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CHAPTER-5

STRUCTURE OF DUO Γ-SEMIGROUPS

DOROFEEVA, MANNEPALLI and SATYANARAYANA [16] established

several equivalent characterizations of commutative cancellative chained semigroups.

SATYANARAYANA [44] characterized commutative chained semigroups using

TAMURA’s concentric condition (< a >w = for all a ∈ S)[52]. He proved that

noetherian cancellative chained semigroups without idempotents has this property. Also

he [44] characterized commutative cancellative chained semigroups. Further in [45], an

ideal theory for commutative semigroups which are unions of a finite number of principal

ideals has been developed and an analogue of HILBERT basis theorem was proved.

ANJANEYULU [4] made a study on the structure of duo chained semigroups and duo

noetherian semigroups. He showed that every noetherian cancellative duo chained

semigroup without idempotents which is not globally idempotent is cyclic and satisfies

TAMURA’s concentric condition [52]. He characterized archimedian duo chained

semigroups without idempotents. Further he proved that in a duo semigroup which is a

union of finite number of principal ideals, if every proper prime ideal is principal, then

every ideal is an intersection of a principal ideal and an S-primary ideal, and if this

semigroup is globally idempotent, then every proper ideal is prime. Also he proved that

every noetherian cancellative duo semigroup without identity is finitely generated. Finally

he showed that every cancellative noetherian duo semigroup is a direct product of the

additive semigroup of nonnegative integers and a group and also he extend ‘The analogue

of HILBERT basis theorem’ to duo semigroups. In this thesis we made a study on the

structure of duo chained Γ-semigroups and duo noetherian Γ-semigroups. We obtained

some characterizations of duo chained Γ-semigroups and duo noetherian Γ-semigroups.

Also we extend ‘The analogue of HILBERT basis theorem’ to duo Γ-semigroups.

This chapter is divided into 2 sections. In section 1, the terms; chained

Γ-semigroup and Γ-group are introduced. It is proved that if P is a prime -ideal of a duo

chained Γ-semigroup S and Px then P Px n

n)(

1

. It is also proved that every duo

chained Γ-semigroup is a semiprimary -semigroup. It is proved that (1) if Sa is a

semisimple element of a duo chained Γ-semigroup S, then wa , (2) if a duo chained

Γ-semigroup S has no -idempotent elements, then for any a ∈ S, < a > w = ∅ or < a >

w is

Structure and ideal theory of duo Γ-semigroup Duo chained Γ-semigroups

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a prime -ideal. In a duo chained Γ-semigroup S if S S S then S\S S = { x } for some

x S. Further it is proved that in a duo chained -semigroup S, if S S S such that

S\S S = { x } for some x S, then (1) S = x S1 = S

1 x and S S = x S = S x is

the unique maximal -ideal of S, (2) If aS and a < x >w then xxa n 1)( for some

natural number n > 1, and (3) If S contains strongly Γ-cancellable elements then x is a

strongly Γ-cancellable element and < x >w is either empty or a prime -ideal of S. It is

proved that, if S is a duo chained Γ-semigroup, then S is an Archimedean Γ-semigroup

without Γ-idempotents if and only if < a >w = ∅ for every a ∈ S. It is proved that if S is a

strongly Γ-cancellative Archimedean duo chained Γ-semigroup with wa for some

a ∈ S, then S is a Γ-group. Also it is proved that, if S is a duo chained Γ-semigroup

containing strongly Γ-cancellative elements and < a >w = ∅ for every a ∈ S, then S is a

strongly Γ-cancellative Γ-semigroup.

The contents of section 1 of chapter 5 are published in “International Journal of

Mathematical Sciences, Technology and Humanities” under the title ‘Duo chained -semigroups’ [19]

In section 2, the terms; noetherian Γ-semigroup, Γ-closed Γ-semigroup and centre

of a Γ-semigroup are introduced. It is proved that if S is a noetherian Γ-semigroup

containing proper Γ-ideals, then S has a maximal Γ-ideal. It is proved that if H is the

collection of all Γ-ideals in a Γ-closed duo Γ-semigroup S which are not principal and

H ≠ ∅, then there exists a prime Γ-ideal of S which is not a principal Γ-ideal. It is proved

that if every prime Γ-ideal including S is principal in a Γ-closed duo Γ-semigroup S, then

every Γ-ideal in S is principal. It is proved that if S is a Γ-closed duo Γ-semigroup, which

is a union of finite number of principal Γ-ideals and every proper prime Γ-ideal is

principal, then every Γ-ideal is an intersection of a principal Γ-ideal and an S-Primary

Γ-ideal. Also it is proved that if S is a Γ-closed duo Γ-semigroup, which is a union of

finite number of principal Γ-ideals and every proper prime Γ-ideal of S is principal and

S = S ΓS then every proper Γ-ideal is principal.

If S is a duo Γ-semigroup such that S and every maximal Γ-ideal is

principal then it is proved that (1) S has at most two maximal Γ-ideals and (2) if P is a

proper prime Γ-ideal of S then either P is a principal Γ-ideal or P = xΓP for some . If

every maximal Γ-ideal in a Γ-closed duo Γ-semigroup S is principal and S ,

for every , then it is proved that S is a union of two principal Γ-ideals

and every Γ-ideal is an intersection of a prime Γ-ideal and an S-primary Γ-ideal. If S is a

114

noetherian or Archimedean duo Γ-semigroup such that S = 1

n

i

i

x

and

suppose ia x a for all , which is not a product of power of , then it is

proved that S is finitely generated and in particular if S is noetherian strongly

Γ-cancellative Γ-semigroup without identity then S is finitely generated. If S is a duo

Γ-semigroup which is a union of finite number of principal Γ-ideals and if S = , then it

is proved that S contains Γ-idempotent elements. If S is a strongly Γ-cancellable duo

Γ-semigroup which is a union of finite number of principal Γ-ideals, then it is proved that

S contains identity if and only if S = .

In an Archimedean duo Γ-semigroup S, if S is a union of finite number of

principal Γ-ideals or S contains a maximal Γ-ideal which is finitely generated, then it is

proved that every proper Γ-ideal is principal and S is a union of at most two principal

Γ-ideals. It is proved that if A is a finitely generated Γ-ideal of a duo Γ-semigroup S,

A = AΓB for some Γ-ideal B and then for some . If S is a duo

Γ-semigroup containing no Γ-idempotents except perhaps the identity 1 and P is a finitely

generated prime Γ-ideal contained properly in for some and , then it is

proved that (1) P does not contain any strongly Γ-cancellable element and (2) if A is

finitely generated Γ-ideal containing a strongly Γ-cancellable element then A for

any proper Γ-ideal B. It is proved that if A is a finitely generated Γ-ideal of a duo

Γ-semigroup S and Aw = B such that A ΓB = where are primary Γ-ideals, then

AΓB = B. If S is a noetherian duo Γ-semigroup without Γ-idempotents except perhaps

identity, then it is proved that for any Γ-ideal A, Aw

where Z is the set of all

nonstrongly Γ-cancellable elements and Aw = if S is strongly Γ-cancellative. If S is a

noetherian Γ-closed duo Γ-monoid with a unique maximal Γ-ideal M = < m > for some m

and if x then it is proved that , u is a unit or with Γx

Γs. If S is a noetherian duo Γ-monoid with a unique maximal Γ-ideal M = < m > for some

m and if P is a proper prime Γ-ideal of S such that P M, then it is proved that

P .

If S is a noetherian duo Γ-monoid with a unique maximal Γ-ideal M = < m > for

some m and if S has no Γ-idempotents except 1, then it is proved that is a prime

Γ-ideal and also if Z M where Z is the set of all non cancellable elements of S, then

Z = If T is a Γ-closed duo Γ-semigroup and S is a duo Γ-semigroup such that S is a

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Γ-subsemigroup of T and T = xΓS1 for some x and if S is noetherian then it is proved

that T is noetherian. Further an analogue of HILBERT basis theorem is obtained for duo

Γ-semigroups.

The contents of section 2 of chapter 5 are published in “International eJournal of Mathematics

and Engineering” under the title ‘Duo Noetherian Γ-semigroups’ [20].

5.1. DUO CHAINED -SEMIGROUPS

In this section, the terms; chained Γ-semigroup and Γ-group are introduced. It is

proved that if P is a prime -ideal of a duo chained Γ-semigroup S and Px then

P Px n

n)(

1

. It is also proved that every duo chained Γ-semigroup is a semiprimary

-semigroup. It is proved that (1) if Sa is a semisimple element of a duo chained

Γ-semigroup S, then wa , (2) if a duo chained Γ-semigroup S has no -idempotent

elements, then for any a ∈ S, < a > w = ∅ or < a >

w is a prime -ideal. In a duo chained

Γ-semigroup S if S S S then S\S S = { x } for some x S. Further it is proved that

in a duo chained -semigroup S, if S S S such that S\S S = { x } for some x S,

then (1) S = x S1 = S

1 x and S S = x S = S x is the unique maximal -ideal of

S, (2) If aS and a < x >w then xxa n 1)( for some natural number n > 1, and (3) If

S contains strongly Γ-cancellable elements then x is a strongly Γ-cancellable element and

< x >w is either empty or a prime -ideal of S. It is proved that, if S is a duo chained

Γ-semigroup, then S is an Archimedean Γ-semigroup without Γ-idempotents if and only if

< a >w = ∅ for every a ∈ S. It is proved that if S is a strongly Γ-cancellative Archimedean

duo chained Γ-semigroup with wa for some a ∈ S, then S is a Γ-group. Also it is

proved that, if S is a duo chained Γ-semigroup containing strongly Γ-cancellative elements

and < a >w = ∅ for every a ∈ S, then S is a strongly Γ-cancellative Γ-semigroup.

We now introduce the notion of a chained Γ-semigroup.

DEFINITION 5.1.1 : A -semigroup S is said to be a chained -semigroup if the

-ideals in S are linearly ordered by set inclusion.

Now we characterize chained Γ-semigroup.

THEOREM 5.1.2 : Let S be a duo chained -semigroup and Sx . If P is a prime

-ideal of S and Px then P 1( )n

nx P

.

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Proof : Since Px and P is prime, 1( )nx x P Ú for all natural numbers n. Since

1( )nx x S and P is a -ideal of S, it follows that 1( )nx x P P for all natural

numbers n and hence ( )nx P P for all natural numbers n. Therefore 1( )n

nx P

P .

Since S is a duo chained -semigroup, 1( )nx S is a -ideal of S. Since 1( )nx x P Ú , we

get 1( )nx S P Ú and since S is a chained -semigroup, P 1( )nx S for all natural

numbers n. Let Py . Then y 1( )nx S . Therefore y ( )nx z for some 1Sz .

Therefore y = 1 2..... nx x x z for some z ∈ S1 and 1 2, ,....., n . Since P is prime, y

= 1 2..... nx x x z ∈ P, x ∉ P, we get z ∈ P. Therefore y ∈ ( )nx P for all natural numbers

n. Hence P 1( )n

nx P

. Therefore P

1( )n

nx P

.

THEOREM 5.1.3 : If S is a duo chained -semigroup, then S is a semiprimary

-semigroup.

Proof : Let A be a -ideal of S. We have A

P = Intersection of all prime

Γ-ideals of S containing A. Since S is a duo chained -semigroup, we have :P

forms a chain. By Zorn`s lemma :P has a minimal element say P . Therefore

A P and P is a prime Γ-ideal of S and hence A is prime. Therefore A is a

semiprimary -ideal of S and hence S is a semiprimary -semigroup.

NOTE 5.1.4 : If S is a -semigroup and a ∈ S then we denote

aaa n

n

w 1

1)( .

NOTE 5.1.5 : If S is a duo -semigroup then 1 1 1

1 1

( ) ( )w n n

n n

a a a a a S

THEOREM 5.1.6 : Let S be duo chained -semigroup. If Sa is a semisimple

element of S, then wa .

Proof : Suppose that a is a semisimple element of S. Therefore aaa , implies

that < a > = aa . Therefore aa = aa n 1)( for all natural

numbers n. Hence

aaa n

n

1

1)( wa and hence wa .

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THEOREM 5.1.7 : Let S be a duo chained Γ-semigroup. If wa for all a ∈S,

then S has no semisimple elements.

Proof : Suppose that < a >w = ∅ for all a ∈ S. Suppose if possible S has a semisimple

element a. By theorem 5.1.6, < a >w ≠ ∅. It is a contradiction. Therefore S has no

semisimple elements.

THEOREM 5.1.8 : Let S be a duo chained -semigroup. If S has no -idempotents

elements, then for any a ∈ S, < a > w = ∅ or < a >

w is a prime -ideal of S.

Proof : Suppose that S has no Γ-idempotent elements and a ∈ S. We have

1

1

( )w n

n

a a a

. Assume that < a > w ≠ ∅. If possible suppose that <a>

w is

not prime. Then there exists x, y ∈ S such that xΓy ⊆ <a>w, and w wx a y a .

By theorem 3.2.7, < x > Γ < y > = < xΓy > ⊆ < a > w.

Now x, y ∉ < a > w, implies that there exists natural numbers n, m such that

x ∉ ( <a >Γ)n-1

< a >, y ∉ ( <a >Γ)m-1

< a >. Consider k = min {n, m}.

Then x, y ∉ (<a >Γ)k-1

< a >. Since S is a duo chained Γ-semigroup,

we have ( <a >Γ)k-1

< a >⊆ < x > and ( <a >Γ)k-1

< a > ⊆ < y >.

Therefore ( <a >Γ)2k-1

< a > = ( <a >Γ)k-1

< a >Γ ( <a >Γ)k-1

< a > ⊆ < x>Γ< y >

⊆ < x Γy > ⊆ < a > w ⊆ ( <a >Γ)

4k-1< a > ⊆ ( <a >Γ)

2k-1< a >Γ ( <a >Γ)

2k-1< a >.

Therefore ( <a >Γ)2k-1

< a > ⊆ ( <a >Γ)2k-1

< a >Γ ( <a >Γ)2k-1

< a >.

Therefore a2k

is a semisimple element of S. By theorem 3.5.8, a 2k

is a regular element of

S. Therefore a 2k

= a2k

ΓxΓa2k

for some x ∈ S, implies that (a2k

Γx)Γ(a2k

Γx) = a2k

Γx and

hence 2ka x is a Γ-idempotent of S. So S has Γ-idempotent elements.

It is a contradiction. Hence < a > w is a prime Γ-ideal of S.

THEOREM 5.1.9 : If S is a duo chained strongly -cancellative -semigroup with an

identity then for every nonunit a, <a>w is either empty or a prime -ideal of S.

Proof : Suppose that a is a nonunit in S. If <a>w

= ∅ then the proof is trivial.

Let <a>w

≠ ∅. If possible suppose that <a>w is not a prime -ideal of S.

Then there exists ,x y S such that wx y a and , wx y a .


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By theorem 3.2.7, x y x y wa . Now , wx y a , implies that there

exists natural numbers n, m such that 1( )nx a a and 1( )my a a .

Consider min{ , }k n m . Then 1, ( )kx y a a .

Since S is duo chained -semigroup, we have 1( )ka a x and

1( )ka a y .

Therefore 2 1( ) ka a = 1( )ka a Γ 1( )ka a ⊆ x y

⊆ x y ⊆ wa ⊆ 4 1( ) ka a .

Then 2 1( ) ka a ⊆ 4 1( ) ka a ⊆ 2 1( ) ka a Γ 2 1( ) ka a

and hence 2ka ∈ 2 1( ) ka a Γ 2 1( ) ka a .

Therefore 2ka is a semisimple element of S.

By theorem 3.5.8, 2ka is a regular element of S.

Therefore 2ka = 2ka ΓxΓ 2ka for some x S implies that ( 2ka x )Γ( 2ka x ) = 2ka x

and hence S has Γ-idempotent elements. Since S is strongly cancellative and

2ka x Γe = ( 2ka x )Γ( 2ka x ) implies that 2ka x = e and hence aΓ(a2k-1

x) = e.

Hence a is a unit in S. It is a contradiction. Thus wa is a prime Γ-ideal of S.

Hence wa = ∅ or wa is a prime Γ-ideal of S.

THEOREM 5.1.10 : Let S be a duo chained -semigroup. If S S S then

S\S S = { x } for some x S.

Proof : Suppose if possible x, y S\S S and x y. Since S is a chained

-semigroup, yx or xy . If yx then yx and hence

syx for some Ss . Therefore SSx , which is not true. If xy , then

xy and hence sxy for some Ss . Therefore SSy , which is not true. It is

a contradiction. Therefore x = y. So there exists unique x S such that SSx .

Therefore S\ S S = { x } for some x S.

THEOREM 5.1.11 : Let S be a duo chained -semigroup with S\ S S = { x } for

some x S. Then S\{ x } is a -ideal of S.

Proof : Let a S\{ x } and s S. Since { }x S SÚ we have }{xsa and hence

a s S\{x}. Therefore S\{x} is a right -ideal of S. Since S is a duo -semigroup,

S\{x} is a -ideal of S.

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THEOREM 5.1.12 : Let S be a duo chained -semigroup. If S S S such that

S\S S = { x } for some x S then S = x S1 = S

1 x and S S = x S = S x is the

unique maximal -ideal of S.

Proof : Since S\S S = { x }, S S = S\{ x }. Now x S1 is a -ideal of S and S S is a

-ideal of S. Since { }x S SÚ and since S is a chained -semigroup,

S S x S1. So x S S S (x S) { x } and { }x S SÚ . Thus S S = x S.

So S = x S1 = S

1 x and S S = x S = S x.

Since S S is trivial, S S = x S = S x is a maximal -ideal.

∵ S is a chained -semigroup, S S = x S = S x is the unique maximal -ideal of S.

THEOREM 5.1.13 : Let S be a duo chained -semigroup with S S S such that

S\S S = { x } for some x S. If aS and a < x >w then xxa n 1)( for some

natural number n > 1.

Proof : Since S is a duo chained -semigroup with S S S such that S\S S = { x } for

some x S, by theorem 5.1.12, S S = x S = S x = S\{x}. Since

a wx = 1

1

( )n

n

x x

, there exists a natural number k such that a 1( )kx x .

Let n be the least positive integer such that a 1( )nx x and a 2( )nx x .

Now a 2 1( )nx x S and a 1 1( )nx x S .

Now a 2 1( )nx x S ⇒ a 2( )nx x S for some s ∈ S. s ∈ S, s ≠ x ⇒ s x S

Therefore a 12( )

nnx x x S x x S . It is a contradiction

and hence s = x. Therefore a 12( )

nnx x x x x .

THEOREM 5.1.14 : Let S be a duo chained -semigroup with S S S such that

S\S S = { x } for some x S. If aS and a < x >w then a xx r 1)( for some

natural number r or a 1( )n

nx x s , w

ns x for all natural numbers n.

Proof : Since S is a duo chained -semigroup with S S S such that x S\S S, by

theorem 5.1.12, S S = x S = S x = S\{ x }. Let aS. Suppose that a < x >w Then

a 1 1

1( )n

nx x S

. Therefore a 1 1( )nx x S for all natural numbers n. Hence

a ∈ 1n

x x

or a 1( )n

nx x s for some sn ∈ S. If sn ∉ < x >w then by theorem 5.1.13,

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1( )r

ns x x for some natural number r and hence a 11 1( ) ( )

n rn rx x x x x x .

If sn ∈ < x >w then a 1( )n

nx x s .

THEOREM 5.1.15 : Let S be a duo chained -semigroup with S\S S = { x } for

some x S. If S contains strongly -cancelable elements then x is a strongly

-cancellable element and < x >w is either empty or a prime -ideal of S.

Proof : Suppose if possible x is not strongly -cancellable element in S. Let Z be the set

of all non strongly -cancellable elements of S. Clearly x Z. So Z is nonempty subset

of S. Let a Z and s S. Since aZ, a is not strongly -cancellable in S. So there

exists b, c S such that caba and cb . Now caba implies

)()( casbas and hence casbas )()( and cb .

Therefore sΓa is a set of nonstrongly -cencellable elements of S.

Therefore s a Z and hence Z is left -ideal of S. Since S is a duo -semigroup, Z is

a -ideal of S. Since S\S S = { x }, by theorem 5.1.12, S = x S1. Since xZ and Z is

a -ideal of S, Z = S. It is a contradiction. Therefore x is a strongly -cancellable

element in S. Suppose that < x >w . Let a, b S and wxba .

Suppose if possible a wx and b wx . Now a, b wx , by theorem 5.1.14,

a xx n 1)( , b xx m 1)( for some natural numbers n, m.

Therefore xx mn 1)( = 1 1n m

x x x x

= a b wx ( )n mx S .

Therefore x x S S S. It is a contradiction.

Therefore either a < x >w or b < x >

w and hence < x >

w is a prime -ideal of S.

THEOREM 5.1.16 : Let S be a duo chained Γ-semigroup. Then S is an archemedian

Γ-semigroup without Γ-idempotents if and only if < a >w = ∅ for every a ∈ S.

Proof : Suppose that S is an archemedian Γ-semigroup without Γ-idempotents. If possible

suppose that < a >w ≠ ∅ for some a ∈ S. By theorem 5.1.8, < a >

w is a prime Γ-ideal of

S. Since S is an archemedian duo Γ-semigroup, by theorem 3.4.4, S has no proper prime

Γ-ideals. Therefore < a >w = S. Now wa a a a . Thus a is semisimple

element. By theorem 3.5.8, a is regular element. By theorem 1.3.30, S has Γ-idempotent

elements. It is a contradiction. Hence < a >w = ∅ for every a ∈ S. Conversely suppose

that < a >w = ∅ for every a ∈ S. Since < a >

w = ∅ for every a ∈ S, by corollary 5.1.6, S

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has no semi simple elements. By theorem 3.5.8, S has no regular elements. By theorem

1.3.30, S has no Γ-idempotent elements. If possible, suppose that P is proper prime

Γ-ideal of S. Let x ∈ S such that x P. Since x P by theorem 5.1.2, P 1( )n

nx P

.

Therefore P ⊆ < a >w = ∅. It is a contradiction. Hence S has no proper prime Γ-ideals.

By theorem 3.4.4, S is an archemedian Γ-semigroup.

Now we characterize Γ-group.

THEOREM 5.1,17 : Let S be a strongly cancellative archemedian duo chained

Γ-semigroup with wa for some a ∈ S, then S is a Γ-group.

Proof : Suppose that S is a strongly Γ-cancellative archemedian duo chained Γ-semigroup

with wa for some a ∈ S. Suppose if possible S has no Γ-idempotent elements.

Since wa , by theorem 5.1.8, < a >w is a prime Γ-ideal of S. Since S is an

archemedian duo Γ-semigroup, by theorem 3.4.4, S has no proper prime Γ-ideals. It is a

contradiction. Hence S has Γ-idempotent elements. Let e be a Γ-idempotent element in S.

Then x e e x e for every x ∈ S and , ∈ Γ. Since S is strongly Γ-cancellative, we

have x e x for every x ∈ S, α ∈ Γ. Similarly e x x for every x ∈ S, α ∈ Γ.

Therefore e x x e x . Hence e is the identity element in S. Let a ∈ S. Since e, a ∈ S

and S is archemedian Γ-semigroup, 1n

e e S a S

. Therefore . Since S is

duo Γ-semigroup = = . Therefore and hence

e x a for some x ∈ S. and α ∈ Γ. Now e e e x a x a implies that

x a x a e x a x e a . Since S is strongly cancelllative, we have a x e .

Similarily x a e Therefore x a a x e and hence x is the -inverse of a in S.

Therefore S is a Γ-group.

THEOREM 5.1.19 : Let S be a duo chained Γ-semigroup containing strongly

Γ-cancellative elements and < a >w = ∅ for every a ∈ S, then S is a strongly

cancellative Γ-semigroup.

Proof : Let S be a duo chained Γ-semigroup containing strongly Γ-cancellable elements.

Suppose that < a >w = ∅ for every a ∈ S. Let Z be the set of all non strongly

Γ-cancellative elements in S. Suppose if possible Z is a nonempty subset of S. If

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x ∈ Z, then there exists y, z ∈ S, , ∈ Γ such that x y = x z and y ≠ z. Therefore for any

s ∈ S, γ ∈ Γ s (x y) = s (x z) ⇒ (s x) y = (s x) z and y ≠ z. Hence s x ∈ Z. Therefore

Z is a left Γ-ideal of S and hence Z is a Γ-ideal of S. If possible, suppose that Z is not

prime. Then there exists a, b ∈ S such that a b ∈ Z and a, b ∉ Z. Since

a ∉ Z, b c = b d for some c, d ∈ S, , . Since b ∉ Z, c = d. It is a contradiction.

Therefore Z is a prime Γ-ideal of S. Since < a >w = ∅ for every a ∈ S, by theorem 5.1.16,

we have S is an archimedian Γ-semigroup without Γ-idempotents. Therefore by theorem

3.4.4, S has no prime Γ-ideals and hence Z = S. It is a contradiction to S contains strongly

Γ-cancellable elements. Hence Z = ∅. Thus S is a strongly Γ-cancellative Γ-semigorup.

5.2. DUO NOETHERIAN -SEMIGROUPS

In this section, the terms; noetherian Γ-semigroup, Γ-closed Γ-semigroup

and centre of a Γ-semigroup are introduced. It is proved that if S is a noetherian

Γ-semigroup containing proper Γ-ideals, then S has a maximal Γ-ideal. It is proved that if

H is the collection of all Γ-ideals in a Γ-closed duo Γ-semigroup S which are not principal

and H ≠ ∅, then there exists a prime Γ-ideal of S which is not a principal Γ-ideal. It is

proved that if every prime Γ-ideal including S is principal in a Γ-closed duo Γ-semigroup

S, then every Γ-ideal in S is principal. It is proved that if S is a Γ-closed duo Γ-semigroup,

which is a union of finite number of principal Γ-ideals and every proper prime Γ-ideal is

principal, then every Γ-ideal is an intersection of a principal Γ-ideal and an S-Primary

Γ-ideal. Also it is proved that if S is a Γ-closed duo Γ-semigroup, which is a union of

finite number of principal Γ-ideals and every proper prime Γ-ideal of S is principal and

S = S ΓS then every proper Γ-ideal is principal. If S is a duo Γ-semigroup such that

S and every maximal Γ-ideal is principal then it is proved that (1) S has at most

two maximal Γ-ideals and (2) if P is a proper prime Γ-ideal of S then either P is a principal

Γ-ideal or P = xΓP for some . If every maximal Γ-ideal in a Γ-closed duo

Γ-semigroup S is principal and S , for every , then it is proved

that S is a union of two principal Γ-ideals and every Γ-ideal is an intersection of a prime

Γ-ideal and an S-primary Γ-ideal. If S is a noetherian or Archimedean duo Γ-semigroup

such that S = 1

n

i

i

x

and suppose ia x a for all , which is not a product of

power of , then it is proved that S is finitely generated and in particular if S is

noetherian strongly Γ-cancellative Γ-semigroup without identity then S is finitely

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generated. If S is a duo Γ-semigroup which is a union of finite number of principal

Γ-ideals and if S = , then it is proved that S contains Γ-idempotent elements. If S is a

strongly Γ-cancellable duo Γ-semigroup which is a union of finite number of principal

Γ-ideals, then it is proved that S contains identity if and only if S = . In an

Archimedean duo Γ-semigroup S, if S is a union of finite number of principal Γ-ideals or S

contains a maximal Γ-ideal which is finitely generated, then it is proved that every proper

Γ-ideal is principal and S is a union of at most two principal Γ-ideals. It is proved that if A

is a finitely generated Γ-ideal of a duo Γ-semigroup S, A = AΓB for some Γ-ideal B and

then for some . If S is a duo Γ-semigroup containing no

Γ-idempotents except perhaps the identity 1 and P is a finitely generated prime Γ-ideal

contained properly in for some and , then it is proved that (1) P does

not contain any strongly Γ-cancellable element and (2) if A is finitely generated Γ-ideal

containing a strongly Γ-cancellable element then A for any proper Γ-ideal B. It is

proved that if A is a finitely generated Γ-ideal of a duo Γ-semigroup S and Aw = B such

that A ΓB = where are primary Γ-ideals, then AΓB = B. If S is a noetherian

duo Γ-semigroup without Γ-idempotents except perhaps identity, then it is proved that for

any Γ-ideal A, Aw

where Z is the set of all nonstrongly Γ-cancellable elements and

Aw = if S is strongly Γ-cancellative. If S is a noetherian Γ-closed duo Γ-monoid with a

unique maximal Γ-ideal M = < m > for some m and if x then it is proved that

, u is a unit or with Γx Γs. If S is a noetherian duo Γ-monoid

with a unique maximal Γ-ideal M = < m > for some m and if P is a proper prime

Γ-ideal of S such that P M, then it is proved that P . If S is a noetherian duo

Γ-monoid with a unique maximal Γ-ideal M = < m > for some m and if S has no

Γ-idempotents except 1, then it is proved that is a prime Γ-ideal and also if Z M

where Z is the set of all non cancellable elements of S, then Z = If T is a Γ-closed

duo Γ-semigroup and S is a duo Γ-semigroup such that S is a Γ-subsemigroup of T and

T = xΓS1 for some x and if S is noetherian then it is proved that T is noetherian.

Further an analogue of HILBERT basis theorem is obtained for duo Γ-semigroups.

Now we introduce the notion of noetherian Γ-semigroup.

DEFINITION 5.2.1 : A Γ-semigroup S is said to be a noetherian Γ-semigroup if

ascending chain of Γ-ideals becomes stationary. i.e., if is an ascending

chain of Γ-ideals of S, then there exists a natural number m such that for all

natural numbers .

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NOTE 5.2.2 : A Γ-semigroup S is noetherian if and only if every Γ-ideal of S is a union

of finite number of principal Γ-ideals of S.

We charectarize the noetherian Γ-semigroup.

THEOREM 5.2.3 : If S is a noetherian Γ-semigroup containing proper Γ-ideals then

S has a maximal Γ-ideal.

Proof : Let be a proper Γ-ideal of S. If is not a maximal Γ-ideal of S, then there

exists a proper Γ-ideal of S such that . If is not a maximal Γ-ideal of S,

then there exists a proper Γ-ideal of S such that . By continuing this

process we get an ascending chain of proper Γ-ideals of S. Since S is noetherian, the chain

… is stationary. It is a contradiction. Therefore there exists a maximal

Γ-ideal of S.

Now we introduce the notions of Γ-closed Γ-ideal and Γ-closed Γ-semigroup.

DEFINITION 5.2.4 : A Γ-ideal A of a Γ-semigroup S is said to be Γ-closed if a, b ∈ S,

∈ Γ, aαb ∈ A ⇒ a Γb ⊆ A.

DEFINITION 5.2.5 : A Γ-semigroup S is said to be Γ-closed if every Γ-ideal of S is

Γ-closed.

NOTE 5.2.6 : In a Γ-closed Γ-semigroup S, < aαb > = < aΓb > where a, b ∈ S and ∈ Γ.

THEOREM 5.2.7 : Let H be the collection of all Γ-ideals in a Γ-closed duo

Γ-semigroup S which are not principal. If H ≠ ∅ then there exists a prime Γ-ideal

which is not a principal Γ-ideal.

Proof : Let H = { Aα : α ∈ Δ } be the collection of all Γ-ideals in a duo Γ-semigroup S,

which are not principal. If A

= < x > for some , then for some .

Therefore < x > ⊆ and hence . Then ∉ H. It is a

contradiction. Hence . So H. Thus H satisfies all

the conditions of Zorn`s lemma. By Zorn`s lemma, H has a maximal element say P.

Suppose if possible P is not a prime Γ-ideal. Then there exists such that

and and . Since P is maximal in H, . Therefore

is a principal Γ-ideal. Then for some . If

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then we get P = < x > and hence . It is not true. Hence . Therefore

and hence = < x >. Hence P ⊆ . Now { : }P s S s b P is a

Γ-ideal of S. Then clearly a P and . Therefore P P and P P . By the

maximality of P in H, we get P H . Therefore P y for some y S . Now

y P ⇒ y b P ⇒ y b P Let t P . Since P b , we have t s b for some

s S , T . Now s b P . Since S is Γ-closed, s b P . Hence s P y .

Therefore s r y for some ,r S .

Now t s b r y b r y b y b ⇒ t y b y b .

Therefore we have P y b . Hence P y b . Thus P ∉ H. It is a contradiction.

Therefore P is a prime Γ-ideal.

COROLLARY 5.2.8 : If H is the collection of all Γ-ideals in a -closed duo

Γ-semigroup S, which are not finitely generated and H≠∅, then there exists a prime

Γ-ideal which is not finitely generated.

THEOREM 5.2.9 : If every prime Γ-ideal including S is principal in a -closed duo

Γ-semigroup S, then every Γ-ideal in S is principal.

Proof : Let H be the collection of all Γ-ideals in S which are not principal. If H≠∅ then by

theorem 5.2.7, H contains a proper prime Γ-ideal which is not principal. It is a

contradiction. Hence H = ∅. Therefore every Γ-ideal in S is principal.

COROLLARY 5.2.10 : If every prime Γ-ideal including S is finitely generated in a

-closed duo Γ-semigroup S, then every Γ-ideal in S is finitely generated.

THEOREM 5.2.11 : If S is a -closed duo Γ-semigroup, which is a union of finite

number of principal Γ-ideals and every proper prime Γ-ideal is principal, then every

Γ-ideal is an intersection of a principal Γ-ideal and an S – Primary Γ-ideal.

Proof : First we prove that every primary Γ-ideal Q such that is a principal

Γ-ideal. Now is a proper prime Γ-ideal of S. Since every prime Γ-ideal is

principal, we have P = < a > for some a Therefore by theorem 3.3.13, there exists

n such that .

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Hence 1 1 1

n n n

P P a a a a Q

. Now in the case,

when Q is contained in every power of P, we have Q = On

the other hand, there exists m such that Q and

Q⊈ a Γ-ideal of S and

Q = A . Since , we get A ⊈ . Since Q is a primary Γ-ideal,

and hence Q = = < Therefore Q is a principal

Γ-ideal. By note 5.2.2, S is a noetherian Γ-semigroup. Thus by theorem 4.3.4, every

Γ-ideal A is of the form where each a primary Γ-ideal such that

for i j. We may assume that for i = 1, 2, .., m and

for . Clearly = S. Therefore by theorem

4.3.2, S-primary Γ-ideal. Now we claim

that = , which proves that is

principal. Without loss of generality, we may assume that P1 is maximal in 1

m

iiP

.

P2 is maximal in 2

m

iiP

and so on. This means that Pi Pj for all .

Now assume for r = .

Therefore = for some

a ∈ . Now .

If a Pr+1, then = 1rP and thus since

Pr+1 is prime, for some . It is a contradiction. So . Since Qr+1

is a primary Γ-ideal, we have and hence .

So .

Therefore by induction = .

Thus every Γ-ideal is an intersection of a principal Γ-ideal and an S-primary Γ-ideal.

THEOREM 5.2.12 : Let S be a -closed duo Γ-semigroup, which is a union of finite

number of principal Γ-ideals. If every proper prime Γ-ideal of S is principal and

S = S ΓS then every proper Γ-ideal is principal.

Proof : Since S is a -closed duo Γ-semigroup which is a union of finite number of

principal Γ-ideals, S = 1

n

i

i

x

where for all i . Since S = S ΓS,

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Γ for i = 1, 2, . ., n. Thus is semi simple and hence by theorem 3.5.8,

is regular. By theorem 1.3.30, < > = < > for some Γ – idempotent in S. Let A be

any proper Γ-ideal such that = S. Therefore for all i = 1, 2, . . .,

n. Therefore and hence S = A. It is a contradiction. Therefore there

exists no Γ-ideal of A of S such that = S. By theorem 5.2.9, every proper Γ-ideal is

principal.

THEOREM 5.2.13 : If S is a duo Γ-semigroup such that S and every maximal

Γ-ideal is principal then S has at most two maximal Γ-ideals.

Proof : Let S be a duo Γ-semigroup such that S Suppose that every maximal

Γ-ideal is principal. Let ΓS. Then S\{ a } is a maximal Γ-ideal. Therefore

S\{ a } = for some b . Clearly a b. Let b ΓS. Then S\{ a } = 

and hence S\{ a } = . Let M be a maximal Γ-ideal of S. Then M = < c > for some

c . If c ΓS then M . Since M is maximal, M = ΓS = S\{ a }. If c ΓS

then c S\{ a } and hence c = a. Thus M = < a >. So if b ΓS, S can have at most two

maximal Γ-ideals, namely S\{a} and < a >. Let b ΓS. Then S = { a }

= { a } { b } ΓS. Let M = < c > be a maximal Γ-ideal. If c ΓS then c = a or

c = b. Then M = S\{ a } or M = S\{ b }. If c ΓS then ΓS and hence M is

properly contained in a proper Γ-ideal S\{ a }. It is a contradiction. Hence S has at most

two maximal Γ-ideals.

THEOREM 5.2.14 : Let S be a duo Γ-semigroup such that S and every

maximal Γ-ideal is principal. If P is a proper prime Γ-ideal of S then either P is a

principal Γ-ideal or P = x ΓP for some .

Proof : Let P be any proper prime Γ-ideal and ΓS. Now S\{ a } is a maximal

Γ-ideal. Therefore S\{ a } = for some b . If a P then P S\{ a } = . If b

then P = . If b then P = b ΓP, since P is a prime Γ-ideal. Let .

If b then P = S. If b P then P S\{ b }. Since S\{ b } is maximal Γ-ideal, we have

P S\{ b }= < x > for some x . If x then P = < x >. If x P, let y . Then

y . So y for some . Since P is prime, . Hence

y . Clearly x P P . Hence P = < x > or P = x ΓP for some x .

128

THEOREM 5.2.15 : If every maximal Γ-ideal in a -closed duo Γ-semigroup S is

principal and S , for every , then S is a union of two principal

Γ-ideals and every Γ-ideal is an intersection of a prime Γ-ideal and an S-primary

Γ-ideal.

Proof : Let P be any proper prime Γ-ideal of S. By theorem 5.2.14, either P is a principal

Γ-ideal or P = x ΓP for some . If P = x ΓP for some , then for all

natural numbers n. Thus 1

1 1

n n w

n n

P x P x x x

.

It is a contradiction. Therefore P = < x > for some . Thus every proper prime

Γ-ideal is a principal Γ-ideal. If ΓS then by hypothesis, the maximal Γ-ideal S\{a}

is of the form for some b . Therefore S = { a } = < a > . By

theorem 5.2.11, every Γ-ideal of S is an intersection of a prime Γ-ideal and an S-primary

Γ-ideal of S.

THEOREM 5.2.16 : Let S be a duo noetherian Γ-semigroup such that S = 1

n

i

i

x

.

Suppose for all , which is not a product of power of . Then S

is finitely generated. In particular if S is noetherian strongly -cancellative

Γ-semigroup without identity then S is finitely generated.

Proof : Suppose that there exists an element a such that a is not a product of .

If a = for Γ, where is not a product of power of . Hence

for Γ, where is not product of powers of . If then

for some and hence Γ , which

is not true. Hence . By continuing this process, we get a nonterminating

chain of Γ-ideals Since S is noetherian, it is a

contradiction. So S is finitely generated. If S is a strongly Γ-cancellative Γ-semigroup and

if for Γ, then is an identity in S. It is a contradiction. So

Γa for all . As above, we have S is finitely generated.

THEOREM 5.2.17 : Let S be a duo Γ-semigroup which is a union of finite number of

principal Γ-ideals. If S = , then S contains Γ-idempotent elements.

Proof : Suppose that S = 1

n

i

i

x

and for and S = . Since S = ,

we have for each i = 1, 2, 3, . . . . , n. Therefore each is semi

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simple in S. By theorem 3.5.8, is regular in S and hence by theorem 1.3.30, S contains

Γ-idempotents.

THEOREM 5.2.18 : Let S be a strongly -cancellable duo Γ-semigroup which is a

union of finite number of principal Γ-ideals. Then S contains identity if and only if

S = .

Proof : Suppose that S is a strongly Γ-cancellable duo Γ-semigroup and S = . By

theorem 5.2.17, S contains Γ-idempotent element say e. Let a . Then .

Since S is strongly Γ-cancellative, . Similarly . Then e is the identity in

S. Therefore S contains the identity. Conversely suppose that S contains the identity.

Then clearly S = .

THEOREM 5.2.19 : Let S be a duo archemedian Γ-semigroup. If S is a union of

finite number of principal Γ-ideals, then every proper Γ-ideal is principal and S is a

union of at most two principal Γ-ideals.

Proof : Suppose that S =1

n

i

i

x

. Let H be the collection of all proper Γ-ideals which

are not principal. If H then clearly H is a partially ordered set under set inclusion.

Let be a chain of Γ-ideals in H. If S = then for some natural number i.

If we take j = max then for i = 1, 2, 3, …, n.

So S = 1

n

i

i

x

. and hence Aj= S. It is a contradiction. Hence S A . If

A = < a > for some a , then a iA for some i and hence iA = < a >, which is not

true. Thus A H. Therefore H satisfies the hypothesis of Zorn`s lemma. By Zorn`s

lemma, there exists a maximal element P in H. By corollary 5.2.8, P is a prime Γ-ideal of

S. Since S is a duo archemedian Γ-semigroup, by theorem 3.4.4, S has no proper prime

Γ-ideals. It is a contradiction. Hence H = . Therefore every proper Γ-ideal of S is a

principal Γ-ideal. Let S = with for . If , then

. Since ;is a proper Γ-ideal, is a

principal Γ-ideal. Thus either or . This contradicts the

choice of . Thus ,


130

THEOREM 5.2.20 : Let S be an archemedian duo Γ-semigroup. If S contains a

maximal Γ-ideal which is finitely generated, then every proper Γ-ideal is principal

and S is a union of at most two principal Γ-ideals.

Proof : Suppose that S contains a maximal Γ-ideal M which is finitely generated. Let

. Since M is maximal, . So S is a union of finite number of

principal Γ-ideals. Therefore by theorem 5.2.19, every Γ-ideal is principal and S is a

union of at most two principal Γ-ideals.

THEOREM 5.2.21 : Let S be an archemedian duo Γ-semigroup with S =1

n

i

i

x

.

If Γa> for all , which is not a product of powers of , then S is

finitely generated.

Proof : Let S be an archemedian duo Γ- semigroup with S =1

n

i

i

x

. By theorem

5.2.19, S is a union of at most two principal Γ-ideals. By theorem 5.2.16, S is finitely

generated.

THEOREM 5.2.22 : Let A be a finitely generated Γ-ideals of a duo Γ-semigroup S. If

A = A ΓB for some Γ-ideal B and if then for some .

Proof : Suppose that A is a finitely generated Γ-ideal of a duo Γ- semigroup S. Without

loss of generality, assume that A =1

n

i

i

x

with for . Suppose that

A = A ΓB for some Γ-ideal B of S. Then A = A ΓB = (1

n

i

i

x

) ΓB = ΓB).

Let . If a = for some i where , then for . So

ΓB. Thus . If a for all , then for some .

Therefore for some s S . So ΓB = . Therefore

for some .

THEOREM 5.2.23 : Let S be a duo Γ-semigroup containing no Γ-idempotents except

perhaps the identity 1. If P is a finitely generated prime Γ-ideal contained properly

in for some and , then P does not contain any strongly

-cancellable element.


131

Proof : Suppose that S is a duo Γ- semigroup containing no Γ-idempotents except the

identity 1 and P is a finitely generated prime Γ-ideal such that ΓS for some

and ΓS . Since ΓS, x . Clearly ΓP. Let . Since ΓS,

ΓS. So for some . Now , P is prime .

Therefore ΓP and hence ΓP. Therefore ΓP. Assume that a is a

strongly cancellable element in P. By theorem 5.2.22, a a b , . Therefore

= = . Since a is strongly Γ-cancellative, we have . It is a

contradiction. Hence P does not contain strongly cancellable elements.

THEOREM 5.2.24 : Let S be a duo Γ-semigroup containing no Γ-idempotents except

perhaps the identity 1 and P be a finitely generated prime Γ-ideal contained properly

in for some and . If A is finitely generated Γ-ideal containing a

strongly -cancellable element then A for any proper Γ-ideal B.

Proof : Suppose that A is a finitely generated Γ-ideal containing a strongly Γ-cancellable

element say a. Suppose if possible A for some proper Γ-ideal B. Now

implies that for some .

Therefore = = . Since a is strongly Γ-cancellative, b b b .

Therefore B contains Γ-idempotent elements. It is a contradiction. Hence A .

THEOREM 5.2.25 : Let A be a finitely generated Γ-ideal of a duo Γ-semigroup S and

Aw = B such that A ΓB = where are primary Γ-ideals. Then A ΓB = B.

Proof : Since A, B are two Γ-ideals of a duo Γ- semigroup S, clearly we have .

Let for each . Since each is a primary Γ-ideal of S, is a prime

Γ-ideal of S for each . Now for each . Let . Since

for each . = B . Let = . Since A =

, = for i = 1, 2, 3, . . ., n. Then for i = 1, 2, 3,

. . ., n. Let m = Max . Then 1m

A A Q

.

Since B = = 1 1

1

m m

m

A A A A Q

. Thus B = . ∴ .

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THEOREM 5.2.26 : Let S be a noetherian duo Γ-semigroup without Γ-idempotents

except perhaps identity. Then for any Γ-ideal A, Aw

where Z is the set of all non-

strongly -cancellable elements and Aw = if S is strongly cancellative.

Proof : If Aw = , then clearly A

w If A

w , by theorem 5.2.25, A .

By theorem 5.2.24, we have does not contain strongly Γ-cancellable elements.

Thus Aw

If S is strongly Γ-cancellative then Z = and hence Aw

Aw = .

THEOREM 5.2.27 : Let S be a noetherian -closed duo monoid with a unique

maximal Γ-ideal M = < m > for some m . If x then , u is a unit or

with Γx Γs.

Proof : Let . Now implies that for some . If

is not a unit, then for some and hence .

By proceding the same process, we get for some natural number n, for some

. If is a unit for some natural number n, then where , is a

unit. If is not a unit, then for n = 1, 2, 3, . . .and

hence 1

1

n n

n

n

x m t m m

1

1

n

n

m m

= w wm M .

Therefore . Since S is noetherian, the chain < < . . . is

stationary. Therefore there exists a natural number n such that for

all natural numbers n. Therefore for some s .

Now = = x .

∴ x = = = m .

THEOREM 5.2.28 : Let S be a noetherian duo monoid with a unique maximal

Γ-ideal M = < m > for some m . If P is a proper prime Γ-ideal of S such that P

M, then P .

Proof : Let P be any proper prime -ideal of S such that P M. Then there exists M

such that . By theorem 5.2.14, P = x . Thus for all natural numbers

n. Therefore 1

1 1

n n w

n n

P x P x x M

and hence P

THEOREM 5.2.29 : Let S be a noetherian duo Γ-monoid with a unique maximal

Γ-ideal M = < m > for some m . If S has no Γ-idempotents except 1, then is a

prime Γ-ideal and also if Z M where Z is the set of all non strongly Γ-cancellable

elements of S, then Z =


133

Proof : Suppose that S has no -idempotent elements except 1 and Z is the set of all non

strongly -cancellable elements of S. Therefore Z is a proper prime -ideal of S. If

Z M, then by theorem 5.2.28, Z

. Since Z is a prime -ideal of S and M is -ideal

of S, by theorem 5.2.26, Mw

and hence Z

.

THEOREM 5.2.30 : Let T be a -closed duo Γ-semigroup and S be a duo

-semigroup such that S is a Γ-subsemigroup of T and T = x ΓS1 for some x . If S

is noetherian then T is noetherian.

Proof : Suppose that S is noetherian duo - semigroup. Let A1 be a proper -ideal of T.

Let A = {a : A1 }. Let and . Now

A

1.

Now T T. Since A1 is a -ideal of T, A

1. So A

1.

Thus . Therefore A is a right -ideal of S. Since S is a duo – semigroup, A is

a Γ-ideal of S. Since S is noetherian, A =1

n

i

i

a

= 1

1

n

i

i

a S

. Write B = 1

1

n

i

i

x a S

. Let

y A1. Clearly and 1y T x S . So for some , . Since

= y A1 and A

1 is Γ-closed we get

1x s A and hence and so 1

i is a a S

for some i. Therefore 1

iy x s x a S B . Therefore A1 B. Let z .

Then 1

iz x a S . Since 1

ia S A , we have 1 1

ix a S A and hence z A1.

Therefore B A1 and hence A

1 B. Now 1 1 1

1 1

n n

i i

i i

x a S A B x a S

.

Hence A1 1

1

n

i

i

x a S

. ∴ T is noetherian.

Now we introduce the notion of center of S.

DEFINITION 5.2.31 : Let S be a Γ-semigroup. The set

ℭ(S) = : for all sa S a s s a S is called the centre of S.

A close analogue of HILBERT’s basis theorem for commutative semigroups has

been proved by SATYANARAYANA[44]. ANJANEYULU[4] extended the HILBERT’s

baisis theorem for duo semigroups. Here we extend it for duo Γ-semigroups.


134

THEOREM 5.2.32 : (ANALOGUE OF HILBERT BASIS THEOREM) :

Let T be a -closed duo Γ-semigroup and S be a duo Γ-semigroup such that S is a

-subsemigroup of T. Suppose T =1

n

i

i

x S

, are in the centre of T,

or for some Γ, and for every i. Then T is noetherian if S is

noetherian.

Proof : We prove the theorem by using induction on the number n of generators of T. i.e.

the number of . By theorem 5.2.30, the result is evident for all T with one generator.

Suppose that the theorem is true for all T with the number of generators . Let A1

be a -ideal of T. Let 1:A a S x a A . Clearly A is a -ideal of S and hence A

= 1

1

n

i

i

a S

, . If T1 =2

n

i

i

x S

and A1

, then is a subsemigroup of T

containing S and L1 is a -ideal in . Then by induction, L1 =1

1

r

i

i

b T

. We first claim

that A1 = B where B = 1

1

1

m

i

i

x a S

. We claim that 1A B L . If

then 1 ix x a A or 1 ix x a S A and hence A1 . Thus .B A Clearly

1L A . Therefore 1B L A . Let x A . If 1x T then 1 1x A T L and hence

1x B L . If 1x T

then 1x x S and hence 1x x s for some

γ ∈ Γ, s ∈ S. Since T is Γ-closed, A is a Γ-closed Γ-ideal of T. So

1 1x s x A x S A s A . 1 1 ix x s x a S for some i x B . Therefore

1A B L . Hence 1A B L . Now 1A B L = 1 1

1 1

m r

i i i

i i

x a S b T

. Therefore A is

finitely generated Γ-ideal. Hence T is noetherian.

* * * * *

STRUCTURE OF DUO Γ-SEMIGROUPSshodhganga.inflibnet.ac.in/bitstream/10603/9855/13/13_chapter...

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