Weak inverses modulo Green's relation $$\mathcal{{H}}$$ on E-inversive and group-closed semigroups.

In this paper we study semigroups defined either by means of weak inverses modulo Green's relation $$\mathcal{{H}}$$ , or by means of the set of completely regular elements. Basic properties of E-inversive semigroups are analyzed, and semigroups whose completely regular elements form a subsemigroup or a rectangular band modulo $$\mathcal{{H}}$$ are considered. We obtain results that generalize the corresponding results for E-semigroups or semigroups whose idempotents form a rectangular band, and also new results for E-inversive semigroups. [ABSTRACT FROM AUTHOR]