Evaluating the Concept of Idempotent Separating Congruence on a Regular Semigroup with a Regular Idempotent

This paper introduces the concept of idempotent separating congruence on a regular semigroup. Let S be a regular semigroup. A congruence ρ on S is called idempotent separating if the associated projection homomorphism ρ#:S➙∣ρ, is idempotent separating. Hall shows that if u is an idempotent of a regular semigroup S then every idempotent-separating congruence on uSu extends to a unique idempotent separating congruence on SuS. An idempotent u of a regular semigroup S is called regular if fuR fL uf for each fε E(S) . In this paper, we proved that if u is a regular idempotent of S then S = SuS. Also, we find the relationship between the idempotent separating congruence on S and uSu, when u is a regular idempotent of S.