Congruence-free restriction semigroups

Restriction semigroups are common generalizations of ample semigroups and inverse semigroups. The main aim of this paper is to probe restriction semigroups with certain congruence properties. In this paper we give some characterizations of restriction semigroups each of whose proper (2,1,1)-congruences are reduced, so called H-reduced restriction semigroups. In particular, the classification of congruence-free restriction semigroups is obtained; that is, it is proved that a restriction semigroup is congruence-free if and only if it is either a simple group or an H-reduced restriction semigroup without nontrivial reduced restriction monoid (2,1,1)-congruences. These results extend and enrich the related results of inverse semigroups.