Epimorphisms, dominions and regular semigroups

Abstract

We show that a regular semigroups satisfying certain conditions in the containing semigroup is closed. As immediate corollaries, we have got that the special semigroup amalgam ~${\cal U} = [\{S,S'\};~U;~\{i,\alpha\mid U\}]$ within the class of left [right] quasi-normal orthodox semigroups, $\cal{R}$[$\cal{L}$]-unipotent semigroups and left[right] Clifford semigroups is embeddable in a left [right] quasi-normal orthodox semigroup, $\cal{R}$[$\cal{L}$]-unipotent semigroup and left[right] Clifford semigroup respectively. Finally we have shown that the class of all semigroups satisfying the identity $xyz=xz$ and the class of all semigroups satisfying the identity $xy=xyx[yx=xyx]$ are closed within the class of all semigroups satisfying the identities $xyz=xz$ and $xy=xyx[yx=xyx]$ repectively.