Volume 15, Number 1, July 2021, 231-253. https://doi.org/10.52547/cgasa.15.1.231 On epimorphisms and structurally regular semigroups A.H. Shah*, S. Bano, S.A. Ahanger, and W. Ashraf Abstract. In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n, m)-regular semigroups for any pair (n, m) of positive integers. In Section 2, after a brief introduction of these semigroups, we prove that the class of structurallly (n, m)-generalized inverse semigroups is closed under morphic images. We then prove the main result of this section that the class of structurally (n, m)-generalized inverse semigroups is saturated and, thus, in the category of all semigroups, epimor- phisms in this class are precisely surjective morphisms. Finally, in the last section, we prove that the variety of structurally (o, n)-left regular bands is saturated in the variety of structurally (o, k)-left regular bands for all positive integers k and n with 1  k  n. C Introduction and preliminaries A morphism α : S → T in the category of all semigroups is called an epi- morphism (epi for short) if for all morphisms β,γ with αβ = αγ implies * Corresponding author Keywords : Dominions, epimorphisms, zigzag, saturated, structurally regular. Mathematics Subject Classiﬁcation [2010]: Primary 06F07, 20M07, 20M10. Received: 15 December 2020, Accepted: 26 April 2021. ISSN: Print 2345-5853, Online 2345-5861. © Shahid Beheshti University 231