**InternationaleJournals**

**Chained Duo Ternary ****Semigroups**

^{1}** C. Sreemannarayana, ^{2}G.
Hanumanta Rao, ^{3}A. Anjaneyulu, ^{4}A. Gangadhara Rao**

^{1}*Department of Mathematics, T.J.P.S. College, Guntur*,* A.P.
India.*, * Email : **csnnrt@gmail.com*

^{
2}*Department of
Mathematics, S.V.R.M. College, Nagaram, Guntur (dt) A.P. India.*, *Email : ghr@svrmc.edu.in
*

*Email
: ^{3}*

**ABSTRACT
**

In this paper, the terms chained ternary
semigroup, cancellable clement , cancellative ternary semigroup, A-regular
element, π- regular element, π- invertible element are introduced. It
is proved that in a duo chained ternary semigroup T, i) if P is a prime ideal of T and x ∉ P then ^{w}
≠ f. iv) If

<
a >^{w }= 휙; for all a ∊ T, then
T has no semisimple elements. v) T has no regular elements, then for any a ∊ T,

< a >^{w }= 휙; or < a >^{w } is a prime ideal. vi) If T is a duo
chained cancellative ternary semigroup then for every non π-invertible
element a, < a >^{w} is either empty or a prime ideal of T.** **Further it is proved that if T is a chained ternary semigroup with T\T3= { x } for some x
∊ T, then i) T\
{ x } is an ideal of T. ii) T = xT^{1}T^{1 }=
T^{1}xT^{1} = T^{1}T^{1}x
and T 3 = xTT
= TxT = TTx is the unique maximal ideal of T.
iii) If
a Î T and a Ï < x >^{w}
then

a = xn for some odd natural number n > 1.iv) T\ < x >^{w} = { x,
x 3, x5, . . . . .}
or T\< x >^{w} ={x, x 3, . . . , xr} for some odd natural number
r. v) If a Î T and a Î < x >^{w} then a = xr for some odd natural number r or a
= xn sn tn
and snÎ < x >^{w} or tn Î < x
>^{w} for every odd natural number n. vi) If T contains cancellable
elements then x is cancellable
element and < x >^{w} is either empty or a prime ideal of T. It is also prove that, in a duo chained
ternary semigroup T, T is archemedian
ternary semigroup without idempotent elements if and only if

< a >^{w} = f for every
aÎ T.* *

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

**Keywords :**** ****-**** **chained ternary
semigroup, cancellable clement and cancellative ternary semigroup.

**International
eJournal of Mathematics and Engineering**

Volume**5**,
**Issue** **2**, **Pages**: ** **2371 - 2380