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Chained Duo Ternary Semigroups

1 C. Sreemannarayana, 2G. Hanumanta Rao, 3A. Anjaneyulu, 4A. Gangadhara Rao

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

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

3,4 Department of Mathematics, V.S.R & N.V.R.College, Tenali, A.P. India.

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  = P for all odd natural numbers n . ii) T is a semiprimary ternary semigroup. iii) If a T is a semisimple element of T, then < a > 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 = xT1T1 = T1xT1 = T1T1x 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