Regular Congruences on an Idempotent-Regular-Surjective Semigroup

Abstract

A semigroup S is called idempotent-surjective (respectively, regular-surjective) if whenever ρ is a congruence on S and aρ is idempotent (respectively, regular) in S /ρ, then there is e ∈ E S ∩ aρ (respectively, r ∈ Reg(S ) ∩ aρ), where E S (respectively, Reg(S )) denotes the set of all idempotents (respectively, regular elements) of S . Moreover, a semigroup S is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.2010 Mathematics subject classification: primary 20M99; secondary 06B10.