**InternationaleJournals**

**DUO Chained ****r****-Semigroup**

**A. Gangadhara Rao ^{1}, A.
Anjaneyulu^{2}, D. Madhusudhana Rao^{3}**

Dept. of Mathematics, V S R & N V R College, Tenali, A.P. India.

^{1}raoag1967@gmail.com, ^{2 }anjaneyulu.addala@gmail.com, ^{3
}dmrmaths@gmail.com

**ABSTRACT**

In this paper
we introduce the terms left α-cancelltive, right α-cancellative,
α-cancellative, left Γ-cancellative, right Γ-cancellative, Γ-cancellative,
strongly left cancellative, strongly right cancellative, strongly cancellative
elements, α-inverse,

Γ-inverse, complete inverse of an element, unit in a Γ-semigroup and a duo
chained

Γ-semigroup. It is proved that if P is a
prime -ideal of a duo chained Γ-semigroup S and then . It is also proved
that every duo chained

Γ-semigroup is a semiprimary -semigroup. It is proved that (1) if is a semisimple
elements of a duo chained Γ-semigroup S , then . (2) if a duo
chained Γ-semigroup S has no 횪;-idempotent
elements, then for any *a* ∈ S, < *a* > ^{w} = ∅
or < *a* > ^{w} is a
prime 횪;-ideal. In
a duo chained Γ-semigroup S if SSS then S\SS = { *x *}
for some *x* S. ** **Further
it is proved that ** **in a duo chained -semigroup S, if
SSS such that S\SS = { *x *}
for some *x* S, then (1) S = *x*
S^{1} = S^{1} *x* and S S = *x* S = S *x* is the
unique maximal -ideal of S.
(2) If *a*S and *a* < *x* >^{w} then for some
natural number *n* > 1. (3) If S
contains strongly cancelable elements then *x*
is a strongly cancelable 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 archemedian Γ-semigroup without Γ-idempotents if and
only if < a >* ^{w} * = ∅ for
every

Γ-semigroup with for some

**Mathematical subject
classification (2010)** : 20M07;
20M11; 20M12.

Γ-inverse, complete inverse of an element, unit in a Γ-semigroup,

**International
eJournal of Mathematical Sciences, Technology and Humanities**