Congruences on N-semigroups

The study of a semigroup in terms of its congruence relations has been used many times in the past. In the case of ^-semigroups Tamura initiated this study with a paper [9] determining the ^-congruences of an ^-semigroup. Recently, Dickinson [4] has determined the congruences which correspond to homomorphic images having no idempotents as refinements of 9?-congruences. Group congruences on a commutative semigroup have been determined by Tamura and the author [11], but here they are determined for an ^-semigroup from the group of quotients of the ^-semigroup, ^-congruences and group congruences on ^-semigroups are of fundamental importance in characterizing ^-semigroups. In this paper we make a study of all types of congruences on ^-semigroups.