Certain Characterizations of Ordered Semigroups by Uni-Soft Generalized Bi-Ideals

(1)

\

and Biological Sciences www.textroad.com

Certain Characterizations of Ordered Semigroups by Uni-Soft Generalized Bi-Ideals

Raees Khan^*1, Asghar Khan², M.Uzair Khan¹, Zia Ur Rahman³, Imaran Khan¹, Zahoor Ahmad⁴

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

4Department of Geology & Geophysics, Bacha Khan University, Charsadda, KPK, Pakistan

Received: April 3, 2017 Accepted: June 19, 2017

ABSTRACT

In this paper, the soft version of generalized bi-ideals in ordered semigroups is considered. The concept of uni- soft generalized bi-ideals is introduced and some related properties are investigated. Characterizations of uni- soft generalized bi-ideals are discussed. Using the notion of uni-soft generalized bi-ideals, characterizations of regular ordered semigroups (see Theorem 1, 2 and Proposition 8) completely regular and left weakly regular ordered semigroups (see Theorems 3 and 4) are provided. Moreover, a relation between uni-soft generalized bi- ideals and uni-soft bi-ideals is established.

KEYWORDS:ordered semigroups, -exclusive set, soft set, uni-soft generalized bi-ideal, uni-soft bi-ideal.

1. INTRODUCTION

“

Fuzzy set theory was initiated by Zadeh in 1965 [23]. This theory was developed in view of the fact that the classical sets are not suitable in describing real-life problems. Following the discovery of fuzzy sets, great attention has been given to generalize the basic concepts of classical algebra in a fuzzy setting, and which develops a theory of fuzzy algebra. In recent some years, great attention is shown to generalize algebraic structures of groups, rings, modules, vector spaces, etc.”

Molodtsov [18] in 1999 introduced the fundamental concept of soft set which provides a natural framework for generalizing several basic notions of algebra. Many related concepts with soft sets, especially soft set operations, have also undergone tremendous studies. Later, Maji et al. [15] studied the theoretical aspect of soft set and introduced several binary operations such as intersection, union, AND-operation, and OR-operation of soft sets and investigated them in more detail. Feng et al. [4] defined the bi- intersection of two soft sets as an alternative to the definition of soft sets intersection by Maji et al. [15]. Maji at al. [16, 17] presented the theory of soft sets and fuzzy soft sets and worked on the applications of soft set theory in decision making problems.

The applications of the soft theory in algebraic structures were introduce by Aktas and Cagman [19]. They presented the idea of soft groups as a generalization of the fuzzy groups. Then in 2009, Shabir at al. [20] studied soft semigroups and soft ideals over a semigroups. Ali et al. [1] introduced several operations of soft sets. Khan et al. [13, 14] characterized different classes of ordered semigroups in terms of uni-soft quasi-ideals and uni-soft ideals. The theory of soft set has also wide-ranging applications as in the following studies [25, 26].”

In this paper, the notion of uni-soft generalized bi-ideals of ordered semigroups are introduced and some characterizations of uni-soft generalized bi-ideals are discussed. Using the notion of uni-soft generalized bi- ideals, we provide characterizations of regular ordered semigroups, completely regular and left weakly regular ordered semigroups. Moreover, it is shown that uni-soft generalized bi-ideal and uni-soft bi-ideal coincide in regular and left weakly regular ordered semigroups.

2. Basic definitions and preliminaries

“An ordered semigoup is a semigroup

( ⋅ S , )

together with a partial orderd

≤

such that for all

a , b , x ∈ S

,

b

a ≤

implies

ax ≤ bx

and

xa ≤ xb

. Let

A

and

B

be subsets of

S

. Then the multiplication of

A

and

B

is defined as follows”

}.

,

|

{ ab S a A b B

AB = ∈ ∈ ∈

A subset

X

of

S

is called

(1) a subsemigroup of

S

if

XX ⊆ X .

(2) a right

( resp., left )

ideal of

S

if:

(2)

(i)

( ∀ a ∈ X )( ∀ b ∈ S )( b ≤ a

implies

b ∈ X )

, (ii)

XS ⊆ X (

resp.,

SX ⊆ X ).

(3) two-sided ideal

( or simply ideal ) ,

if it is both a left and a right ideal of

S

. (4) a generalized bi-ideal of

S

if:

(iii)

( ∀ a ∈ X )( ∀ b ∈ S )( b ≤ a

implies

b ∈ X ),

(iv)

XSX ⊆ X

.

(5) a bi-ideal of

S

if it is

( 1 )

and

( 4 )

.

By

R a ( ), L a ( ), I a ( ),

and

B a ( ),

we mean the principal right, left, two-sided ideal, and bi- ideal of

S

generated by

x ( x ∈ S ),

respectively. Note that

(

²

( ) ( ],

( ) .

R a a aS L a a Sa

I a a Sa aS SaS

B a a a aSa

= ∪

= ∪ ∪ ∪

= ∪ ∪  

“”The soft set theory introduced by Molodtsov (see [18]), Cağman and Enginoğlu (see [2]) provided new definitions and various results in this theory. In what follows, let

U

be an common universe set and

E

be a set of parameters. The power set of

U

denoted by

P U ( )

and

A , B , C

are subsets of

E

.

The concept of a soft set is given in the following:”

Definition 1.[18]”A soft set

( f

_S

, S )

over

U

is defined to be the set of ordered pairs

{ ^{( ,} ^{( )) |} ^, ^{( )} ^{( ) ,} }

A A A

f = x f x x ∈ E f x ∈ P U

Where

f

_A

: E → P U ( )

such that

f

_A

(x ) = φ

if

x ∉ E .

The function

f

_A is called approximate function of the soft set

( f

_S

, S )

over

U

.

“Definition 2.[18].”Let

f

_A

, f

_B

∈ S ( U ).

Then

(1)

f

_Ais called soft subset of

f

_B and denoted by

( f

_A

, S ) ⊆ ~ ( f

_B

, S )

if

. ),

( )

( x f x x E

f

_A

⊆

_B

∀ ∈

(2)soft union of

f

_A and

f

_B

,

denoted by

( f

_A

∪ ~ f

_B

, S )

and is defined as

. ),

( ) ( ) ( )

~ )(

( f

_A

∪ f

_B

x = f

_A_∪^~_B

x = f

_A

x ∪ f

_B

x ∀ x ∈ E

(3) soft intersection of

f

_A and

f

_B

,

denoted by

( f

_A

∩ ~ f

_B

, S )

and is defined as

. ),

( ) ( ) ( )

~ )(

( f

_A

∩ f

_B

x = f

_A_∩^~_B

x = f

_A

x ∩ f

_B

x ∀ x ∈ E

(4)theuni-soft product, denoted by

( f

_S

◊ g

_S

, S ) ,

is defined by

{ }

( , )

( ) ( ) if ,

: ( ),

if ,

x

S S x

y z A

S S

x

f y g z A

f g S P U x

U A

ϕ ϕ



∈

∪ ≠

◊ →  

 = I a

where

A

_x

= { ( ) y , z ∈ S × S x ≤ yz }

for all

x , y , z ∈ S .

_”

“For a subset

A

of

S ,

we define mapping

χ

_A as follows

if ,

: ( ),

if \ .

A

U x A

S P U x

x S A

χ ϕ

 ∈

→ a   ∈

Which is called the characteristic soft set over

U .

For a subset

A

of

S

, the soft set

( χ

^cA

, ^S )

over

U

in which

χ

^c_A is define as follows

if ,

: ( ),

if \ .

c A

x A S P U x

U x S A

χ → a ^   ϕ ∈ ^∈

For a soft set

( f

_S

, S )

over

U

and a subset

δ

of

U

, the

δ

-exclusive set of

( f

_S

, S )

, denoted by

e

_S

( f

_S

; δ )

, is defined to be the set

(3)

{ ( ) } . )

;

( f δ = x ∈ A f x ⊆ δ

e

_A _S _S

” Definition 3.[5]” Let

( ^f

S

, ^S )

be a soft set over

U

. Then

(1)

( f

_S

, S )

is called a uni-soft semigroup over

U

, if

. , ), ( ) ( )

( xy f x f y x y S

f

_S

⊆

_S

∪

_S

∀ ∈

(2)

( f

_S

, S )

is called a uni-soft left

( resp., right )

ideal over

U

if (i)

x ≤ y ⇒ f

_S

( x ) ⊆ f

_S

( y ),

(ii)

( ∀ ^x ^, ^y ∈ ^S ) ( ^f

S

⁽ ^xy ⁾ ⊆ ^f

S

⁽ ^y ⁾ ( ^resp., ^f

S

⁽ ^xy ⁾ ⊆ ^f

S

⁽ ^x ⁾ ) ) ^.

If a soft set

( f

_S

, S )

over

U

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

U

, we say that

( ^f

S

, ^S )

is a uni-soft two-sided ideal over

U

.”

Definition 4.[13]A uni-soft semigroup

( f

_S

, S )

over

U

is called a uni-soft bi-ideal over

U

if it satisfies (i)

x ≤ y ⇒ f

_S

( x ) ⊆ f

_S

( y ),

(ii)

( ∀ x , y , z ∈ S )( f

_S

( xyz ) ⊆ f

_S

( x ) ∪ f

_S

( z ) )

.

Proposition 1.[5].For subset

A

and

B

of

S

, the following properties hold (i)

( χ

^c_A

◊ χ

_B^c

^, S ) ( = χ

(^c_AB]

^, S ) ^.

(ii)

( ^~ ^, ^S ) (

^cA B

^, ^S ) ^.

c B c

A

∪ χ = χ

∪

χ

Lemma 1.Let

A

be non-empty subst of

S

, then the soft set

( χ

₍^cA_]

, ^S )

over

U

has the following property.

( ]

( x )

( ]

( y ), x , y S . y

x ≤ ⇒ χ

^c_A

⊆ χ

^c_A

∀ ∈

Proof .Let

x , y ∈ S

, be such that

x ≤ y

. If

y ∉ ( A ]

, then

χ

₍^c_A_]

( y ) = U

and so (^c_A]

( x ) U χ

(^c_A]

( y )

χ ⊆ =

. Let

y ∈ ( A ]

, then

χ

₍^c_A_]

( y ) = φ

. Since

y ∈ ( A ]

, so there exists

z ∈ A

such that

y ≤ z

. We have

x ≤ z

, hence

x ∈ ( A ]

. Then

χ

₍^c_A_]

(x ) = φ

and

χ

₍^c_A_]

( ) x ⊆ = ϕ χ

₍^c_A_]

( ). y

. Lemma 2.Let

A

be a subset of

S ,

then

A = ( A ]

if and only if the soft set

( χ

^cA

, ^S )

over

U

has the property.

. , ), ( )

( x y x y S

y

x ≤ ⇒ χ

^c_A

⊆ χ

_A^c

∀ ∈

Proof

⇒ .

It follows from Lemma

1

.

⇐ .

Let

x ∈ ( A ]

, then there exists

y ∈ A

such that

x ≤ y

. By hypothesis, we have

χ

_A^c

( x ) ⊆ χ

^c_A

( y )

. Since

y ∈ A

, we have

χ

^c_A

( y ) = φ

. Then

χ

^c_A

(x ) = φ

and

x∈ A

. Hence

( A ⊆ ] A

. On the other hand

A ⊆ ( A ].

Therefore

A = ( A ].

Lemma 3.[5]Let

A

be any subset of

S .

Then,

A

is a right

( left, two - sided )

ideal of

S

if and only if

( χ

^cA

, ^S )

is an uni-soft right

( left, two - sided )

over

U .

Lemma 4.A soft set

( f

_S

, S )

is a uni-soft semigroup over

U

if and only if

).

,

~ ( ) ,

( f

_S

◊ f

_S

S ⊇ f

_S

S

Proof. Assume that

( f

_S

, S )

U .

Let

a

be an element of

S .

If

A

_a

≠ φ ,

then there exists

x , y ∈ S

such that

a ≤ xy .

Since

( f

_S

, S )

U ,

then we have

( )( ) { ( ) ( ) }

( ) ( ) ^a ^.

f

xy f

y f x f a

f f

S xy S a

S xy S

S a S

⊇

∪

=

◊

≤

I I

Thus,

( f

_S

◊ f

_S

, S ) ⊇ ~ ( f

_S

, S ).

(4)

Conversely, assume that

( f

_S

◊ f

_S

, S ) ⊇ ~ ( f

_S

, S ).

. Let

x , y ∈ S

, we have

( ) ( )( )

( ) ( )

{ }

( ) x f ( ) y . f

q f p f

xy f f xy f

S S

S pq S

xy S S S

∪

⊆

∪

=

◊

⊆

≤

I

Hence,

( f

_S

, S )

is a uni-soft semigroupover

U .

_”

3. Uni-soft generalized bi-ideals of ordered semigroups

In this section, we define uni-soft generalized bi-ideals and study different properties of uni-soft generalized bi- ideals as a regards of uni-soft product.

Definition 5.A soft set

( f

_S

, S )

over

U

is called a uni-soft generalized bi-ideal over

U

if it satisfies.

(i)

x ≤ y ⇒ f

_S

( x ) ⊆ f

_S

( y ),

(ii)

( ∀ x , y , z ∈ S )( f

_S

( xyz ) ⊆ f

_S

( x ) ∪ f

_S

( z ) )

.

“Example1.Let

S = { a , b , c , d , f }

be an ordered semigroup defined by the following caylays table:

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 a b a c a d b b c c c d d d

≤

Let

( f S

_S

, )

be a soft set over

^{U =} { ^{0,1, 2,3} }

in which

f

S is given by

{0}, if , : ( ), {0,1}, if , {0,1, 2}, if { , }

S

x a

f S P U x x c

x b d

 =

→   =

 ∈



a

Then by routine checking it is easy to verify that

f

S

, S

is a uni-soft generalized bi-ideal over

U

.”

Theorem 1.A soft set

( f

_S

, S )

over

U

is a uni-soft generalized bi-ideal over

U

if and only if

).

,

~ ( ) ,

( f

_S

◊ φ

_S

◊ f

_S

S ⊇ f

_S

S . , ), ( )

( x f y x y S f

y

x ≤ ⇒

_S

⊆

_S

∀ ∈

Proof. Assume that

( f

_S

, S )

U

. Let

x

be an element of

S .

In the case, when

A

_x

≠ φ .

Then there exists

p , q , u , v ∈ S ,

such that

x ≤ pq

and

p ≤ uv .

Then, since

( f

_S

, S )

U

, we have

( ) x f ( ) pq f ( ) uvq f ( ) u f ( ) q .

f

_S

⊆

_S

⊆

_S

⊆

_S

∪

_S

Thus,

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

( ) ( )

{ } ( )

( )

{ } ( )

( ) ( )

{ }

( )

x. f

q f u f

q f u

f

q f v u f

q f p f x

f f

S

S uvq S

x

S uv S

p pq x

S S

S uv pq p x

S S

pq S S x

S S

⊇

∪

=



 ∪ ∪

=



 



 ∪ ∪

=

∪

◊

=

◊

≤

≤ ≤

≤

I I I I I

φ φ φ φ

(5)

Hence,

( f

_S

◊ φ

_S

◊ f

_S

, S ) ⊇ ~ ( f

_S

, S ).

Conversely, assume that

( f

_S

◊ φ

_S

◊ f

_S

, S ) ⊇ ~ ( f

_S

, S ).

. Let

a ∈ S

, then there exists

x , y , z ∈ S

such that

.

xyz

a =

Then, we have

( ) ( )

( )( )

( )( ) ( )

{ }

( )( ) ( )

( ) ( )

{ } ( )

( ) ( ) ( )

( ) ( )

( ) ^x ^f ( ) ^z ^.

f

z f x

f

z f y x

f

z f v u

f

z f xy f

q f p f

a f f

a f xyz f

S S

S

S S

uv S xy

S S

S

S S

pq S a

S S S S S

∪

=

∪

=

∪

⊆

∪

=

∪

◊

⊆

∪

◊

=

◊

⊆

=

≤

φ φ

I I

Thus,

( f

_S

, S )

U

.

“Proposition 2.A subset

A

of

S

is a generalized bi-ideal of

S

if and only if

( χ

A^c

, ^S )

U .

_”

“Proof.Assume that

X

S

, that is,

XSX ⊆ X

. Then we have

( ]

~

^c

,

X c

XSX c

X c S c X c X S c

X

φ χ χ χ χ χ χ

χ ◊ ◊ = ◊ ◊ = ⊇

since

XSX ⊆ X .

Furthermore, Let

x , y ∈ S

be such that

S ∋ x ≤ y ∈ X .

If

χ

^c_X

( y ) = φ ,

then

y ∈ X

. Since

x ≤ y

and

X

S

, we have

x ∈ X

. Hence

χ

^c_X

( x ) = φ ⊆ χ

^c_X

( y ).

If

, ) ( y U

c

X

=

χ

then obviously,

χ

^c_X

( x ) ⊆ U = χ

^c_X

( y ).

This means that

( χ

^cX

, ^S )

U .

Conversely, assume that X

c

, S

U .

_Let

x X S X .

Then, we have

( ) ( ^χ ^φ ^χ ) ( ) ( ^χ ^χ ^χ ) ( ) ^χ

₍ ]

( ) ^φ ^,

χ

_X^c

x ⊆

^c_X

◊

_S

◊

_X^c

x =

^c_X

◊

^c_S

◊

^c_X

x =

^c_XSX

x =

and so

x ∈ X

. Thus,

XSX ⊆ X

and

X

S

.”

“Theorem 2.A soft set

( f

_S

, S )

over

U

if and only if the non- empty

δ

-exclusive set of

( f

_S

, S )

S

for all

δ ⊆ U

.”

Proof.First assume that

( f

_S

, S )

U

. Let

δ ⊆ U

be such that

(

_S

; δ ) ≠ φ .

S

f

e

Let

y ∈ S

and

x , z ∈ e

_S

( f

_S

; δ ) ,

then

f

_S

( ) x ⊆ δ

and

f

_S

( ) z ⊆ δ

. Then by hypothesis, we have

. ) ( ) ( )

( xyz ⊆ f x ∪ f z ⊆ δ

f

_S _S _S

It follows that

xyz ∈ e

_S

( f

_S

; δ ) .

”

“Furthermore, let

x , y ∈ S

be such that

x ≤ y .

Let if

y ∈ e

_S

( f

_S

; δ ) ,

then by hypothesis we have

( ) x ⊆ f ( ) y ⊆ δ ,

f

_S _S

which means that

x ∈ e

_S

( f

_S

; δ ) .

Thus,

e

_S

( f

_S

; δ )

S

.

Conversely, assume that the non-empty

δ

-exclusive set of

( f

_S

, S )

S

for all

.

⊆ U

δ

Let

x , y , z ∈ S

be such that

f

_S

( ) x = δ

_x and

f

_S

( ) z = δ

_z

.

Taking

δ = δ

_x

∪ δ

_z implies that

( ; ) . , z e

_S

f

_S

δ

x ∈

Hence

xyz ∈ e

_S

( f

_S

; δ )

, and so

).

( ) ( )

( xyz f x f z

f

_S

⊆ δ = δ

_x

∪ δ

_z

=

_S

∪

_S

(6)

Let

x , y ∈ S

,

x ≤ y

be such that

f

_S

( ) y = δ ,

implies that

y ∈ e

_S

( f

_S

; δ ) .

Since

e

_S

( f

_S

; δ )

S

for all

δ ⊆ U ,

so

x ∈ e

_S

( f

_S

; δ ) .

Thus

( ) x f ( y ).

f

_S

⊆ δ =

_S

Therefore,

( f

_S

, S )

U

.”

Proposition 3.In an ordered semigroup

S

, every uni-soft bi-ideal is a uni-soft generalized bi-ideal.

Proof. It is straight farward.

The converse of Proposition 3 does not hold in general as shown in the following example.

Example 2.Consider the ordered semigroup in Example

1

, define a soft set

( f

_S

, S )

over

^U ⁼ { ⁰ ^, ¹ ^, ² ^, ³ }

ⁱⁿ

which

f

_S is given by

{0}, if , : ( ), {0, 2}, if , {0, 2, 3}, if { , }.

S

x a

f S P U x x c

x b d

 =

→   =

 ∈



a

Then

( f

_S

, S )

U

. However since

).

( ) ( ) ( )

( cc f b f c f c

f

_S

=

_S

⊆/

_S

∪

_S

So

( f

_S

, S )

is not a uni-soft bi-ideal over

U

.”

Uni-soft generalized bi-ideals of regular ordered semigroups

In this section, we study different properties of uni-soft generalized bi-ideals as a regards of uni-soft product.

We also characterize regular ordered semigroups in terms of uni-soft generalized bi-ideals. An element

a ∈ S

is called regular, if there exists some

x ∈ S

such that

a ≤ axa .

An ordered semigroup

S

is called regular if every element of

S

is regular.”

Proposition 4.[21].In a regular ordered semigroup

S

, every generalized bi-ideal is a subsemigroup, that is, a bi-ideal of

S

.

Proposition 5.In a regular ordered semigroup

S

, every uni-soft generalized bi-ideal is a uni-soft semigroup, that is, a uni-soft bi-ideal over

U

.”

Proof.”Let

( f

_S

, S )

be a uni-soft generalized bi-ideal over

U

. Let

a , b ∈ S .

Since

S

is regular, so there exists

x ∈ S ,

such that

a ≤ axa ⇒ ab ≤ ( ) axa b .

Then

( ) ( ( ) ) ( ( ) ) ( ) a f ( ) b .

f

b xa a f b axa f ab f

S S

S

∪

⊆

=

⊆

Thus,

( f

_S

, S )

U

.””

By Propositions

3

and

5

, we have the following:

Remark.In regular ordered semigroups the concepts of uni-soft generalized bi-ideals and uni-soft bi-ideals coincide.

Proposition 6.Let

( f

_S

, S )

be a uni-soft generalized bi-ideal of a regular ordered semigroup over

U .

Then for all

a ∈ S

such that

a ≤ a

²

,

we have

( ) ^a ^f ( ) ^a

²

^.

f

_S

=

_S

Proof.”Sppose that

( f

_S

, S )

U .

Let

a ∈ S

be such that

a ≤ a

²

.

Then

( ) ( ) ^{( )}

( ) ( ) ( ) .

2

a f

a f a f

aa f a f a f

S

S S

S

=

∪

⊆

=

⊆

Hence

^f

S

( ) ^a = ^f

S

( ) ^a

²

^.

”

(7)

“Theorem 3. For an ordered semigroup

S

, the following conditions are equivalent:

(1)

S

is regular.

(2)

( f

_S

, S ) ( = f

_S

◊ φ

_S

◊ f

_S

, S )

for every uni-soft generalized bi-ideal

( f

_S

, S )

over

U

.”

“Proof. First assume that

( 1 )

holds. Let

( f

_S

, S )

U

and

a ∈ S

. Since

S

is regular, then there exists

x ∈ S

such that

a ≤ axa

. Thus, we have

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

( )( ) ( )

( ) ( )

{ } ( )

( ) ( ) ( )

( ) ( )

( ) a . f

a f a

f

a f x a

f

a f v u

f

a f ax f

q f p f

a f f

S

S S

S

S S

uv S ax

S S

S

S S

pq S S a

S S

=

∪

=

∪

⊆

∪

=

∪

◊

⊆

∪

◊

=

◊

≤

φ φ

I I

Which means that

( f

_S

◊ φ

_S

◊ f

_S

, S ) ⊆ ~ ( f

_S

, S )

and by theorem

1 ,

we have

( f

_S

◊ φ

_S

◊ f

_S

, S ) ⊇ ~ ( f

_S

, S )

Thus,

( f

_S

, S ) ( = f

_S

◊ φ

_S

◊ f

_S

, S )

.

Conversely, assume that (

2

) holds. To prove that

S

is regular, we need to illustrate that

X ⊆ ( XSX ]

. For this let

a ∈ X

. Then, by Proposition

2

, the soft set

( χ

^cX

, ^S )

is a uni-soft generalized bi-ideal of

S

over

U

. Thus, we have

( ]

, ) )(

(

) )(

(

) )(

( ) )(

(

φ χ

χ φ χ

χ χ χ χ

=

◊

=

◊

=

a a

c X

c X c X

c X c S c X c

XSX

which means that

a ∈ ( XSX ]

. Thus,

X ⊆ ( XSX ]

. It follows that

S

is regular.”

Lemma 5.[21]For an ordered semigroup, the following conditions are equivalent:

“(1)

S

is regular.

(2)

B ∩ ⊆ L ( BL ],

for every generalized bi-ideal

B

and left ideal

L

of

S

. (3)

B x ( ) ∩ L x ( ) ⊆ ( ( ) ( )], B x L x ∀ ∈ x S

.”

“Theorem 4.For an ordered semigroup, the following conditions are equivalent:

(1)

S

is regular.

(2)

( f

_S

∪ ~ g

_S

, S ) ⊇ ~ ( f

_S

◊ g

_S

, S )

( f

_S

, S )

and uni-soft left ideal

( g

_S

, S )

over

U

.””

“Proof .First Assume that (1) holds. Let

( f

_S

, S )

be a uni-soft generalized bi-ideal and

( g

_S

, S )

be a uni-soft left ideal over

U

. Let

a

be an element in

S

, since

S

is regular. Then there exists

x ∈ S

, such that

. axaxa axa

a ≤ ≤

Then

( axa , xa ) ∈ A

_a

.

Since

A

_a

≠ φ

, we have

( )( ) { ( ) ( ) }

( ) ( )

{ f axa g xa } . z g y f a

g f

S S

S yz S

S a S

∪

⊆

∪

=

◊ I

≤

Since

( f

_S

, S )

U ,

we have

f

_S

( ) axa ⊆ f

_S

( ) a ∪ f

_S

( ) a = f

_S

( ) a

and

( g

_S

, S )

is a uni-soft left ideal over

U

, we have

g

_S

( xa ) ⊆ g

_S

( a ).

Thus

( )( ) { ( ) ( ) }

).

~ )(

(

) ( ) (

.

a g f

a g a f

xa g axa f a g f

S S

∪

=

∪

⊆

∪

⊆

◊

Therefore,

( f

_S

∪ ~ g

_S

, S ) ⊇ ~ ( f

_S

◊ g

_S

, S )

.

(8)

Conversely, assume tha (2) holds. To prove that

S

is regular, by Lemma

5

, it is enough to prove that

( ) ( ) ( ( ) ( )], .

B x ∩ L x ⊆ B x L x ∀ ∈ x S

Let

y ∈ B x ( ) ∩ L x ( ).

Then

y ∈ ( ( ) ( )]. B x L x

Since

B x ( )

is the generalized bi-ideal and

L x ( )

is the left ideal of

S

generated by

x

, respectively. Then by Lemma

3

,

( χ

_L^c₍x₎

, ^S )

is a uni-soft left ideal and by Proposition

2

,

( χ

_B^c₍x₎

, ^S )

a uni-soft generalized bi-ideal over

U

. Then by Proposition 1 and hypothesis

,

we have

( ₍ ₎ ₍ ₎]

( ) ( χ

₍ ₎

χ

₍ ₎

)( ) ( χ

₍ ₎

~ χ

₍ ₎

)( ) χ

₍ ₎

( ) χ

₍ ₎

( ) φ , χ

^c_B_{x L}_x

y =

_B^c _x

◊

^c_L_x

y ⊆

_B^c _x

∪

_L^c _x

y =

_B_x

y ∪

_L^c _x

y =

which implies that

^y ^∈ ( ^{B x L x} ^{( ) ( ) .} ]

Thus

^{B x} ^{( )} ^∩ ^{L x} ^{( )} ^⊆ ( ^{B x L x} ^{( ) ( ) .} ]

It follows from Lemma

5 ,

that is

S

is regular.”

“Proposition 7.Let

S

be an ordered semigroup,

( f

_S

, S )

is a uni-soft generalized bi-ideal and

( g

_S

, S )

is a uni-soft ideal over

U

. Then we have

( ~ , ) . ) ~

,

( f

_S

◊ g

_S

◊ f

_S

S ⊇ f

_S

∪ g

_S

S

”

“Proof.Let

( f

_S

, S )

( g

_S

, S )

be a uni-soft ideal over

U

. Let

a ∈ S

. Then

( f

_S

◊ g

_S

◊ f

_S

, S ) ⊇ ~ ( f

_S

∪ ~ g

_S

, S ) .

In fact: If

A

_a

= φ ,

then

( ) ( ) ( ) ( ~ , ) .

)

( f

_S

◊ g

_S

◊ f

_S

a = U ⊇ f

_S

a ∪ g

_S

a = f

_S

∪ g

_S

S

Let

A

_a

≠ φ ,

then

( ) { ( ) ( )( ) }

( ) { ( ) ( ) }

( ) ( ) ( )

{ } , ( 1 ) )

(

q f p g y f

q f p g y

f

z f g y f a

f g f

S S

pq S z yz a

S pq S

S z yz a

S S yz S

S a S S

∪

=

 



 

 



 



 ∪

∪

=

◊

∪

=

◊

≤

I I

I

For each

( ) y , z ∈ A

_a and

( p , q ) ∈ A

_z

,

we have

. Thus

.

and z pq a ypq

yz

a ≤ ≤ ≤

Since

( f

_S

, S )

U

, so we have

) ( ) ( ) ( )

( a f ypq f y f q

f

_S

⊆

_S

⊆

_S

∪

_S

And

( g

_S

, S )

U

, so we have

).

( ) ( )

( a g ypq g p

g

_S

⊆

_S

⊆

_S

Thus

{ }

).

( ) ( ) (

) ( ) ( ) ( ) ( ) (

q g p f y f

p g q f y f a g a f

S S

S

S S

S

∪

=

∪

⊆

∪

Hence from

( 1 )

we get

( f

_S

◊ g

_S

◊ f

_S

) ( a ) ⊇ f

_S

( a ) ∪ g

_S

( a ) = ( f

_S

∪ ~ g

_S

)( a ).

Thus,

( f

_S

◊ g

_S

◊ f

_S

, S ) ⊇ ~ ( f

_S

∪ ~ g

_S

, S ) .

”

Lemma6.[21] For an ordered semigroup, the following conditions are equivalent:

(1)

S

is regular.

(2)

B ∩ = I ( BIB ],

for every generalized bi-ideal

B

and ideal

I

of

S

. (3)

B x ( ) ∩ I x ( ) = ( ( ) ( ) ( )], B x I x B x ∀ ∈ x S

.

“Theorem 5.”For an ordered semigroup, the following conditions are equivalent:

(1)

S

is regular.

(2)

( f

_S

◊ g

_S

◊ f

_S

, S ) = ( f

_S

∪ ~ g

_S

, S ) .

( f

_S

, S )

and every uni-soft ideal

( g

_S

, S )

over

U

.”

“Proof.Assume that (1) holds. Let

( f

_S

, S )

( g

_S

, S )

be a uni-soft ideal over

U

. Since

S

is regular, then for each element

a

in

S ,

there exists

x ∈ S

, such that

(9)

( ) axa xa . axa

a ≤ ≤

Thus,

( axa , xa ) ∈ A

_a

.

Hence

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

( ) ( )( )

( ) { ( ) ( ) }

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

( ) ( ) ^.

since xax

g axa f

axa xax axa

x xa axa

f xax g axa f

q f p g axa

f

xa f g axa f

z f g y f a

f g f

S S

S

S pq S

S xa

S S S

S S yz S

S a S S

∪

=

≤

∪

⊆

 



 

 ∪

∪

=

◊

∪

⊆

◊

∪

=

◊

≤

I I

As

( f

_S

, S )

is a uni-soft generalized bi-ideal and

( g

_S

, S )

U

, so we have

) ( ) ( ) ( )

( axa f a f a f a

f

_S

⊆

_S

∪

_S

=

_S ,

g

_S

( xax ) ⊆ g

_S

( a ).

Thus

( ) axa g ( ) xax f ( a ) g ( a ) ( f ~ g )( a ).

f

_S

∪

_S

⊆

_S

∪

_S

=

_S

∪

_S

Therefore,

( f

_S

◊ g

_S

◊ f

_S

, S ) ⊆ ~ ( f

_S

∪ ~ g

_S

, S ) .

On the other hand, by Proposition

7

, we have

( f

_S

◊ g

_S

◊ f

_S

, S ) ⊇ ~ ( f

_S

∪ ~ g

_S

, S ) .

Thus

( ~ , ) . )

,

( f

_S

◊ g

_S

◊ f

_S

S = f

_S

∪ g

_S

S

Conversely, assume that (2) holds. To prove that

S

is regular, by Lemma

6

, it is enough to show that

( ) ( ) ( ( ) ( ) ( )], .

B x ∩ I x = B x I x B x ∀ ∈ x S

Let

y ∈ B x ( ) ∩ I x ( )

. Then

y ∈ ( ( ) ( ) ( )] B x I x B x

. Since

B x ( )

is the generalized bi-ideal and I

(x )

^is the ideal of

S

, generated by

x

, respectively. Then by Lemma

3

,

( ^χ

^{I x}^c^{( )}

^, ^S )

is a uni-soft ideal and by Proposition

2

,

( χ

_{B x}^c( )

, S )

U

. Then by Proposition

1

and hypothesis

,

_{we have}

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

( ) ( )

( )( ) ( )( )

( )( )

( ) ( )

,

c c c c

B x I x B x

B x I x B x

c c

B x I x

c c

B x I x

y y

y

y y

χ χ χ χ

χ χ

ϕ

= ◊ ◊

= ∪

=

%

which means that

y ∈ ( ( ) ( ) ( )]. B x I x B x

Thus

B x ( ) ∩ I x ( ) ⊆ ( ( ) ( ) ( )]. B x I x B x

It follows Lemma

6 ,

that is

S

is regular.”

“Lemma 7.[21] For an ordered semigroup, the following conditions are equivalent:

(1)

S

is regular.

(2)

R ∩ ∩ ⊆ B L ( RBL ],

for every right ideal

R

, generalized bi-ideal

B

and every left ideal

L

of

S

. (3)

R x ( ) ∩ B x ( ) ∩ L x ( ) ⊆ ( ( ) ( ) ( )], R x B x L x ∀ ∈ x S

.”

“Theorem 5.For an ordered semigroup, the following conditions are equivalent:

(1)

S

is regular.

(2)

( f

_S

∪ ~ g

_S

∪ ~ h

_S

, S ) ⊇ ~ ( f

_S

◊ g

_S

◊ h

_S

, S )

for every uni-soft right ideal

( f

_S

, S )

, every uni-soft generalized bi-ideal

( g

_S

, S )

and every uni-soft left ideal

( h

_S

, S )

over

U .

”

“Proof. Firstassumes that (1) holds. Let

( f

_S

, S ) ( , g

_S

, S )

and

( h

_S

, S )

be uni-soft right, uni-soft generalized bi-ideal and uni-soft left ideal over

U

respectively. Since

S

is regular, then for every

a ∈ S

, there exists

S

x ∈

, such that

a ≤ axa .

Hence

( ax , a ) ∈ A

_a . Since

A

_a

≠ φ

, we have

Certain Characterizations of Ordered Semigroups by Uni-Soft Generalized Bi-Ideals