CHAPTER 1

Γ-SEMIGROUPS

Γ-semigroups

35

CHAPTER – 1 횪-SEMIGROUPS

As a generalization of a semigroup, SEN [41], introduced the notion of

Γ-semigroup in 1981 and developed some theory on Γ-semigroups [42], [43], [44] and

[45]. JIROJKUL, SRIPAKORN, CHINRAM [21], extended many classical notions of

semigroups to Γ-semigroups. The study of bi-ideals was made by AIYARED IAMPAN

[6], BRAJA ISLAM [9] in Γ-semigroups and by KAUSHIK and KHAN MOIN [22] in

Γ-semirings. IAMPAN[6] and BRAJA[9] introduced and characterized 0-minimal and

0-maximal bi-ideals in Γ-semigroups. BRAJA ISLAM [8], [9]; JAGATAP and PAWAR

[20] studied about quasi ideals in Γ-semirings. CHINRAM [10] studied about quasi-ideals

and obtained some characterizations of regular Γ-semigroups. CHINRAM and SIAMMAI

[11] generalized the green’s relations in semigroups to Γ-semigroups and to reductive

Γ-semigroups. DUTTA and CHATTERZEE [15] also studied the properties of green’s

relations in Γ-semigroups and generalized the notions; idempotent elements, regular

elements and semisimple elements in Γ-semigroups. In this chapter we develop the

algebraic theory of Γ-subsemigroups and Γ-ideals in Γ-semigroups.

This chapter is divided into 4 sections. In section 1, the notion of a Γ-semigroup is

introduced and some examples are given. Further the terms; commutative Γ-semigroup,

quasi commutative Γ-semigroup, normal Γ-semigroup, left pseudo commutative

Γ-semigroup, right pseudo commutative Γ-semigroup are introduced. It is proved that

(1) if S is a commutative Γ-semigroup then S is a quasi commutative Γ-semigroup, (2) if S

is a quasi commutative Γ-semigroup then S is a normal Γ-semigroup, (3) if S is a

commutative Γ-semigroup, then S is both a left pseudo commutative and a right pseudo

commutative Γ-semigroup. Further the terms; left identity, right identity, identity, left

zero, right zero, zero of a Γ-semigroup are introduced. It is proved that if a is a left

identity and b is a right identity of a Γ-semigroup, then a = b. It is also proved that any

Γ-semigroup has at most one identity. It is proved that if a is a left zero and b is a right

zero of a Γ-semigroup, then a = b and also it is proved that any Γ-semigroup has at most

one zero element.

In section 2, the terms; Γ-subsemigroup, Γ-subsemigroup generated by a subset,

cyclic Γ-subsemigroup of a Γ-semigroup and cyclic Γ-semigroup are introduced. It is

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36

proved that (1) the nonempty intersection of any two 횪-subsemigroups of a 횪-semigroup S

is a 횪-subsemigroup of S, (2) the nonempty intersection of any family of 횪-subsemigroups

of a 횪-semigroup S is a 횪-subsemigroup of S. It is also proved that if A is a nonempty

subset of a Γ-semigroup S, then the Γ-subsemigroup of S generated by A is the

intersection of all Γ-subsemigroups of S containing A.

In section 3, the terms; left Γ-ideal, right Γ-ideal, Γ-ideal, proper Γ-ideal, trivial

Γ-ideal, maximal left Γ-ideal, maximal right Γ-ideal, maximal Γ-ideal, Γ-ideal generated

by a subset A, principal left Γ-ideal, principal right Γ-ideal, principal Γ-ideal of a

Γ-semigroup are introduced. Also left duo Γ-semigroup, right duo Γ-semigroup, duo

Γ-semigroup, left simple Γ-semigroup, right simple Γ-semigroup, simple Γ-semigroup are

introduced. It is proved that (1) the nonempty intersection of two left Γ-ideals of a

Γ-semigroup S is a left Γ-ideal of S, (2) the nonempty intersection of any family of left

횪-ideals of a 횪-semigroup S is a left 횪-ideal of S, (3) the union of two left 횪-ideals of a

횪-semigroup S is a left 횪-ideal of S and (4) the union of any family of left 횪-ideals of a

횪-semigroup S is a left 횪-ideal of S. It is also proved that (1) the nonempty intersection of

two right Γ-ideals of a Γ-semigroup S is a right Γ-ideal of S, (2) the nonempty intersection

of any family of right 횪-ideals of a 횪-semigroup S is a right 횪-ideal of S, (3) the union of

two right 횪-ideals of a 횪-semigroup S is a right 횪-ideal of S and (4) the union of any

family of right 횪-ideals of a 횪-semigroup S is a right 횪-ideal of S. Further it is proved that

(1) the nonempty intersection of two Γ-ideals of a Γ-semigroup S is a Γ-ideal of S, (2) the

nonempty intersection of any family of 횪-ideals of a 횪-semigroup S is a 횪-ideal of S,

(3) the union of two 횪-ideals of a 횪-semigroup S is a 횪-ideal of S and (4) the union of any

family of 횪-ideals of a 횪-semigroup S is a 횪-ideal of S.

It is proved that if S is a Γ-semigroup and a ∈ S then (i) L(a) = a∪S횪a,

(ii) R(a) = a∪a횪S, (iii) J(a) = a ∪ aΓS ∪ SΓa ∪ SΓaΓS. It is proved that a Γ- semigroup

S is a duo Γ- semigroup if and only if x횪S1 = S1횪x for all x ∈ S. Further it is also proved

that every normal Γ- semigroup is a duo Γ- semigroup. It is proved that (1) a 횪-semigroup

S is a left simple 횪-semigroup if and only if S횪a = S for all a ∈ S, (2) a 횪-semigroup S is a

right simple 횪-semigroup if and only if a횪S = S for all a ∈ S, (3) a 횪-semigroup S is a

simple 횪-semigroup if and only if SΓaΓS = S for all a ∈ S.

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In section 4, the terms α-idempotent, Γ-idempotent, strongly idempotent, midunit,

r-element, regular element, left regular element, right regular element, completely regular

element, (α, β)-inverse of an element, semisimple element and intra regular element in a

Γ-semigroup are introduced. Further the terms, idempotent Γ-semigroup and generalized

commutative Γ-semigroup are introduced. It is proved that every generalized commutative

Γ-semigroup is a left duo Γ-semigroup. It is proved that every

훂-idempotent element of a Γ-semigroup is regular. It is proved that every 횪-ideal of a

regular 횪-semigroup S is a regular 횪-subsemigroup of S. It is proved that a Γ-semigroup S

is regular Γ-semigroup if and only if every principal Γ-ideal is generated by an

idempotent. Further it is also proved that, in a Γ-semigroup, a is a regular element if and

only if a has an (훼, 훽)-inverse. It is proved that, (1) if a is a completely regular element of

a Γ- semigroup then a is both left regular and right regular, (2) if ‘a’ is a completely

regular element of a Γ-semigroup S, then a is regular and semisimple, (3) if ‘a’ is a left

regular element of a Γ-semigroup S, then a is semisimple, (4) if ‘a’ is a right regular

element of a Γ- semigroup S, then a is semisimple, (5) if ‘a’ is a regular element of a

Γ- semigroup S, then a is semisimple and (6) if ‘a’ is a intra regular element of a

Γ- semigroup S, then a is semisimple. It is also proved that if a is an element of a duo

횪-semigroup, then (1) a is regular (2) a is left regular, (3) a is right regular, (4) a is intra

regular, (5) a is semisimple, are equivalent.

The contents of section 4 in chapter 1 are accepted for publication in “International Journal of Mathematical Sciences, Technology and Humanities” under the title ‘Primary and semiprimary 횪-ideals in 횪-semigroups’ [32].

1.1 . 횪-SEMIGROUP

In this section, the notion of a Γ-semigroup [40] is introduced and some examples are

given. Further the terms commutative Γ-semigroup, quasi commutative Γ-semigroup,

normal Γ-semigroup, left pseudo commutative Γ-semigroup, right pseudo commutative

Γ-semigroup are introduced. It is proved that (1) if S is a commutative Γ-semigroup then

S is a quasi commutative Γ-semigroup, (2) if S is a quasi commutative Γ-semigroup then S

is a normal Γ-semigroup, (3) if S is a commutative Γ-semigroup, then S is both a left

pseudo commutative and a right pseudo commutative Γ-semigroup. Further the terms; left

identity, right identity, identity, left zero, right zero, zero of a Γ-semigroup are introduced.

It is proved that if a is a left identity and b is a right identity of a Γ-semigroup S,

then a = b. It is also proved that any Γ-semigroup S has at most one identity. It is proved

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38

that if a is a left zero and b is a right zero of a Γ-semigroup S, then a = b and it is also

proved that any Γ-semigroup S has at most one zero element.

We now introduce the notion of a Γ-semigroup which is due to SEN [41].

DEFINITION 1.1.1 : Let S and Γ be two non-empty sets. Then S is called a

Γ-semigroup if there exist a mapping from S S to S which maps (a, , b) a b

satisfying the condition : (aγb)µc = aγ(bµc) for all a, b, c ∈ S and γ, µ ∈ Γ.

NOTE 1.1.2 : Let S be a Γ-semigroup. If A and B are two subsets of S, we shall denote

the set { a훾b : a ∈ A , b ∈ B and 훾 ∈ Γ } by AΓB.

In the following some examples of Γ-semigroups are given

EXAMPLE: 1.1.3 : Let S be the set of all non-positive integers and Γ be the set of all non-

positive even integers. If a훼b denote as usual multiplication of integers for a, b ∈ S and

훼 ∈ Γ, then S is a Γ-semigroup.

EXAMPLE: 1.1.4 : Let Q be the set of rational numbers and Γ = N be the set of natural

numbers. Define a mapping from Q × Γ × Q to Q by a훼b = usual product of a, 훼, b;

for a, b ∈ Q, 훼 ∈ Γ. Then Q is a Γ-semigroup.

EXAMPLE: 1.1.5 : Let S = { 5n + 4 : n is a positive integer} and Γ = { 5n + 1 : n is a

positive integer}. Then S is a Γ-semigroup with the operation defined by a훼b = a + 훼 + b

where a, b ∈ S, 훼 ∈ Γ and + is the usual addition of integers.

EXAMPLE: 1.1.6 : Let S be the set of all integers of the form 4n+1 where n is an integer

and Γ denote the set of all integers of the form 4n+3. If aγb is a+γ+b , for all a, b ∈ S and

γ ∈ Γ, then S is a Γ-semigroup.

EXAMPLE: 1.1.7 : Let S = { ∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} and

Γ = {∅, {a}, {a, b, c}}. If for all A, C ∈ S and B ∈ Γ, ABC = A ∩ B ∩ C, then S is a

Γ-semigroup.

EXAMPLE: 1.1.8 : Let S = {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} and

Γ = {{a, b, c}}. If for all A, C ∈ S and B ∈ Γ, ABC = A ∩ B ∩ C, then S is a Γ-semigroup.

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39

EXAMPLE: 1.1.9 : Let S be the set of all 2 × 3 matrices over Q, the set of rational

numbers and Γ be the set of all 3 × 2 matrices over Q. Define AαB = usual matrix product

of A, α, B; for all A, B ∈ S and for all α ∈ Γ. Then S is a Γ-semigroup. Note that S is not

a semigroup.

EXAMPLE: 1.1.10 : Let S = { - i , 0, i } and Γ = S. Then S is a Γ-semigroup under the

multiplication of complex numbers, while S is not a semigroup under multiplication of

complex numbers.

EXAMPLE: 1.1.11 : Let S be a Γ-semigroup and α a fixed element in Γ. We define

a.b = aαb for all a,b ∈ S. We can show that (S,.) is a semigroup and we denote this

semigroup by Sα .

EXAMPLE: 1.1.12 : Let S be a semigroup and Γ be a nonempty set. Define a mapping

from S Γ S to S as aαb = ab, for all a, b ∈ S and α ∈ Γ. Then S is a Γ-semigroup.

Verification: Let a, b, c ∈ S and α, 훽 ∈ Γ.

Then ( aαb)βc = (ab)βc = (ab)c = a(bc) = a훼(bc) = a훼(b훽c). Therefore S is a Γ-semigroup.

Note 1.1.13: Every semigroup can be considered to be a Γ-semigroup. Thus the class of all

Γ-semigroups includes the class of all semigroups.

EXAMPLE: 1.1.14 (FREE 횪-SEMIGROUP) : Let X and Γ be two nonempty sets. A

sequence of elements a1α1a2α2 …… an-1αn-1an where a1,a2,a3, ……..,an ∈ X and

α1,α2,α3,……….αn ∈ Γ is called a word over the alphabet X relative to Γ. The set S of all

words with the operation defined from S × Γ × S to S as (a1α1a2α2 .... an-1αn-1an)

γ(b1β1b2β2 …….. bm-1βm-1bm) = a1α1a2α2 ……..an-1αn-1an γb1β1b2β2 ……… bm-1βm-1bm is a

Γ-semigroup. This Γ-semigroup is called free Γ-semigroup over the alphabet

X relative to Γ.

In the following we introduce the notion of a commutative Γ-semigroup.

DEFINITION 1.1.15: A Γ-semigroup S is said to be commutative provided aγb = bγa

for all a,b ∈ S and γ ∈ Γ.

NOTE 1.1.16 : If S is a commutative Γ-semigroup then a Γb = b Γa for all a, b ∈ S.

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40

NOTE 1.1.17 : Let S be a Γ-semigroup and a, b ∈ S and α ∈ Γ. Then aαaαb is denoted by

(aα)2b and consequently a α a α a α…..(n terms)b is denoted by (aα)nb.

In the following we introduce a quasi commutative Γ-semigroup.

DEFINITION 1.1.18 : A Γ-semigroup S is said to be quasi commutative provided for

each a,b ∈ S, there exists a natural number n such that ( ) na b b a .

NOTE 1.1.19 : If a Γ-semigroup S is quasi commutative then for each a,b ∈ S , there

exists a natural number n such that, aΓb = (b Γ)na.

THEOREM 1.1.20: If S is a commutative Γ-semigroup then S is a quasi commutative

Γ-semigroup.

Proof: Suppose that S is a commutative Γ-semigroup. Let ., Sba Now )( 1 abbaabba . Therefore S is a quasi commutative Γ-semigroup.

In the following we introduce the notion of a normal Γ-semigroup.

DEFINITION 1.1.21 : A Γ-semigroup S is said to be normal provided

S S and a a a S .

NOTE 1.1.22 : If a Γ-semigroup S is normal then aΓS = SΓa for all a ∈ S.

THEOREM 1.1.23 : If S is a quasi commutative Γ-semigroup then S is a normal

Γ-semigroup.

Proof: Suppose that S is a quasi commutative Γ- semigroup.

n

n 1

Let S and .Let S. Then where S.Since S is quasi commutative, ( ) for some N

( ) ( )(1)

n

ax a x a b b

a b b a nx aαb bα a b b a S a

a S S a

n

n 1

Let S . Then tα for some S.

Since S is quasi commutative t ( ) for some NNow ( ) ( ) .

(2)

n

x a x a ta a t n

x tαa a t a a t a SS a a S

From (1) & (2), S S ,a a a S and hence S is a normal Γ-semigroup.

COROLLARY 1.1.24 : Every commutative 횪-semigroup is a normal 횪-semigroup.

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Proof : Let S be a commutative Γ-semigroup. By theorem 1.1.20, S is a quasi

commutative Γ-semigroup. By theorem 1.1.23, S is a normal Γ-semigroup. Therefore

every commutative Γ-semigroup is a normal Γ-semigroup.

In the following we are introducing left pseudo commutative Γ-semigroup and

right pseudo commutative Γ-semigroup.

DEFINITION 1.1.23 : A Γ-semigroup S is said to be left pseudo commutative provided

aΓbΓc = bΓaΓc for all a,b,c ∈ S.

DEFINITION 1.1.24 : A Γ-semigroup S is said to be right pseudo commutative provided

aΓbΓc = aΓcΓb for all a,b,c ∈ S.

THEOREM 1.1.25 : If S is a commutative Γ-semigroup, then S is both a left pseudo

commutative 횪-semigroup and a right pseudo commutative Γ-semigroup.

Proof : Suppose that S is commutative Γ-semigroup.

Then aΓbΓc = (aΓb)Γc = (bΓa)Γc = bΓaΓc.

Therefore S is a left pseudo commutative Γ-semigroup.

Again aΓbΓc = aΓ(bΓc) = aΓ(cΓb) = aΓcΓb.

Therefore S is a right pseudo commutative Γ-semigroup.

NOTE 1.1.26 : The converse of the above theorem is not true. i.e., if S is a left and right

pseudo commutative Γ-semigroup then S need not be a commutative Γ-semigroup.

EXAMPLE 1.1.27: Let S = {a , b , c } and Γ = { x, y, z }. Define a binary operation ‘.’ in

S as shown in the following table:

. a b c

a a a a

b a a a

c a b c

Define a mapping S × Γ × S → S by aαb = a.b for all a, b ∈ S and α ∈ Γ. It is easy to

see that S is a Γ-semigroup. Now S is a left and right pseudo commutative Γ-semigroup.

But S is not a commutative Γ-semigroup.

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In the following we are introducing left identity, right identity and identity of a

Γ-semigroup.

DEFINITION 1.1.28 : An element a of a Γ-semigroup S is said to be a left identity of S

provided aαs = s for all s ∈ S and α ∈ Γ.

DEFINITION 1.1.29 : An element ‘a’ of a Γ-semigroup S is said to be a right identity of

S provided sαa = s for all s ∈ S and α ∈ Γ.

DEFINITION 1.1.30 : An element ‘a’ of a Γ-semigroup S is said to be a two sided

identity or an identity provided it is both a left identity and a right identity of S.

THEOREM 1.1.31 : If a is a left identity and b is a right identity of a Γ-semigroup S,

then a = b.

Proof: Since a is a left identity of S, a s s for all s ∈ S and Γ and hence a b = b

for all Γ. Since b is a right identity of S, s b s for all s ∈ S and Γ and hence

a b = a for all Γ. Now a = a b = b.

THEOREM 1.1.32 : Any Γ-semigroup S has at most one identity.

Proof : Let a, b be two identity elements of the Γ-semigroup S. Now a can be considered

as a left identity and b can be considered as a right identity of S. By theorem 1.1.31, a = b.

Then S has atmost one identity.

NOTE 1.1.33 : The identity (if exists) of a Γ-semigroup is usually denoted by 1.

DEFINITION 1.1.34 : A Γ-semigroup S with identity is called a Γ-monoid.

In the following we are introducing left zero, right zero and zero of a Γ-semigroup.

DEFINITION 1.1.35 : An element a of a Γ-semigroup S is said to be a left zero of S

provided aαs = a for all s ∈ S and α ∈ Γ.

DEFINITION 1.1.36 : An element a of a Γ-semigroup S is said to be a right zero of S

provided sαa = a for all s ∈ S and α ∈ Γ.

DEFINITION 1.1.37 : An element a of a Γ-semigroup S is said to be a two sided zero or

zero provided it is both a left zero and a right zero of S.

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We are now introducing left zero Γ-semigroup, right zero Γ-semigroup and zero

Γ-semigroup.

DEFINITION 1.1.38 : A Γ-semigroup in which every element is a left zero is called a

left zero Γ-semigroup.

DEFINITION 1.1.39 : A Γ-semigroup in which every element is a right zero is called a

right zero Γ-semigroup.

DEFINITION 1.1.40 : A Γ-semigroup with 0 in which the product of any two elements

equals to 0 is called a zero Γ-semigroup or a null Γ-semigroup.

THEOREM 1.1.41 : If a is a left zero and b is a right zero of a Γ-semigroup S,

then a = b.

Proof: Since a is a left zero of S, a s a for all s ∈ S, and hence a b =a for all

α ∈ Γ. Since b is a right zero of S, s b b for all s ∈ S and α ∈ Γ and hence a b = b for

all α ∈ Γ. Now a = a훼b = b.

THEOREM 1.1.42 : Any Γ-semigroup S has at most one zero element.

Proof : Let a, b be two zeros of the Γ-semigroup S. Now a can be considered as a left

zero and b can be considered as a right zero. By theorem 1.1.41, a = b. Thus S has at

most one zero.

NOTE 1.1.43 : The zero (if exists) of a Γ-semigroup is usually denoted by 0.

NOTATION 1.1.44 : Let S be a Γ-semigroup. If S has an identity, let S1 = S and if S does

not have an identity, let S1 be the Γ-semigroup S with an identity adjoined usually denoted

by the symbol 1. Similarly if S has a zero, let S0 = S and if S does not have a zero, let S0 be

the Γ-semigroup S with zero adjoined usually denoted by the symbol 0.

1.2. 횪-SUBSEMIGROUP

The term subsemigroup plays a special role in the theory of semigroups. In this

section, the terms; Γ-subsemigroup, Γ-subsemigroup generated by a subset, cyclic

Γ-subsemigroup of a Γ-semigroup and cyclic Γ-semigroup are introduced. It is proved

that (1) the nonempty intersection of two 횪-subsemigroups of a 횪-semigroup S is a

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44

횪-subsemigroup of S, (2) the nonempty intersection of any family of 횪-subsemigroups of a

횪-semigroup S is a 횪-subsemigroup of S. It is also proved that if A is a nonempty subset

of a Γ-semigroup S, then the Γ-subsemigroup of S generated by A is the intersection of all

Γ-subsemigroups of S containing A.

DEFINITION 1.2.1 : Let S be a Γ-semigroup. A nonempty subset T of S is said to be a

Γ-subsemigroup of S if aγb ∈ T, for all a, b ∈ T and γ ∈ Γ.

NOTE 1.2.2 : A nonempty subset T of a Γ-semigroup S is a Γ-subsemigroup of S iff

TΓT ⊆ T.

EXAMPLE 1.2.3 : Let S = [ 0,1] and Γ = { 1/n : n is a positive integer}. Then S is a

Γ-semigroup under the usual multiplication. Let T = [0, 1/2]. Now T is a nonempty subset

of S and aγb ∈ T, for all a, b ∈ T and γ ∈ Γ . Then T is a 횪-subsemigroup of S.

THEOREM 1.2.4 : The nonempty intersection of two 횪-subsemigroups of a

횪-semigroup S is a 횪-subsemigroup of S.

Proof : Let T1, T2 be two Γ-subsemigroups of S. Let a, b ∈ T1∩T2 and γ ∈ Γ.

a, b ∈ T1∩T2 ⇒ a, b ∈ T1 and a, b ∈ T2

a, b ∈ T1, γ ∈ Γ, T1 is a Γ-subsemigroup of S ⇒ a훾b ∈ T1.

a, b ∈ T2, γ ∈ Γ, T2 is a Γ-subsemigroup of S ⇒ a훾b ∈ T2.

a훾b ∈ T1, a훾b ∈ T2⇒ a훾b ∈ T1∩T2. Therefore T1∩T2 is a Γ-subsemigroup of S.

THEOREM 1.2.5 : The nonempty intersection of any family of 횪-subsemigroups of a

횪-semigroup S is a 횪-subsemigroup of S.

Proof: Let T be a family of Γ-subsemigroups of S and let T = T

.

Let a, b ∈ T and γ ∈ Γ.

a, b ∈ T ⇒ a, b ∈ T ⇒ a, b ∈ T for all α ∈ Δ.

a, b ∈ T , γ ∈ Γ, T is a Γ-subsemigroup of S ⇒ aγb ∈ T .

aγb ∈ T for all α ∈ Δ ⇒ aγb ∈ T ⇒ aγb ∈ T.

Therefore T is a Γ-subsemigroup of S.

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45

In the following we are introducing a Γ-subsemigroup which is generated by a

subset and a cyclic Γ-subsemigroup of Γsemigroup.

DEFINITION 1.2.6 : Let S be a Γ-semigroup and A be a nonempty subset of S. The

smallest Γ-subsemigroup of S containing A is called a Γ-subsemigroup of S generated by

A. It is denoted by < A >.

THEOREM 1.2.7 : Let S be a 횪-semigroup and A be a nonempty subset of S. Then

< A > = { a1 훂1 a2 훂2…….an-1 훂n-1 an : n ∈ N, a1, a2, ….., an ∈ A, 훂1, 훂2, …., 훂n-1 ∈ 횪 }.

Proof : Let T = { a1 α1 a2 α2…an-1 αn-1 an : n ∈ N, a1, a2, …an ∈ A, α1, α2, …, αn-1 ∈ Γ}.

Let a, b ∈ T and γ ∈ Γ.

a ∈ T ⇒ a = a1 α1 a2 α2……am-1 αm-1 am where a1, a2, …am ∈ A, α1, α2, …, αm-1 ∈ Γ.

b ∈ T ⇒ b = b1 β1 b2 β2……bn-1 βn-1 bn where b1, b2, …bn ∈ A, β 1, β 2, …, β n-1 ∈ Γ.

Now a훾b = (a1 α1 a2 α2……am-1 αm-1 am) 훾 (b1 β1 b2 β2……bn-1 βn-1 bn) ∈ T.

Therefore T is a Γ-subsemigroup of S.

Let K be a Γ-subsemigroup of S such that A ⊆ K.

Let a ∈ T. Then a = a1 α1 a2 α2……an-1 αn-1 an where a1, a2, …an ∈ A, α1, α2, …, αn-1 ∈ Γ

a1, a2, ….an ∈ A, A ⊆ K ⇒ a1, a2, ….an ∈ K.

a1, a2, ….an ∈ K, α1, α2, …, αn-1 ∈ Γ, K is a Γ-subsemigroup

⇒ a1 α1 a2 α2……an-1 αn-1 an ∈ K ⇒ a ∈ K. Therefore T ⊆ K.

So T is the smallest Γ-subsemigroup of S containing A. Hence < A > = T.

THEOREM 1.2.8 : Let S be a 횪-semigroup and A be a nonempty subset of S. Then

< A > = The intersection of all 횪-subsemigroups of S containing A.

Proof : Let Δ be the set of all Γ-subsemigroups of S containing A.

Since S is a Γ-subsemigroup of S containing A, S ∈ Δ. So Δ ≠ ∅.

Let T

T T

. Since A ⊆ T for all T ∈ Δ, A T .

By theorem 1.2.4, T is a Γ-subsemigroup of S.

Since T ⊆ T for all T ∈ Δ, T is the smallest Γ-subsemigroup of S containing A.

Therefore T A .

DEFINITION 1.2.9 : Let S be a Γ-semigroup. A Γ-subsemigroup T of S is said to be

cyclic Γ-subsemigroup of S if T is generated by a single element subset of S.

Γ-semigroups

46

NOTE 1.2.10 : Let T be a Γ-subsemigroup of Γ-semigroup S. Then T is cyclic iff

T = 1n

n Na a

for some a ∈ S.

DEFINITION 1.2.11 : A Γ-semigroup S is said to be a cyclic Γ-semigroup if S is a cyclic

Γ-subsemigroup of S itself.

1.3. Γ-IDEALS

Ideals has greater importance in the theory of semigroups. In this section, the

terms left Γ-ideal, right Γ-ideal, Γ-ideal, proper Γ-ideal, trivial Γ-ideal, maximal left

Γ-ideal, maximal right Γ-ideal, maximal Γ-ideal, left Γ-ideal generated by a subset, right

Γ-ideal generated by a subset, Γ-ideal generated by a subset, principal left Γ-ideal,

principal right Γ-ideal, principal Γ-ideal of a Γ-semigroup are introduced. Also left duo

Γ-semigroup, right duo Γ-semigroup, duo Γ-semigroup, left simple Γ-semigroup, right

simple Γ-semigroup, simple Γ-semigroup are introduced. It is proved that (1) the

nonempty intersection of any two left Γ-ideals of a Γ-semigroup S is a left Γ-ideal of S,

(2) the nonempty intersection of any family of left 횪-ideals of a 횪-semigroup S is a left

횪-ideal of S, (3) the union of any two left 횪-ideals of a 횪-semigroup S is a left 횪-ideal of S

and (4) the union of any family of left 횪-ideals of a 횪-semigroup S is a left 횪-ideal of S. It

is also proved that (1) the nonempty intersection of any two right Γ-ideals of a

Γ-semigroup S is a right Γ-ideal of S, (2) the nonempty intersection of any family of right

횪-ideals of a 횪-semigroup S is a right 횪-ideal of S, (3) the union of any two right 횪-ideals

of a 횪-semigroup S is a right 횪-ideal of S and (4) the union of any family of right 횪-ideals

of a 횪-semigroup S is a right 횪-ideal of S. Further it is proved that (1) the nonempty

intersection of any two Γ-ideals of a Γ-semigroup S is a Γ-ideal of S, (2) the nonempty

intersection of any family of 횪-ideals of a 횪-semigroup S is a 횪-ideal of S, (3) the union of

any two 횪-ideals of a 횪-semigroup S is a 횪-ideal of S and (4) the union of any family of

횪-ideals of a 횪-semigroup S is a 횪-ideal of S. It is proved that if S is a Γ-semigroup and a

∈ S then (i) L(a) = a∪S횪a, (ii) R(a) = a∪a횪S, (iii) J(a) = a ∪ aΓS ∪ SΓa ∪ SΓaΓS. It is

proved that a Γ-semigroup S is a duo Γ-semigroup if and only if x횪S1 = S1횪x for all

x ∈ S. Further it is also proved that every normal Γ-semigroup is a duo Γ-semigroup. It is

proved that (1) a 횪-semigroup S is a left simple 횪-semigroup if and only if S횪a = S for all

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47

a ∈ S, (2) a 횪-semigroup S is right a simple 횪-semigroup if and only if a횪S = S for all

a ∈ S, (3) a 횪-semigroup S is a simple 횪-semigroup if and only if SΓaΓS = S for all a ∈ S.

We now introduce the term of a left Γ-ideal in a Γ-semigroup.

DEFINITION 1.3.1 : A nonempty subset A of a Γ-semigroup S is said to be a left Γ-ideal

of S if , ,s S a A implies s a A .

NOTE 1.3.2 : A nonempty subset A of a Γ-semigroup S is a left Γ- ideal of S iff SΓA⊆A.

THEOREM 1.3.3 : The nonempty intersection of any two left Γ-ideals of a

Γ-semigroup S is a left Γ-ideal of S.

Proof : Let A, B be two left Γ-ideals of S. Let a ∈ A∩B and s ∈ S, γ ∈ Γ.

a ∈ A∩B ⇒ a ∈ A and a ∈ B.

a ∈ A, s ∈ S, γ ∈ Γ, A is a left Γ-ideal of S ⇒ s훾a ∈ A.

a ∈ B, s ∈ S, γ ∈ Γ, B is a left Γ-ideal of S ⇒ s훾a ∈ B.

s훾a ∈ A, s훾a ∈ B ⇒ s훾a ∈ A∩B. Therefore A∩B is a left Γ-ideal of S.

THEOREM 1.3.4 : The nonempty intersection of any family of left 횪-ideals of a

횪-semigroup S is a left 횪-ideal of S.

Proof: Let { }A be a family of left Γ-ideals of S and let A A

.

Let a ∈ A, s ∈ S, γ ∈ Γ.

a ∈ A ⇒ a ∈ A ⇒ a A for each α ∈ Δ.

a A , s ∈ S, γ ∈ Γ, A is a left Γ-ideal of S ⇒ s훾a ∈ A .

s훾a ∈ A for all α ∈ Δ ⇒ s훾a ∈ A ⇒ s훾a ∈ A. Therefore A is a left Γ-ideal of S.

THEOREM 1.3.5 : The union of any two left 횪-ideals of a 횪-semigroup S is a left

횪-ideal of S.

Proof : Let A1, A2 be two left Γ-ideals of a Γ-semigroup S. Let A = A1∪A2.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S and γ ∈ Γ.

a ∈ A ⇒ a ∈ A1∪A2 ⇒ a ∈ A1 or a ∈ A2.

If a ∈ A1 then a ∈ A1, s ∈ S, γ ∈ Γ, A1 is a left Γ-ideal of S ⇒ s훾a ∈ A1 ⊆ A1∪A2 = A

⇒ s훾a ∈ A.

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48

If a ∈ A2 then a ∈ A2, s ∈ S, γ ∈ Γ, A2 is a left Γ-ideal of S ⇒ s훾a ∈ A2 ⊆ A1∪A2 = A

⇒ s훾a ∈ A.

Therefore a ∈ A, s ∈ S, γ ∈ Γ ⇒ s훾a ∈ A and hence A is a left Γ-ideal of S.

THEOREM 1.3.6 : The union of any family of left 횪-ideals of a 횪-semigroup S is a left

횪-ideal of S.

Proof: Let { }A be a family of left Γ-ideals of a Γ-semigroup S and let A A

.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, 훾 ∈ Γ.

a ∈ A ⇒ a ∈ A ⇒ a ∈ Aα for some 훼 ∈ Δ.

a ∈ Aα, s ∈ S, 훾 ∈ Γ, Aα is a left Γ-ideal of S ⇒ s훾a ∈ Aα ⊆ A = A ⇒ s훾a ∈ A.

Therefore A is a left Γ-ideal of S.

We now introduce the notion of a right Γ-ideal in a Γ-semigroup.

DEFINITION 1.3.7 : A nonempty subset A of a Γ-semigroup S is said to be a right

Γ-ideal of S if , ,s S a A implies a s A .

NOTE 1.3.8 : A nonempty subset A of a Γ-semigroup S is a right Γ- ideal of S iff

AΓS⊆A.

THEOREM 1.3.9 : The nonempty intersection of any two right Γ-ideals of a

Γ-semigroup S is a right Γ-ideal of S.

Proof : Let A, B be two right Γ-ideals of S. Let a ∈ A∩B, s ∈ S and γ ∈ Γ.

a ∈ A∩B ⇒ a ∈ A and a ∈ B.

a ∈ A, s ∈ S, γ ∈ Γ, A is a right Γ-ideal of S ⇒ a훼s ∈ A.

a ∈ B, s ∈ S, γ ∈ Γ, B is a right Γ-ideal of S ⇒ a훼s ∈ B.

a훼s ∈ A, a훼s ∈ B ⇒ a훼s ∈ A∩B and hence A∩B is a right Γ-ideal of S.

THEOREM 1.3.10 : The nonempty intersection of any family of right 횪-ideals of a

횪-semigroup S is a right 횪-ideal of S.

Proof: Let { }A be a family of right Γ-ideals of S and let A A

.

Let a ∈ A, s ∈ S, γ ∈ Γ.

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49

a ∈ A ⇒ a ∈ A ⇒ a A for each α ∈ Δ.

a A , s ∈ S, γ ∈ Γ, A is a right Γ-ideal of S ⇒ a훾s ∈ A .

a훾s ∈ A for all α ∈ Δ ⇒ a훾s ∈ A ⇒ a훾s ∈ A. Therefore A is a right Γ-ideal of S.

THEOREM 1.3.11 : The union of any two right 횪-ideals of a 횪-semigroup S is a right

횪-ideal of S.

Proof : Let A1, A2 be two right Γ-ideals of a Γ-semigroup S. Let A = A1∪A2.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, γ ∈ Γ.

a ∈ A ⇒ a ∈ A1∪A2 ⇒ a ∈ A1 or a ∈ A2.

If a ∈ A1 then a ∈ A1, s ∈ S, γ ∈ Γ, A1 is a right Γ-ideal of S

⇒ a훾s ∈ A1 ⊆ A1∪A2 = A ⇒ a훾s ∈ A.

If a ∈ A2 then a ∈ A2, s ∈ S, γ ∈ Γ, A2 is a right Γ-ideal of S

⇒ a훾s ∈ A2 ⊆ A1∪A2 = A ⇒ a훾s ∈ A.

∴ a ∈ A, s ∈ S, γ ∈ Γ then a훾s ∈ A. Therefore A is a right Γ-ideal of S.

THEOREM 1.3.12 : The union of any family of right 횪-ideals of a 횪-semigroup S is a

right 횪-ideal of S.

Proof: Let { }A be a family of right Γ-ideals of S and let A A

.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, 훾 ∈ Γ.

a ∈ A ⇒ a ∈ A ⇒ a ∈ Aα for some 훼 ∈ Δ.

a ∈ Aα, s ∈ S, 훾 ∈ Γ, Aα is a right Γ-ideal of S ⇒ a훾s ∈ Aα ⊆ A = A ⇒ a훾s ∈ A.

Therefore A is a right Γ-ideal of S.

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

DEFINITION 1.3.13 : A nonempty subset A of a Γ-semigroup S is said to be a two sided

Γ- ideal or simply a Γ- ideal of S if s ∈ S, a ∈ A, 훼 ∈ Γ imply s훼a ∈ A, a훼s ∈ A.

NOTE 1.3.14 : A nonempty subset A of a Γ-semigroup S is a two sided Γ-ideal iff it is

both a left Γ-ideal and a right Γ- ideal of S.

EXAMPLE 1.3.15 : Let N be the set of natural numbers and Γ = 2N. Then N is a

Γ-semigroup and A = 3N is a Γ- ideal of the Γ-semigroup N.

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THEOREM 1.3.16 : The nonempty intersection of any two Γ-ideals of a Γ-semigroup

S is a Γ-ideal of S.

Proof : Let A, B be two Γ-ideals of S. Let a ∈ A∩B and s ∈ S, γ ∈ Γ.

a ∈ A∩B ⇒ a ∈ A and a ∈ B.

a ∈ A, s ∈ S, γ ∈ Γ, A is a Γ-ideal of S ⇒ s훼a, a훼s ∈ A.

a ∈ B, s ∈ S, γ ∈ Γ, B is a Γ-ideal of S ⇒ s훼a, a훼s ∈ B.

∴ s훼a, a훼s ∈ A, s훼a, a훼s ∈ B ⇒ s훼a, a훼s ∈ A∩B.

Therefore A∩B is a Γ-ideal of S.

THEOREM 1.3.17 : The nonempty intersection of any family of 횪-ideals of a

횪-semigroup S is a 횪-ideal of S.

Proof: Let { }A be a family of Γ-ideals of S and let A A

.

Let a ∈ A, s ∈ S, γ ∈ Γ.

a ∈ A ⇒ a ∈ A ⇒ a A for each α ∈ Δ.

a A , s ∈ S, γ ∈ Γ, A is a Γ-ideal of S ⇒ s훾a, a훾s ∈ A .

s훾a, a훾s ∈ A for all α ∈ Δ ⇒ s훾a, a훾s ∈ A ⇒ s훾a, a훾s ∈ A.

Therefore A is a Γ-ideal of S.

THEOREM 1.3.18 : The union of any two 횪-ideals of a 횪-semigroup S is a 횪-ideal of

S.

Proof : Let A1, A2 be two Γ-ideals of a Γ-semigroup S. Let A = A1∪A2.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, γ ∈ Γ.

a ∈ A ⇒ a ∈ A1∪A2 ⇒ a ∈ A1 or a ∈ A2.

If a ∈ A1 then a ∈ A1, s ∈ S, γ ∈ Γ, A1 is a Γ-ideal of S ⇒ s훾a, a훾s ∈ A1 ⊆ A1∪A2 = A

⇒ s훾a, a훾s ∈ A.

If a ∈ A2 then a ∈ A2, s ∈ S, γ ∈ Γ, A2 is a Γ-ideal of S ⇒ s훾a, a훾s ∈ A2 ⊆ A1∪A2 = A

⇒ s훾a, a훾s ∈ A.

Thus a ∈ A, s ∈ S, γ ∈ Γ ⇒ s훾a, a훾s ∈ A. Therefore A is a Γ-ideal of S.

THEOREM 1.3.19 : The union of any family of 횪-ideals of a 횪-semigroup S is a

횪-ideal of S.

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51

Proof: Let { }A be a family of Γ-ideals of S and let A A

.

Clearly A is a nonempty subset of S. Let a ∈ A, s ∈ S, 훾 ∈ Γ.

a ∈ A ⇒ a ∈ A ⇒ a ∈ Aα for some 훼 ∈ Δ.

Now a ∈ Aα, s ∈ S, 훾 ∈ Γ, Aα is a Γ-ideal of S ⇒ s훾a, a훾s ∈ Aα ⊆ A = A

Thus a ∈ A, s ∈ S, 훾 ∈ Γ ⇒ s훾a, a훾s ∈ A. Therefore A is a Γ-ideal of S.

We now introduce a proper Γ-ideal, trivial Γ-ideal, maximal left Γ-ideal, maximal

right Γ-ideal, maximal Γ-ideal and globally idempotent Γ-ideal of a Γ-semigroup.

DEFINITION 1.3.20 : A Γ-ideal A of a Γ-semigroup S is said to be a proper Γ-ideal of S

if A is different from S.

DEFINITION 1.3.21 : A Γ-ideal A of a Γ-semigroup S is said to be a trivial Γ-ideal

provided S\A is singleton.

DEFINITION 1.3.22 : A Γ-ideal A of a Γ-semigroup S is said to be a maximal left

횪-ideal provided A is a proper left Γ-ideal of S and is not properly contained in any proper

left Γ-ideal of S.

DEFINITION 1.3.23 : A Γ-ideal A of a Γ-semigroup S is said to be a maximal right

횪-ideal provided A is a proper right Γ-ideal of S and is not properly contained in any

proper right Γ-ideal of S.

DEFINITION 1.3.24 : A Γ-ideal A of a Γ-semigroup S is said to be a maximal 횪-ideal

provided A is a proper Γ-ideal of S and is not properly contained in any proper

Γ-ideal of S.

DEFINITION 1.3. 25 : A Γ-ideal A of a Γ-semigroup S is said to be globally idempotent

if AΓA = A.

THEOREM 1.3.26 : If A is a 횪-ideal of a 횪-semigroup S with unity 1 and 1 ∈ A then

A = S.

Proof : Clearly A ⊆ S. Let s ∈ S.

1 ∈ A, s ∈ S, A is a Γ-ideal of S ⇒ 1Γs ⊆ A ⇒ s ∈ A. Thus S ⊆ A.

A ⊆ S, S ⊆ A ⇒ S = A.

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52

THEOREM 1.3.27 : If S is a 횪-semigroup with unity 1 then the union of all proper

횪-ideals of S is the unique maximal 횪-ideal of S.

Proof : Let M be the union of all proper Γ-ideals of S. Since 1 is not an element of any

proper Γ-ideal of S, 1 ∉ M. Therefore M is a proper subset of S. By theorem 1.3.19, M is

a Γ-ideal of S. Thus M is a proper Γ-ideal of S. Since M contains all proper Γ-ideals of S,

M is a maximal Γ-ideal of S. If M1 is any maximal Γ-ideal of S, then M1 ⊆ M ⊂ S and

hence M1 = M. Therefore M is the unique maximal Γ-ideal of S.

We now introducing left Γ-ideal generated by a subset, right Γ-ideal generated by a

subset, Γ-ideal generated by a subset of a Γ-semigroup.

DEFINITION 1.3.28 : Let S be a Γ-semigroup and A be a nonempty subset of S. The

smallest left Γ-ideal of S containing A is called left Γ-ideal of S generated by A.

THEOREM 1.3.29 : The left Γ-ideal of a Γ-semigroup S generated by a nonempty

subset A is the intersection of all left Γ-ideals of S containing A.

Proof : Let Δ be the set of all left Γ-ideals of S containing A.

Since S itself is a left Γ-ideal of S containing A, S ∈ Δ. So Δ ≠ ∅.

Let T

T T

. Since A ⊆ T for all T ∈ Δ, A T .

By theorem 1.3.4, T* is a left Γ-ideal of S.

Let K is a left Γ-ideal of S containing A.

Clearly A ⊆ K and K is a left Γ-ideal of S.

Therefore K ∈ Δ ⇒T* ⊆ K.

Therefore T* is the left Γ-ideal of S generated by A.

DEFINITION 1.3.30 : Let S be a Γ-semigroup and A be a nonempty subset of S. The

smallest right Γ-ideal of S containing A is called right Γ-ideal of S generated by A.

THEOREM 1.3.31 : The right Γ-ideal of a Γ-semigroup S generated by a nonempty

subset A is the intersection of all right Γ-ideals of S containing A.

Proof : Let Δ be the set of all right Γ-ideals of S containing A.

Since S itself is a right Γ-ideal of S containing A, S ∈ Δ. So Δ ≠ ∅.

Let T

T T

. Since A ⊆ T for all T ∈ Δ, A T .

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53

By theorem 1.3.10, T* is a right Γ-ideal of S.

Let K is a right Γ-ideal of S containing A.

Clearly A ⊆ K and K is a right Γ-ideal of S.

Therefore K ∈ Δ ⇒T* ⊆ K.

Therefore T* is the right Γ-ideal of S generated by A.

DEFINITION 1.3.32 : Let S be a Γ-semigroup and A be a nonempty subset of S. The

smallest Γ-ideal of S containing A is called Γ-ideal of S generated by A.

THEOREM 1.3.33 : The Γ-ideal of a Γ-semigroup S generated by a nonempty subset

A is the intersection of all Γ-ideals of S containing A.

Proof : Let Δ be the set of all Γ-ideals of S containing A.

Since S itself is a Γ-ideal of S containing A, S ∈ Δ. So Δ ≠ ∅.

Let T

T T

. Since A ⊆ T for all T ∈ Δ, A T .

By theorem 1.3.17, T* is a Γ-ideal of S.

Let K is a Γ-ideal of S containing A.

Clearly A ⊆ K and K is a Γ-ideal of S.

Therefore K ∈ Δ ⇒T* ⊆ K.

Therefore T* is the Γ-ideal of S generated by A.

We now introduce a principal left Γ-ideal of a Γ-semigroup and characterize

principal left Γ-ideal.

DEFINITION 1.3.34 : A left Γ-ideal A of a Γ-semigroup S is said to be the principal left

Γ-ideal generated by a if A is a left Γ-ideal generated by {a} for some a ∈ S. It is denoted

by L(a).

THEOREM 1.3.35 : If S is a 횪-semigroup and a ∈ S then L(a) = a ∪ S횪a.

Proof: Let s ∈ S, r ∈ a ∪ SΓa and γ ∈ Γ.

r ∈ a ∪ SΓa ⇒ r = a or r = t훼a for some t ∈ S, α ∈ Γ.

If r = a then s훾r = s훾a ∈ SΓa ⊆ a ∪ SΓa.

If r = t훼a then s훾r = s훾(t훼a) = (s훾t)훼a ∈ SΓa ⊆ a ∪ SΓa.

Therefore s훾a ∈ a ∪ SΓa and hence a ∪ SΓa is a left Γ-ideal of S.

Let L be a left Γ-ideal of S containing a.

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54

Let r ∈ a ∪ SΓa. Then r = a or r = t훼a for some t ∈ S, α ∈ Γ.

If r = a then r = a ∈ L. If r = t훼a then r = t훼a ∈ L.

Therefore a∪SΓa ⊆ L and hence a∪SΓa is the smallest left Γ-ideal containing a.

Therefore L(a) = a∪SΓa.

NOTE 1.3.36 : If S is a Γ-semigroup and a ∈ S then L (a) = S1Γa.

We now introduce principal right Γ-ideal of a Γ-semigroup and characterize

principal right Γ-ideal.

DEFINITION 1.3.37 : A right Γ-ideal A of a Γ-semigroup S is said to be the principal

right Γ-ideal generated by a if A is a right Γ-ideal generated by {a} for some a ∈ S. It is

denoted R(a).

THEOREM 1.3.38 : If S is a 횪-semigroup and a ∈ S then R(a) = a ∪ a횪S.

Proof: Let s ∈ S, r ∈ a ∪ aΓS.

Now r ∈ a ∪ aΓS ⇒ r = a or r = a훼t for some t ∈ S, α ∈ Γ.

If r = a then r훾s = a훾s ∈ aΓS ⊆ a ∪ aΓS.

If r = a훼t then r훾s = (a훼t)훾s = a훼(t훾s) ∈ aΓS⊆ a ∪ aΓS.

Therefore r훾s ∈ a ∪ aΓS and hence a ∪ aΓS is a right Γ-ideal of S.

Let R be a right Γ-ideal of S containing a.

Let r ∈ a ∪ aΓS. Then r = a or r = a훼t for some t ∈ S, α ∈ Γ.

If r = a then r = a ∈ R. If r = a훼t then r = a훼t ∈ R.

Therefore a ∪ aΓS ⊆ R and hence a ∪ SΓa is the smallest right Γ-ideal containing a.

Therefore R(a) = a∪aΓS.

NOTE 1.3.39 : If S is a Γ-semigroup and a ∈ S then R (a) = aΓS1.

We now introduce a principal Γ-ideal of a Γ-semigroup and characterize principal

Γ-ideal.

DEFINITION 1.3.40 : A Γ- ideal A of a Γ-semigroup S is said to be a principal Γ- ideal

provided A is a Γ- ideal generated by {a} for some a ∈ S. It is denoted by J[a] or <a>.

THEOREM 1.3.41 : If S is a 횪-semigroup and a ∈ S then

J(a) = a ∪ aΓS ∪ SΓa ∪ SΓaΓS.

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Proof: Let s ∈ S, r ∈ a ∪ aΓS ∪ SΓa ∪ SΓaΓS and γ ∈ Γ.

r ∈ a ∪ aΓS ∪ SΓa ∪ SΓaΓS ⇒ r = a or r = aαt or r = tαa or r = tαa훽u for some

t, u ∈ S and α, 훽 ∈ Γ.

If r = a then rγs = aγs ∈ aΓS and sγr = sγa ∈ SΓa.

If r = aαt then rγs = (aαt)γs = aα(tγs) ∈ aΓS and sγr = sγ(aαt) = sγaαt ∈ SΓaΓS.

If r = tαa then rγs = (tαa)γs = tαaγs ∈ SΓaΓS or sγr = sγ(tαa) = (sγt)αa ∈ SΓa.

If r = tαa훽u then rγs = (tαa훽u)γs = tαa훽(uγs) ∈ SΓaΓS

and sγr = sγ(tαa훽u) = (sγt)αa훽u ∈ SΓaΓS.

But aΓS, SΓa, SΓaΓS are all subsets of aΓS ∪ SΓa ∪ SΓaΓS.

Therefore rγs, sγr ∈ a ∪ aΓS ∪ SΓa ∪ SΓaΓS and hence a ∪ aΓS ∪ SΓa ∪ SΓaΓS is a

Γ-ideal of S.

Let J be a Γ-ideal of S containing a. Let r ∈ a ∪ aΓS ∪ SΓa ∪ SΓaΓS.

Then r = a or r = aαt or r = tαa or r = tαa훽u for some t, u ∈ S and α, 훽 ∈ Γ.

If r = a then r = a ∈ J. If r = aαt then r = aαt ∈ J.

If r = tαa then r = tαa ∈ J. If r = tαa훽u then r = tαa훽u ∈ J.

Therefore a ∪ aΓS ∪ SΓa ∪ SΓaΓS ⊆ J.

Hence a ∪ aΓS ∪ SΓa ∪ SΓaΓS is the smallest Γ-ideal of S containing a.

Therefore J(a) = a ∪ aΓS ∪ SΓa ∪ SΓaΓS.

NOTE 1.3.42 : If S is a Γ-semigroup and a ∈ S, then <a> = a ∪ aΓS ∪ SΓa ∪ SΓaΓS

= S1Γa ΓS1.

THEOREM 1.3.43 : In any 횪-semigroup S, the following are equivalent.

(1) Principal 횪-ideals of S form a chain.

(2) 횪-ideals of S form a chain.

Proof : (1) ⇒ (2) : Suppose that principal Γ-ideals of S form a chain.

Let A, B be two Γ-ideals of S. Suppose if possible A ⊈ B, B ⊈ A.

Then there exists a ∈ A\B and b ∈ B\A.

a ∈ A ⇒ < a > ⊆ A and b ∈ B ⇒ < b > ⊆ B.

Since principal Γ-ideals form a chain, either < a > ⊆ < b > or < b > ⊆ < a >.

If < a > ⊆ < b >, then a ∈ < b > ⊆ B. It is a contradiction.

If < b > ⊆ < a >, then b ∈ < a > ⊆ A. It is also a contradiction.

Therefore either A ⊆ B or B ⊆ A and hence Γ-ideals from a chain.

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56

(2) ⇒ (1) : Suppose that Γ-ideals of S form a chain.

Then clearly principal Γ-ideal of S form a chain.

Duo semigroups played an important role in the theory of semigroups. We now

introduce a left duo Γ-semigroup, a right duo Γ-semigroup and a duo Γ-semigroup.

DEFINITION 1.3.44 : A Γ- semigroup S is said to be a left duo Γ- semigroup provided

every left Γ- ideal of S is a two sided Γ- ideal of S.

DEFINITION 1.3.45 : A Γ- semigroup S is said to be a right duo Γ- semigroup provided

every right Γ-ideal of S is a two sided Γ- ideal of S.

DEFINITION 1.3.46 : A Γ- semigroup S is said to be a duo Γ- semigroup provided it is

both a left duo Γ- semigroup and a right duo Γ- semigroup.

We now characterize duo Γ-semigroups.

THEOREM 1.3.47 : A Γ-semigroup S is a duo Γ- semigroup if and only if

x횪S1 = S1횪x for all x ∈ S .

Proof: Suppose that S is a duo Γ-Semigroup and x ∈ S.

Let t ∈ xΓS1. Then t=xγs for some s ∈ S1, γ ∈ Γ.

Since S1Γx is a left Γ-ideal of S, S1Γx is a Γ-ideal of S.

So 1 1 1 1, , , is a -ideal Sx S x s S S x x s x t S x .

Therefore xΓS1⊆ S1Γx. Similarly we can prove that S1Γx ⊆ xΓS1 .Therefore S1Γx = xΓS1.

Conversely suppose that S1Γx = xΓS1 for all x ∈ S. Let A be a left Γ-ideal of S.

Let x ∈ A, s ∈ S and α ∈ Γ. Then xαs∈ xΓS1 = S1Γx ⇒ xαs = tβx for some t ∈S1, β ∈Γ.

, , , A is a left -ideal of S A Ax A t S t x x s .

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

Therefore S is left duo Γ-semigroup.

Similarly we can prove that S is a right duo Γ-semigroup. Hence S is duo Γ-semigroup.

THEOREM 1.3.48 : Every normal Γ- semigroup is a duo Γ- semigroup.

Proof: Suppose that S is normal Γ-semigroup.

Then aΓS=SΓa for all a ∈ S . allfor Sa 11 SaaS

By theorem 1.3.47, S is a duo Γ-semigroup.

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We now introduce a left simple Γ-semigroup and characterize left simple

Γ-semigroups.

DEFINITION 1.3.49 : A Γ-semigroup S is said to be a left simple Γ-semigroup if S is its

only left Γ-ideal.

THEOREM 1.3.50 : A 횪-semigroup S is a left simple 횪-semigroup if and only if

S횪a = S for all a ∈ S.

Proof: Suppose that S is a left simple Γ-semigroup and a ∈ S.

Let t ∈ SΓa, s ∈ S, 훾 ∈ Γ.

t ∈ SΓa ⇒ 1t s a where 1s S and α ∈ Γ.

Now s훾t = s훾( 1s a ) = 1( )s s a S a ⇒ SΓa is a left Γ-ideal of S.

Since S is a left simple Γ-semigroup, SΓa = S.

Therefore SΓa = S for all a ∈ S.

Conversely suppose that SΓa = S for all a ∈ S. Let L be a left Γ-ideal of S.

Let l ∈ L. Then l ∈ S. By assumption SΓl = S.

Let s ∈ S. Then s ∈ SΓl ⇒ s = tαl for some t ∈ S, α ∈ Γ.

l ∈ L, t ∈ S, α ∈ Γ and L is a left Γ-ideal ⇒ tαl ∈ L ⇒ s ∈ L.

Therefore S ⊆ L. Clearly L ⊆ S and hence S = L.

Therefore S is the only left Γ-ideal of S. Hence S is left simple Γ-semigroup.

We now introduce a right simple Γ-semigroup and characterize right simple

Γ-semigroups.

DEFINITION 1.3.51 : A Γ-semigroup S is said to be a right simple Γ-semigroup if S is

its only right Γ-ideal.

THEOREM 1.3.52 : A 횪-semigroup S is a right simple 횪-semigroup if and only if

a횪S= S for all a ∈ S.

Proof: Suppose that S is a right simple Γ-semigroup and a ∈ S. Let t ∈ aΓS, s ∈ S, 훾 ∈ Γ.

t ∈ aΓS ⇒ 1t a s where 1s S and α ∈ Γ.

Now t훾s = ( 1a s )훾 s= 1( )a s s a S ⇒ aΓS is a right Γ-ideal of S.

Since S is a right simple Γ-semigroup, aΓS = S.

Therefore aΓS = S for all a ∈ S.

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58

Conversely suppose that aΓS = S for all a ∈ S.

Let R be a right Γ-ideal of a Γ-semigroup S.

Let r ∈ R. Then r ∈ S. By assumption rΓS = S.

Let s ∈ S. Then s ∈ rΓS ⇒ s = rαt for some t ∈ S, α ∈ Γ.

r∈ R, t ∈ S, α ∈ Γ and R is a right Γ-ideal ⇒ rαt ∈ R⇒ s ∈ R.

Therefore S ⊆ R. Clearly R ⊆ S and hence S = R.

Therefore S is the only right Γ-ideal of S. Hence S is right simple Γ-semigroup.

We now introduce a simple Γ-semigroup and characterize simple Γ-semigroups.

DEFINITION 1.3.53 : A Γ-semigroup S is said to be simple Γ-semigroup if S is its only

two-sided Γ-ideal.

THEOREM 1.3.54 : If S is a left simple Γ-semigroup or a right simple Γ-semigroup

then S is a simple Γ-semigroup.

Proof : Suppose that S is a left simple Γ-semigroup. Then S is the only left Γ-ideal of S.

If A is a Γ-ideal of S, then A is a left Γ-ideal of S and hence A = S.

Therefore S itself is the only Γ-ideal of S and hence S is a simple Γ-semigroup.

Suppose that S is a right simple Γ-semigroup. Then S is the only right Γ-ideal of S.

If A is a Γ-ideal of S, then A is a right Γ-ideal of S and hence A = S.

Therefore S itself is the only Γ-ideal of S and hence S is a simple Γ-semigroup.

THEOREM 1.3.55 : A 횪-semigroup S is simple 횪-semigroup if and only if S횪a횪S = S

for all a ∈ S.

Proof: Suppose that S is a simple Γ-semigroup and a ∈ S.

Let t ∈ SΓaΓS, s ∈ S and 훾 ∈ Γ.

t ∈ SΓaΓS ⇒ 1 2t s a s where 1s , 2s S and α, β ∈ Γ.

Now t훾s = ( 1 2s a s )훾 s= 1 2( )s a s s ∈ SΓaΓS

and s훾t =s훾( 1 2s a s ) = 1 2( )s s a s ∈ SΓaΓS. Therefore SΓaΓS is a Γ-ideal of S.

Since S is a simple Γ-semigroup, S itself is the only Γ-ideal of S and hence SΓaΓS = S .

Conversely suppose that SΓaΓS = S for all a ∈ S. Let I be a Γ-ideal of S.

Let a ∈ I. Then a ∈ S. So SΓaΓS = S.

Let s ∈ S. Then s ∈ SΓaΓS ⇒ s = 1 2t a t for some 1 2,t t ∈ S, α, 훽 ∈ Γ.

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59

a ∈ I, 1 2,t t ∈ S, α ,훽 ∈ Γ , I is a Γ-ideal of S ⇒ 1 2t a t ∈ I⇒ s ∈ I.

Therefore S ⊆ I. Clearly I ⊆ S and hence S = I.

Therefore S is the only Γ-ideal of S. Hence S is a simple Γ-semigroup.

1.4. SPECIAL ELEMENTS OF A 횪-SEMIGROUP

In this section, the terms α-idempotent, Γ-idempotent, strongly idempotent,

midunit, r-element, regular element, left regular element, right regular element, completely

regular element, (α, β)-inverse of an element, semisimple element and intra regular

element in a Γ-semigroup are introduced. Further the terms idempotent Γ-semigroup and

generalized commutative Γ-semigroup are introduced. It is proved that every generalized

commutative Γ-semigroup is a left duo Γ-semigroup. It is proved that every

훂-idempotent element of a Γ-semigroup is regular. It is proved that every 횪-ideal of a

regular 횪-semigroup S is a regular 횪-subsemigroup of S. It is proved that a Γ-semigroup S

is regular Γ-semigroup if and only if every principal Γ-ideal is generated by an

idempotent. Further it is also proved that, in a Γ-semigroup, a is a regular element if and

only if a has an (훼, 훽)-inverse. It is proved that, (1) if a is a completely regular element of

a Γ-semigroup S, then a is both left regular and right regular, (2) if ‘a’ is a completely

regular element of a Γ-semigroup S, then a is regular and semisimple, (3) if ‘a’ is a left

regular element of a Γ-semigroup S, then a is semisimple, (4) if ‘a’ is a right regular

element of a Γ-semigroup S, then a is semisimple, (5) if ‘a’ is a regular element of a

Γ- semigroup S, then a is semisimple and (6) if ‘a’ is a intra regular element of a

Γ- semigroup S, then a is semisimple. It is also proved that if a is an element of a duo

횪-semigroup, then (1) a is regular (2) a is left regular, (3) a is right regular, (4) a is intra

regular and (5) a is semisimple, are equivalent.

We now introduce α-idempotent element and Γ-idempotent element in a

Γ-semigroup.

DEFINITION 1.4.1 : An element a of Γ-semigroup S is said to be a 휶-idempotent

provided aαa = a.

NOTE 1.4.2 : The set of all α-idempotent elements in a Γ- semigroup S is denoted by E휶.

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60

DEFINITION 1.4.3 : An element a of Γ- semigroup S is said to be an idempotent or

Γ-idempotent if aαa = a for all α ∈ Γ.

NOTE 1.4.4 : In a Γ-semigroup S, a is an idempotent iff a is an α-idempotent for all

α ∈ Γ.

NOTE 1.4.5 : If an element a of Γ- semigroup S is an idempotent, then aΓa = a.

We now introduce an idempotent Γ-semigroup and a strongly idempotent

Γ-semigroup.

DEFINITION 1.4.6 : A Γ-semigroup S is said to be an idempotent Γ-semigroup provided

every element of S is α –idempotent for some α ∈ Γ.

DEFINITION 1.4.7 : A Γ-semigroup S is said to be a strongly idempotent

Γ- semigroup provided every element in S is an idempotent.

We now introduce a special element which is known as midunit in a

Γ-semigroup.

DEFINITION 1.4.8 : An element a of Γ-semigroup S is said to be a midunit provided

xΓaΓy = xΓy for all x, y ∈ S.

NOTE 1.4.9 : Identity of a Γ- semigroup S is a midunit.

We now introduce an r-element in a Γ-semigroup and also a generalized

commutative Γ-semigroup.

DEFINITION 1.4.10 : An element ‘a’ of Γ- semigroup S is said to be an r-element

provided aΓs = sΓa for all s ∈ S and if x, y ∈ S, then aΓxΓy = bΓyΓx for some b ∈ S.

DEFINITION 1.4.11 : A Γ- semigroup S with identity 1 is said to be a generalized

commutative Γ- semigroup provided 1 is an r-element in S.

THEOREM 1.4.12 : Every generalized commutative Γ-semigroup is a left duo

Γ-semigroup.

Proof: Let S be a generalized commutative Γ-semigroup. Therefore 1 is an r-element.

Let A be a left Γ-ideal of S. Let x ∈ A and s ∈ S.

Now xΓs = 1ΓxΓs =bΓsΓx = (bΓs)Γx ⊆ A. Therefore A is a Γ-ideal of S.

Therefore S is a left duo Γ-semigroup.

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We now introduce a regular element in a Γ-semigroup and regular Γ-semigroup

which are due to CHINRAM and SIAMMAI [11] and SEN and SAHA [45].

DEFINITION 1.4.13 : An element a of a Γ-semigroup S is said to be regular provided

a = a훼x훽a, for some x ∈ S and α, 훽 ∈ Γ. i.e, a ∈ aΓSΓa.

DEFINITION 1.4.14 : A Γ- semigroup S is said to be a regular Γ- semigroup provided

every element is regular.

The following example is due to SEN and SAHA [45].

EXAMPLE 1.4.15 : Let S be the set of 3×2 matrices and Γ be a set of some 2×3 matrices

over of field. Then S is a regular Γ-semigoup.

Verification: Let A ∈ S, where A =

a bc de f

.

Then we chose B ∈ Γ according to the following cases such that ABABA = ABA = A.

CASE 1: When the submatrix a bc d

is non-singular, then ad - bc ≠ 0.

e, f may both be 0 or one of them is 0 or both of them are non-zero.

Then B = 0

0

d bad bc ad bc

c aad bc ad bc

and we find ABA = A.

CASE 2: af - be ≠ 0. Then B = 0

0

f baf be af be

e aaf be af be

and ABA = A.

CASE 3: cf – de ≠ 0. Then B = 0

0

f dcf de cf de

e ccf de cf de

and ABA = A.

CASE 4: When the submatrices are singular,

then either 0 or{ 0ad bccf be

0{ 0ad bcaf de

.

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62

If all the elements of A are 0, then the case is trivial. Next we consider at least one of the

elements of A is non-zero, say 0ija , 1,2,3i and 1,2.j Then we take the jib th

element of B as 1( )ija and the other elements of B are zero and we find that ABA = A.

Thus A is regular. Hence S is a regular Γ-semigroup.

EXAMPLE 1.4.16: Let S = {0, a, b} and Γ be any nonempty set. If we define a binary

operation on S as the following Cayley table, then S is a semigroup.

. 00 0 0 0

00

a b

a a ab b b

Define a mapping from S × Γ × S to S as a훼b = ab for all a, b ∈ S and α ∈ Γ. Then S is

regular Γ-semigroup.

THEOREM 1.4.17 : Every 훂-idempotent element in a 횪-semigroup is regular

Proof : Let a be an α-idempotent element in a Γ-semigroup S.

Then a = aαa for some α ∈ Γ. Hence a = aαa훼a. Therefore a is a regular element.

We now introduce a regular Γ-ideal of a Γ-semigroup.

DEFINITION 1.4.18 : A Γ-ideal A of a Γ-semigroup S is said to be regular if every

element of A is regular in A.

THEOREM 1.4.19 : Every 횪-ideal of a regular 횪-semigroup S is a regular

횪-ideal of S.

Proof : Let A be a Γ-ideal of S and a ∈ A. Then a ∈ S and hence a is regular in S.

Therefore a = a훼b훽a where b ∈ S and 훼, 훽 ∈ Γ.

Hence a = a훼b훽a = (a훼b훽)(a훼b훽a) = a훼[(b훽a)훼b]훽a.

Let b1 = (b훽a)훼b ∈ SΓAΓS ⊆ A.

Now a훼b1훽a = a훼[(b훽a) 훼b] 훽a = a.

Therefore a is regular in A and hence A is a regular Γ-ideal.

THEOREM 1.4.20 : If a Γ-semigroup S is a regular Γ-semigroup then every

principal Γ-ideal is generated by a 훽-idempotent for some 훽 ∈ Γ.

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63

Proof: Suppose that S is a regular Γ-semigroup. Let < a > be a principal Γ-ideal of S.

Since S is a regular Γ-semigroup, there exists x ∈ S, α, 훽 ∈ Γ such that a = a훼x훽a.

Let a훼x = e. Then e훽e = (a훼x)β(a훼x) = (a훼x훽a)훼x = a훼x = e.

Thus e is a 훽-idempotent element of S.

Now a = a훼x훽a = e훽a ∈ < e > ⇒ < a > ⊆ < e >.

Also e = a훼x ∈ < a > ⇒ < e > ⊆ < a >.

Therefore < a > = < e > and hence every principal Γ-ideal is generated by an idempotent.

We now introduce left regular element, right regular element, completely regular

element in a Γ-semigroup and completely regular Γ-semigroup.

DEFINITION 1.4.21 : An element a of a Γ-semigroup S is said to be left regular

provided a = a훼a훽x, for some x ∈ S and α, 훽 ∈ Γ. i.e, a ∈ aΓaΓS.

DEFINITION 1.4.22 : An element a of a Γ-semigroup S is said to be right regular

provided a = x훼a훽a, for some x ∈ S and α, 훽 ∈ Γ. i.e, a ∈ SΓaΓa.

DEFINITION 1.4.23 : An element a of a Γ-semigroup S is said to be completely regular

provided, there exists an element x ∈ S such that a = a훼x훽a for some α, 훽 ∈ Γ and

a훼x = x훽a i.e., a ∈ aΓxΓa and aΓx = xΓa.

DEFINITION 1.4.24 : A Γ-semigroup S is said to be completely regular Γ-semigroup

provided every element of S is completely regular.

We now introduce (α, 훽)-inverse of an element in a Γ-semigroup which is due to

CHINRAM and SIAMMAI [11].

DEFINITION 1.4.25 : Let S be a Γ-semigroup, a ∈ S and 훼, 훽 ∈ Γ. An element b ∈ S is

said to be an (훼, 훽)-inverse of a if a = a훼b훽a and b = b훽a훼b.

THEOREM 1.4.26 : Let S be a 횪-semigroup and a ∈ S. Then a is a regular element if

and only if a has an (훼, 훽)-inverse.

Proof : Suppose that a is a regular element. Then a = a훼b훽a for some b ∈ S and α, 훽 ∈ Γ.

Let x = b훽a훼b ∈ S.

Now a훼x훽a = a훼(b훽a훼b)βa = (a훼b훽a)훼bβa = a훼bβa = a and

x훽a훼x = (b훽a훼b)훽a훼 (b훽a훼b) = b훽(a훼b훽a)훼 (b훽a훼b) = b훽a훼(b훽a훼b) = b훽(a훼b훽a)훼b

= b훽a훼b = x. Therefore x = b훽a훼b is the (훼, 훽)-inverse of a.

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64

Conversely suppose that b is an (훼, 훽)-inverse of a.

Then a = a훼b훽a and b = b훽a훼b. Therefore a = a훼b훽a and hence a is regular.

We now introduce a semisimple element of a Γ-semigroup and a semisimple

Γ-semigroup.

DEFINITION 1.4.27 : An element a of Γ- semigroup S is said to be semisimple provided

a ∈ < a > Γ < a >, that is, < a > Γ < a > = < a >.

DEFINITION 1.4.28 : A Γ- semigroup S is said to be semisimple Γ- semigroup provided

every element of S is a semisimple element.

We now introduce an intra regular element of a Γ-semigroup.

DEFINITION 1.4.29 : An element a of a Γ-semigroup S is said to be intra regular

provided a = xαaβaγy for some x, y ∈S and α, β, γ ∈ Γ.

EXAMPLE 1.4.30 : The Γ-semigroup given in example 1.4.16 is an intra regular

Γ-semigroup.

THEOREM 1.4.31: If ‘a’ is a completely regular element of a Γ- semigroup S, then a

is regular and semisimple.

Proof : Since a is a completely regular element in the Γ- smigroup S, a = a훼x훽a for some

α, β ∈ Γ and x ∈ S. Therefore a is regular.

Now a = a훼x훽a ∈ aΓxΓa ⊆ < a > Γ < a >. Therefore a is semisimple.

THEOREM 1.4.32 : If ‘a’ is a completely regular element of a Γ- semigroup S, then a

is both a left regular element and a right regular element.

Proof : Suppose that a is completely regular. Then a ∈ aΓSΓa and aΓS = SΓa.

Now a ∈ aΓSΓa = aΓaΓS. Therefore a is left regular.

Also a ∈ aΓSΓa = SΓaΓa. Therefore a is right regular.

THEOREM 1.4.33 : If ‘a’ is a left regular element of a Γ-semigroup S, then a is

semisimple.

Proof: Suppose that a is left regular. Then a ∈ aΓaΓx and hence a ∈ < a > Γ < a >.

Therefore a is semisimple.

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65

THEOREM 1.4.34 : If ‘a’ is a right regular element of a Γ-semigroup S, then a is

semisimple.

Proof: Suppose that a is right regular. Then a ∈ aΓaΓx and hence a ∈ < a > Γ < a >.

Therefore a is semisimple.

THEOREM 1.4.35 : If ‘a’ is a regular element of a Γ-semigroup S, then a is

semisimple.

Proof: Suppose that a is regular element of Γ-semigroup S.

Then a = a훼x훽a, for some x ∈ S, α, 훽 ∈ Γ and hence a ∈ < a > Γ < a >.

Therefore a is semisimple.

THEOREM 1.4.36 : If ‘a’ is a intra regular element of a Γ- semigroup S, then a is

semisimple.

Proof: Suppose that a is intra regular. Then a ∈ xΓaΓaΓy for x, y ∈ S and hence

a ∈ < a > Γ < a > Therefore a is semisimple.

THEOREM 1.4.37 : If S is a duo 횪-semigroup, then the following are equivalent for

any element a ∈ S.

1) a is regular.

2) a is left regular.

3) a is right regular.

4) a is intra regular.

5) a is semisimple.

Proof : Since S is duo Γ-semigroup, aΓS1 = S1Γa.

We have aΓS1Γa = aΓaΓS1 = S1ΓaΓa = < aΓa > = < a > Γ < a >.

(1) ⇒ (2) : Suppose that a is regular. Then a = a훼x훽a for some x ∈ S and α, 훽 ∈ Γ.

Therefore a ∈ aΓS1Γa = aΓaΓS1 ⇒ a = aγa훿y for some y ∈ S1, 훾, 훿 ∈ Γ.

Therefore a is left regular.

(2) ⇒ (3) : Suppose that a is left regular. Then a = a훼a훽x for some x ∈ S and α, 훽 ∈ Γ.

Therefore a ∈ aΓaΓS1 = S1ΓaΓa ⇒ a = y훾a훿a for some y ∈ S1, 훾, 훿 ∈ Γ.

Therefore a is right regular.

(3) ⇒ (4): Suppose that a is right regular. Then for some x ∈ S, α, 훽 ∈ Γ; a = x훼a훽a.

Γ-semigroups

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Therefore a ∈ S1ΓaΓa = < aΓa > ⇒ a = x훼a훽a훾y for some x, y ∈ S1 and 훼, 훽, 훾 ∈ Γ.

Therefore a is intra regular.

(4) ⇒ (5): Suppose that a is intra regular.

Then a = x훼a훽a훾y for some x, y ∈ S1 and 훼, 훽, 훾 ∈ Γ. Therefore a ∈ < a > Γ < a >.

Therefore a is semisimple.

(5) ⇒ (1): Suppose that a is semisimple. Then a ∈ < a > Γ < a > = aΓS1Γa

⇒ a ∈ a훼x훽a for some x ∈ S1and 훼, 훽 ∈ Γ.

Therefore a is a regular element.

* * * * *