CONGRUENCES ON TERNARY SEMIGROUPS
Sukhendu Kar, Bimal Maity · 2007
Abstract. In this paper we introduce the notion of congruence ona ternary semigroup and study some interesting properties. We alsointroduce the notions of cancellative congruence, group congruenceand Rees congruence and characterize these congruences in ternarysemigroups. 1. IntroductionThe notion of congruence was flrst introduced by Karl Fredrich Gaussin the beginning of the nineteenth century. Congruence is a special typeof equivalence relation which plays a vital role in the study of quotientstructures of difierent algebraic structures. We study the quotient struc-ture of ternary semigroup by using the notion of congruence in ternarysemigroup. The notion of ternary semigroup was known to S. Banach.He showed, by an example, that a ternary semigroup does not necessarilyreduce to an ordinary semigroup. In [4], J. Los studied some propertiesof ternary semigroup and proved that every ternary semigroup can beembedded in a semigroup. In [6], F. M. Sioson studied ideal theory internary semigroups. He also introduced the notion of regular ternarysemigroups and characterized them by using the notion of quasi-ideals.In [5], M. L. Santiago developed the theory of ternary semigroups andsemiheaps.The main purpose of this paper is to classify congruences on ternarysemigroups and study some basic properties of congruences on ternarysemigroups. We also introduce the notions of cancellative congruence,group congruene and Rees congruence and characterize these congru-ences in ternary semigroups.