U-Ternary Semigroups And V-Ternary Semigroup

Abstract

In this paper the term U-ternary semigroup is introduced.  It is proved that a ternary semigroup T is a U-ternary semigroup if either T has a left (lateral, right) identity or T is generated by an idempotent.  It is proved that a ternary semigroup is U-ternary semigroup if and only if (1) every proper ideal of T is contained in a proper prime ideal of T, (2) every ideal A of T is semiprime ideal of T, (3) every ideal A of T is the intersection of all prime ideal of T contains A, (4) T\A is an n-system of T or empty where A is an ideal of T, (5) T\A is an m-system of T where A is an ideal of T.  Further it is proved that if T be a U-ternary semigroup.  Then T = T³ and hence every maximal ideal is prime.  Conversely if {P_ð”›‚} is a collection of all prime ideals in T and if P is a maximal element in this collection, then P is a maximal ideal of T.  The term dimension n is introduced and it is proved that if A is a proper ideal of the finite dimensional U-ternary semigroup T.  Then A is contained in maximal ideal.

The term V-ternary semigroup is introduced and proved that a ternary semigroup T is a
V-ternary semigroup if and only if T has at least one proper prime ideal and if {P_𔛂} is the family of all proper prime ideals, then < x > = T for x ∈ T\∪P_𔛂 or T is a simple ternary semigroup.

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