Green's relations on ternary semigroups

Abstract

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces and the like. This provides su_cient motivation to researchers to review various concepts and results. The theory of ternary algebraic system was introduced by D. H. Lehmer [11]. He investigated certain ternary algebraic systems called triplexes which turn out to be commutative ternary groups. The notion of ternary semigroups was introduced by Banach S. He showed by an example that a ternary semigroup does not necessarily reduce to an ordinary semigroups.

In another hand, in mathematics, Green's relations characterise the elements of a semigroup in terms of the principal ideals they generate. John Mackintosh Howie, a prominent semigroup theorist, described this work as so all-pervading that, on encountering a new semigroup, almost the _rst question one asks is "What are the Green relations like?" (Howie 2002). The relations are useful for understanding the nature of divisibility in a semigroup.

In this paper we study Green's relations on ternary semigroup in view of those obtained in binary semigroups. Many interesting results (essentially analogous of Green's lemmas for semigroups [4] and [13]) can be derived for our case. We are also interested in the quality of idempotents with respect to the Green's relations. The particular case of ternary inverse semigroup has been studied and a relationship between the existence of idempotents and the inverse elements has been caracterized.