• DUO TERNARY SEMIGROUPS

Abstract

In this paper the terms left duo teranry semigroup, right duo teranry semigroup, duo teranry semigroup are introduced. it is proved that a ternary semigroup T is a duo teranry semigroup if and only if xT1T1 = T1T1x = T1xT1 for all x ∊ T. Further it is proved that every commutative / quasi commutative ternary semigroup is a duo ternary semigroup. If A is an ideal of a ternary semigroup T and a ∊ T, then 1) Al(a) = { x ∊ T : xua ∊ A } is a left ideal of T for all u ∊ T. 2) Ar(a) = { x ∊ T : aux ∊ A } is a right ideal of T for all u ∊ T. If A is an ideal of a duo ternary semigroup T and a ∊ T, then 1) Al(a) = { x ∊ T : xua ∊ A } is an ideal of T for all u ∊ T. 2) Ar(a) = {x ∊ T: aux∊ A} is an ideal of T for all u ∊ T. It is proved that if A is an ideal of a duo ternary semigroup T, then 1) abc ∊ A if and only if < a > < b > < c > ⊆ A for all a, b, c ∊ T. 2) a1a2 .....an ∊ A if and only if < a1 > < a2 > < a3 >........< an > ⊆ A for all a1 , a2 , a3 ,. . . .an ∊ T. 3) a n ∊ A if and only if < a >n ⊆ A for all a ∊ A. 4) < abc > = < a > < b > < c > for all a, b, c ∊ T. 5) < a n > = < a >n for all a ∊ T. Further it is proved, if A is an ideal of duo ternary semigroup T then 1) A4 = {x : < x >n⊆ A for some odd natural number n } is the minimal semiprime ideal of T containing A. 2) A2 = {x T : xn ∊ A for some odd natural number n} is the minimal completely semiprime ideal of T containing A. It is proved that, 1) An ideal A of a duo ternary semigroup T is completely semiprime if and only if A is semiprime. 2) Every prime ideal P minimal relative to containing an ideal A of a duo ternary semigroup T is completely prime. It is also proved that, if T is a duo ternary semigroup and A is an ideal of T, then A1=A2=A3=A4. It is proved that 1) in a duo semi group T, the following are equivalent 1) T is a strongly archimedean ternary semigroup. 2) T is an archimedean ternary semigroup. 3) T has no proper completely prime ideals. 4) T has no proper completely semiprime ideals. 5) T has no proper prime ideals. 6) T has no proper semiprime ideals. Further it is proved that, if T is a duo ternary semigroup, then 1) S = {a ∊ T: √() ≠ T} is either empty or prime ideal. 2) T\S is either empty or an archemedian ternary sub semigroup of T. It is proved that, if T is a duo ternary semigroup and contains a nontrivial maximal ideal then T contains semisimple elements. Finally, it is proved that, in a duo archimedian ternary semigroup T, an ideal M is maximal if and only if M is trivial. Also if T = T3, then T has no maximal ideals.