### Chained Commutative Ternary Semigroups

G. Hanumanta Rao G. Hanumanta Rao
2013 IOSR Journal of Mathematics
In this paper, the terms chained ternary semigroup, cancellable clement , cancellative ternary semigroup, A-regular element, π-regular element, πinvertible element, noetherian ternary semigroup are introduced. It is proved that in a commutative chained ternary semigroup T, i) if P is a prime ideal of T and x ∉ P then n n1 x PT    = 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 ≠ . iv) If < a > w = for all
more » ... a ∊ T, then T has no semisimple elements. v) T has no regular elements, then for any a ∊ T, < a > w = or < a > w is a prime ideal. vi) If T is a commutative 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\T 3 = { 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  T and a  < x > w then a = x n for some odd natural number x r } for some odd natural number r. v) If a  T and a  < x > w then a = x r for some odd natural number r or a = x n s n t n and s n  < x > w or t n  < 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 commutative chained ternary semigroup T, T is archemedian ternary semigroup without idempotent elements if and only if < a > w =  for every a T. Further it is proved that if T is a commutative chained ternary semigroup containing cancellable elements and < a > w =  for every a  T , then T is a cancellative ternary semigroup. It is proved that if T is a noetherian ternary semigroup containing proper ideals then T has a maximal ideal. Finally it is proved that if T is a commutative ternary semigroup such that T = < x > for some x  T, then the following are equivalent. 1) T = {x, x 2 , x 3 , ............} is infinite. 2) T is a noetherian cancellative ternary semigroup with x  xTT. 3) T is a noetherian cancellative ternary semigroup without idempotents. 4) < a > w =  for all a T. 5) < x > w = . and if T is a commutative chained ternary semigroup with T ≠ T 3 , then the following are equivalent. (1) T={x, x 3 , x 5 , . . . . . . .}, where x T\ T 3 (2) T is Noetherian cancellative ternary semigroup without idempotents. (3) < a > w =  for all a T. Finally, it is proved that If T is a commutative chained noetherian cancellative ternary semigroup without regular elements, then < a > w =  for all a T. Keywordschained ternary semigroup, cancellable clement , cancellative ternary semigroup, noetherian ternary semigroup and ternary group.