

be an intra-regular ordered semigroup,

( f

_S

, S )

U

. Then

( f

_S

, S )

U

.”

Proof.Let

( f

_S

, S )

U .

Let

x y , ∈ S ,

since

S

is a intra-regular then there exist

S

b

a , ∈

such that

x ≤ ax

²

a

and

y ≤ by

²

b .

Since

( f

_S

, S )

is an uni-soft interior ideal of

S

, we have

( ) ⁾ ⁽ ( ) ( ) ⁾ ⁽ ^),

( )

( xy f ax

²

a y f ax x ay f x

f

_S

⊆

_S

=

_S

⊆

_S

and

^f

_S

⁽ ^xy ⁾ ^⊆ ^f

_S

⁽ ^x ( ^by

²

^b ) ⁾ ⁼ ^f

_S

⁽ ( ) ( ) ^xb ^y ^yb ⁾ ^⊆ ^f

_S

⁽ ^y ^).

Let

x , y ∈ S

be such that

x ≤ y

. Then

f

_S

( x ) ⊆ f

_S

( y )

, because

( ^f

S

, ^S )

S

over

U .

Thus

( ^f

_S

^, ^S )

is a uni-soft tw0-sided ideal of

S

over

U .

By Lemma

1

and Theorem

2

(5)

Remark2.In intra regular ordered semigroups, the concepts of uni-soft two-sided ideals and uni-soft interior ideals coincide.”

“Theorem 3. Let

( S , ⋅, ≤ )

be an ordered semigroup with identity

e

. Then every uni-soft interior ideal over

U

is a uni-soft two sided ideal over

U .

_”

Proof. Let

( f

_S

, S )

U

and

x , y ∈ S

. Then

^f

S

( ) ^xy = ^f

S

( ) ^xye ⊆ ^f

S

( ) ^y

and

^f

_S

( ) ^xy ⁼ ^f

_S

( ) ^exy ^⊆ ^f

_S

( ) ^x ^.

Furthermore let

x , y ∈ S

be such that

x ≤ y

. Then

) ( )

( x f y

f

_S

⊆

_S , because

( ^f

S

, ^S )

S

over

U .

Therefore,

( ^f

S

, ^S )

U .

“Proposition 1.(see [17]) Let

( S , ⋅ , ≤ )

be a semi simple ordered semigroup,

( ^f

_S

^, ^S )

a uni-soft interior ideal of

S

over

U

. Then

( ^f

S

, ^S )

is a uni-soft two-sided ideal of

S

over

U .

_”

By Lemma

1

and Proposition

1

“Remark 3.In semisimple ordered semigroups, the concepts of uni-soft ideals and uni-soft interior ideals coincide.”

For a non-empty subset

A

of

S

and

ε , δ ∈ P ( ) U

with

ε ⊄ δ ,

define a soft set

χ

_A⁽^ε^,^δ⁾ as follows:

( )

 

 ∈

→ otherwise.

, if ,

,

:

δ

χ

_A^ε^δ

S x a ε ^x ^A

The soft set

( χ

⁽A^ε^,^δ⁾

, ^S )

is called

( ) ε , δ −

characteristic uni-soft set over

U

. The soft set

( χ

⁽A^ε^,^δ⁾

, ^S )

is called

( ) ε , δ −

identity uni-soft set over

U .

The

( ) ε , δ −

characteristic uni-soft set

( χ

⁽A^ε^,^δ⁾

, ^S )

with

φ

ε =

and

δ = U

is called the characteristic soft set, and is denoted by

( χ

A^c

, ^S )

. The

( ) ε , δ −

identity uni-soft set with

ε = φ

and

δ = φ

is called the empty soft set, and is denoted by

( χ

S^c

^, ^S ) ^.

“Theorem 4. Let

( S , ⋅, ≤ )

be an ordered semigroup. Then for a non-empty subset

A

of

S ,

the following conditions are equivalent:

( ) ¹ ^A

S .

( ) ²

^The

( ) ε , δ −

( χ

⁽A^ε^,^δ⁾

, ^S )

over

U

for any

( ) U

∈ P δ

ε ,

with

ε ⊄ δ .

”

“Proof. Suppose that

A

S

and

x , y , z ∈ S .

Let

ε , δ ∈ P ( ) U

with

ε ⊄ δ

, if

A

y ∉

implies that

χ

⁽_A^ε^,^δ⁾

( ) y = δ

, then either

xyz ∈ A

or

xyz ∉ A

implies that ( ^, )

( ) xyz

⁽_A^, ⁾

( ) y .

A

δ ε δ

ε

δ χ

χ ⊆ =

If

y ∈ A , χ

⁽_A^ε^,^δ⁾

( ) y = ε

and

xyz ∈ SAS ⊆ A .

Hence

( ^, )

( ) xyz

_A⁽ ^, ⁾

( ) y .

A

δ ε δ

ε

ε χ

χ = =

Now let

x , y ∈ S

be such that

x ≤ y .

If

y ∈ A

then

χ

⁽_A^ε^,^δ⁾

( y ) = ε .

Since

A

is a interior ideal of

S

, so

x ∈ A

. Hence

χ

⁽_A^ε^,^δ⁾

( x ) = ε = χ

⁽_A^ε^,^δ⁾

( y ).

If

y ∉ A

then

χ

⁽_A^ε^,^δ⁾

( y ) = δ ,

then obviously,

( ^, )

( ) ^x

A⁽ ^, ⁾

⁽ ^y ^).

A

δ ε δ

ε

δ χ

χ ⊆ =

Therefore,

( χ

⁽A^ε^,^δ⁾

, ^S )

U

for any

( )

, P U

ε δ ∈

with

ε δ ⊄ .

Conversely, suppose that

( ) ε , δ −

( χ

⁽A^ε^,^δ⁾

, ^S )

over

U

for any

ε , δ ∈ P ( ) U

with

ε ⊄ δ .

Let

a

be any element of

SAS .

Then

a = xyz

for some

S

z

x , ∈

and

y ∈ A .

Then

( ^, )

( ) χ

⁽ ^, ⁾

( ) χ

⁽ ^, ⁾

( ) ε , χ

_A^ε^δ

a =

_A^ε ^δ

xyz ⊆

_A^ε^δ

y =

and so

χ

⁽_A^ε^,^δ⁾

( ) a = ε .

Thus

a ∈ A ,

which means that

SAS ⊆ A .

Furthermore, let

x , y ∈ S

with

x ≤ y ∈ A .

Then

^χ

⁽A^ε^,^δ⁾

( ) ^y ⁼ ^ε ^.

By hypothesis we have ( ^, )

( ) χ

⁽ ^, ⁾

( ) ε ,

χ

_A^ε ^δ

x ⊆

_A^ε ^δ

y =

(6)

which means that

χ

⁽_A^ε^,^δ⁾

( ) x = ε .

Hence

x ∈ A .

Therefore

A

S .

”

“Theorem 5. Let

( S , ⋅, ≤ )

be an ordered semigroup. Then for a non-empty subset

A

of

S ,

( ) ¹ ^A

S .

( ) ²

The characteristic soft set

( χ

^cA

, ^S )

over

U

.”

Proof.The proof follows from Theorem 4.

“Theorem 6. Let

( S , ⋅, ≤ )

be an ordered semigroup and

( f

_S

, S )

a soft set of

S

over

U

. Then

( f

_S

, S )

is a uni-soft uni-soft interior ideal over

U

if and only if satisfies that”

(1)

( ^f ^, ^S ) ^~ ( ^f

S

^, ^S ) ^,

c S S c

S

◊ ◊ χ ⊇

χ

(2)

x ≤ y ⇒ f

_S

( x ) ⊆ f

_S

( y ), ∀ x , y ∈ S .

Proof. Suppose that

( ^f

S

, ^S )

U .

Let

x

be any element of

S

and there exist

y , z , p , q ∈ S .

If

x

is not expressed as

x ≤ yz

, then

( χ

_S^c

◊ ^f

_S

◊ χ

_S^c

) ( ) ^x = ^U ⊇ ( )( ) ^f

_S

^x ^.

Assume that

x

is expressed as

x ≤ yz

and

y ≤ pq ,

then

“

( ) ^{( )} { ( ) ^{( )} ^{( )} }

( ) ( )

{ } ^{( )}

( )

{ }

( )

{ }

( ) x , f

q f

z q

f p

z y

f x

f

S pqz S x

pq S y yz x

c S S

c pq S y yz x

c S S

c yz S x c

S S c S

⊇

∪

⊇

 



 

  ∪



 



 ∪

⊇

 



 

  ∪



 



 ∪

=

∪

◊

=

◊

≤

φ

φ φ

χ χ

I I I

”

since

( f

_S

, S )

U .

So

f

_S

( ) x ⊆ f

_S

( pqz ) ⊆ f

_S

( ) q .

Therefore,

( ^f ^, ^S ) ^~ ( ^f

S

^, ^S ) ^.

c S S c

S

◊ ◊ χ ⊇

χ

Conversely, assume that

( χ

_S^c

◊ ^f

_S

◊ χ

_S^c

^, ^S ) ⊇ ^~ ( ^f

_S

^, ^S ) ^.

For any

a , x , y ∈ S ,

we have

( ) ( ) ^{( )}

( ) ^{( )} ^{( )}

{ }

( ) ( ) ( )

( ) ( )

{ }

( ) ( ) ( ) ( ) a . f

a f

a f x

q f p

y xa

f

q p

f xay f

xay f

S S

S c

S

S c

pq S xa

c S S

c S

c S S

c pq S xay

c S S c S S

=

∪

=

∪

⊆

 ∪



 



 ∪

=

∪

◊

⊆

∪

◊

=

◊

⊆

≤

φ χ

χ χ

I I

Therefore,

( f

_S

, S )

U .

“Lemma 2. (see[31]).Let

( S , ⋅, ≤ )

be an ordered semigroup. For any non-empty subset

A

of

S ,

( ) ¹ ^A

is a left

( resp., right )

ideal of

S .

( ) ²

The characteristic uni-soft set

( χ

A^c

, ^S )

over

U

is a uni-soft left

( ^resp., ^right )

ideal over

U .

_”

(7)

4. Uni-soft simple ordered semigroups

In this section, we define uni-soft simple ordered semigroups and characterize simple ordered semigroups in terms of uni-soft interior ideals.

“Definition 4. An ordered semigroup

S

is called uni-soft simple if for any uni-soft ideal of

S

over , we have

( ) ( )

S S

f x ⊆ f y

, for all S ∀ , ∈ .

Lemma 3. (see [13]). An ordered semigroup

S

is left simple if and only if for every

a ∈ S

, we have

]

(Sa S =

.”

“Theorem 7. Let

( S , ⋅, ≤ )

be an ordered semigroup. Then the following conditions are equivalent:

( ) 1 S

is a left

( resp., right )

simple ordered semigroup

.

( ) 2

Every uni-soft left

( resp., right )

ideal

( f

_S

, S )

over

U

is a constant function

.

_”

Proof.Let

( f

_S

, S )

be a uni-soft left ideal over

U

and

a , b ∈ S .

Since

S

is left simple and

b ∈ S ,

by Lemma 3, we have

S = ( Sb ] .

Since

a ∈ S ,

we have

^a ^∈ ( ^Sb ] ^,

^then

^a ^≤ ^xb

for some

x ∈ S .

Since

( f

_S

, S )

is a uni-soft left ideal over

U

, We have

( ) a f ( ) xb f ( ) b ,

f

_S

⊆

_S

⊆

_S

In a similar way, we can prove that

^f

_S

( ) ^b ^⊆ ^f

_S

( ) ^a ^.

^Hence

^f

_S

( ) ^a ⁼ ^f

_S

( ) ^b ^.

This implies that

( ) ^U

P S

f

_S

: →

is constant, since

a

and

b

are arbitrarily in

S .

Similarly if

S

is a right simple ordered semigroup, then every uni-soft right ideal over

U

is a constant function.

Conversely, assume that

( ) 2

holds. Let

A

be a left ideal of

S .

Then the characteristic soft set

( χ

A^c

, ^S )

over

U

is a uni-soft left ideal over

U

by lemma

2,

and so it is constant by supposition. For any

x ∈ S ,

we have

χ

^c_A

( ) x = φ

, since

A

is non-empty, and thus

x∈ A .

This shows that

A = S .

Therefore

S

is left simple.

In the case that

S

is right simple, the proof follows similarly.

“Theorem 8. Let

( S , ⋅, ≤ )

be a simple ordered semigroup. Then every uni-soft interior ideal over

U

is constant.

Proof. Suppose that

S

be a simple ordered semigroup,

( f

_S

, S )

be a uni-soft interior ideal over

U

and

.

, b S

a ∈

Since

S

is simple and

b ∈ S ,

we have

S = ( SbS ] .

For

a ∈ S

we have

a ∈ ( SbS ] ,

it follows that

a ≤ xby

for some

x , y ∈ S .

By hypothesis we have

( ) a f ( ) xby f ( ) b ,

f

_S

⊆

_S

⊆

_S

Similarly we have

^f

S

( ) ^b ⊆ ^f

S

( ) ^a ^,

and so

^f

S

( ) ^a = ^f

S

( ) ^b ^.

This shows that

^f

S

: ^S → ^P ( ) ^U

is constant, since

a

and

b

are arbitrarily in

S .

“Theorem 9.Let

( S , ⋅, ≤ )

( ^f

_S

^, ^S )

over

U

if and only if the non-empty

δ −

exclusive set of

( ^f

S

, ^S )

S

for all

( ) ^U ^.

∈ P

δ

”

Proof. Suppose that

( ^f

S

, ^S )

U .

Let

δ ∈ P (U )

be such that

(

_S

^; δ ) ≠ φ ^.

S

f

e

Let

x , y ∈ S

and

a ∈ e

_S

( f

_S

^; δ ) ^,

then

f

_S

( ) x ⊆ δ ^.

It follows from hypothesis that

( ) xay ⊆ f ( ) a ⊆ δ .

f

_S _S

Thus

xay ∈ e

S

( f

S

^; ^δ ) ^.

Furthermore, let

x , y ∈ S

be such that

x ≤ y ∈ e

S

( f

S

^; ^δ ) ^.

Then

^f

S

( ) ^y ^⊆ ^δ ^.

By hypothesis we have

( ) ^x ^⊆ ^f ( ) ^y ^⊆ ^δ ^.

f

_S _S

Which means that

x ∈ e

S

( f

S

^; ^δ ) ^.

Therefore,

e

_S

( f

_S

; δ )

S .

(8)

Conversely, assume that the non-empty

δ −

exclusive set of

( f

_S

, S )

S

for all

( ) U .

∈ P

δ

Let

x , y , z ∈ S

be such that

^f

S

( ) ^y ⁼ ^δ ^.

Then

y ∈ e

S

( f

S

^; ^δ ) ^.

Since

e

_S

( f

_S

; δ )

S

, so

xyz ∈ e

S

( f

S

^; ^δ ) ^.

Hence

( ) xyz f ( ) y . f

_S

⊆ δ =

_S

Now let

x , y ∈ S

with

x ≤ y

be such that

f

_S

( y ) = δ ,

which means that

y ∈ e

_S

( f

_S

; δ )

. Since

(

_S

; δ )

S

f

e

S

, so

x ∈ e

_S

( f

_S

; δ ) .

Hence

f

_S

( x ) ⊆ δ = f

_S

( y ).

Therefore,

)

,

( f

_S

S

U

.

“Theorem 10.Let

( S , ⋅, ≤ )

be an ordered semigroup. Then the soft union of two uni-soft interior

( resp., left, right )

ideals over

U

is also a uni-soft interior

( resp., left, right )

ideal over

U .

” Proof. Let

( f

_S

, S )

and

( g

_S

, S )

be union soft interior ideals over

U .

For any

x , a , y ∈ S

, we have

( )( ) ( ) ( )

( ) ( ) ( ~ )( ) .

~

a g f

a g a f

xay g xay f xay g f

S S

∪

=

∪

⊆

∪

=

∪

Furthermore, let

x , y ∈ S

such that

x ≤ y .

Since

( f

_S

, S )

and

( g

_S

, S )

are union soft interior ideals over

,

U

so we have

( )( ) ( ) ( )

( ) ( ) ( ~ )( ) .

~

y g f

y g y f

x g x f x g f

S S

∪

=

∪

⊆

∪

=

∪

Therefore,

( f

_S

∪ ~ g

_S

, S )

U .

In a similar way,

( ^f

_S

∪ ^~ ^g

_S

^, ^S )

is a uni-soft left

( resp., right )

ideal over

U .

“Theorem 11.Let

( S , ⋅, ≤ )

be an ordered semigroup. Then for any uni-soft set

( f

_S

, S )

over

U ,

let

( ^f

S^∗

, ^S )

be a uni-soft over

U

defined by

( ) ( ) ( )

 

 ∈

∗

→

otherwise,

,

; if

,

: ρ

S

δ

S S

S

f e x x x f

U P S

f a

where

δ

and

ρ

are subsets of

U

with

( )

( ) .

;

ρ

δ

⊄

∉

x f

_S

f e x _S _S

U

^If

( f

_S

, S )

,

U

then so is

( ^f

S

^{, S} ) ^.

∗ ”

“Proof. Let

x , y , z ∈ S .

If

y ∈ e

S

( f

S

^; ^δ ) ^,

then

xyz ∈ e

S

( f

S

^; ^δ ) ^,

as

e

_S

( f

_S

; δ )

S

by Theorem

9.

Hence

( ) ^xyz ^f ( ) ^xyz ^f ( ) ^y ^f ( ) ^y ^.

f

_S^∗

=

_S

⊆

_S

=

_S^∗ Now if

y ∉ e

S

( f

S

^; ^δ ) ^,

then

f

_S^∗

( ) y = ρ .

Thus

f

_S^∗

( ) xyz ⊆ δ = f

_S^∗

( ) y .

Furthermore, let

x , y ∈ S

with

x ≤ y

be such that

f

_S^∗

( ) y = δ .

Which means that

y ∉ e

S

( f

S

^; ^δ ) ^.

Then either

x ∈ e

S

( f

S

^; ^δ )

or

x ∉ e

S

( f

S

^; ^δ ) ^.

In any case

( ) x f ( ) y . f

_S^∗

⊆ ρ =

_S^∗

Now if

y ∈ e

S

( f

S

^; ^δ )

. Since

e

S

( f

S

^; ^δ )

S

by Theorem

9.

Then

x ∈ e

S

( f

S

^; ^δ ) ^.

Hence

( ) ^x ^f ( ) ^x ^f ( ) ^y ^f ( ) ^y ^.

f

_S^∗

=

_S

⊆

_S

=

_S^∗ Therefore,

( ^f

S^∗

, ^S )

U

.”

Let

( S , ⋅ , ≤ )

and

( T , ⋅ , ≤ )

be two ordered semigroups. Under the coordinatewise multiplication, i.e.,

( xy ab )

b y a

x , )( , ) ,

( =

(9)

where

( x , a ), ( y , b ) ∈ S × T

, the Cartesian product

^S ^× ^T ⁼ { ( ) ^x ^, ^a ^x ^∈ ^S ^, ^a ^∈ ^T }

is a semigroup. Define a partial order

≤

on

S × T

by

b a y x b

y a

x , ) ≤ ( , ) if and only if ≤ and ≤ (

where

( x , a ), ( y , b ) ∈ S × T .

Then,

( S × , T , ⋅ ≤ )

is an ordered semigroup.

For uni-soft sets

( ^f

S

, ^S )

and

( f

_T

, T )

over

U ,

we consider a uni-soft set

( ^f

S_∨T

, ^S × ^T )

over

U

in which

f

_S_∨_T is defined as follows:

( ) ^, ( ) ( ) ^.

), (

: S T P U x a f x f a

f

_S_∨_T

× → a

_S

∪

_T

“Theorem 12. Let

( S , ⋅, ≤ )

be an ordered semigroup. If

( ^f

S

, ^S )

and

( f

_T

, T )

are uni-soft interior

( resp., left, right )

ideals over

U ,

then

( ^f

S_∨T

, ^S × ^T )

is a uni-soft interior

( resp., left, right )

ideal over

U .

”

Proof. Let

( x , a ), ( y , b ), ( ) z , c ∈ S × T .

Then

( )( )( )

( ) ( )

( ) ( ) ( ) ¹

, ,

abc f xyz f

abc xyz f c z b y a x f

T S

T S T

S

∪

=

_∨

∨

Since

( ^f

S

, ^S )

and

( f

_T

, T )

are uni-soft interior ideal over

U .

Then

f

_S

( ) xyz ⊆ f

_S

( ) y

and

( ) ( ) ^.

T T

f abc ⊆ f b

Hence From equation

( ) 1

we have

( ) ( ) ( ) ( )

( ) ^y ^,b ^.

f

b f y f abc f xyz f

T S

=

∨

∪

⊆

∪

Furthermore, Let

( x , a ), ( y , b ) ∈ S × T

be such that

( x , a ) ≤ ( y , b ).

Then

( ) ( ) ( )

( ) ( ) ( ) ^, ^.

,

b y f

b f y f

a f x f a x f

T S

∨

=

∪

⊆

∪

=

Therefore,

( ^f

S_∨T

, ^S × ^T )

U .

Similarly, we show that If

( ^f

S

, ^S )

and

( f

_T

, T )

are uni-soft left

( resp., right )

ideals over

U ,

then

( ^f

S_∨T

, ^S × ^T )

is a uni-soft left

( resp., right )

ideal over

U .

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4-5, pp. 19--31, 1999.

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[3] K. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20, 87-96, (1986).

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[6] H. Aktas and N. Cağman, "Soft sets and soft groups," Information Sciences, vol. 177, no. 13, pp. 2726- -2735, 2007.

[7] D. Chen, E. C. C. Tsang, D. S. Yeung, and X. Wang, "The parameterization reduction of soft sets and its applications," Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 757--763, 2005.

[8] P. K. Maji, R. Biswas and A. R. Roy, “Soft set theory,” Comput. Math. Appl., 45 (2003), 555-562.

[9] C. Goswami, T. J. Neog and D. K. Sut, "Generalized Union and Intersection of Fuzzy Soft Sets,"

International Journal of Mathematics Trends and Technology, Vol. 2 Issue 2 (2011).

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On Uni-Soft Ideals and Uni-Soft Interior Ideals of Ordered Semigroups Raees Khan