On epimorphisms and structurally regular semigroups

Saturated (n, m)-Regular Semigroups

“…In [14], Ahanger and Shah proved that in the variety of all bands any epi from the left [right] seminormal band is surjective and thus extending the result of Alam and Khan [15], that the variety of left [right] seminormal bands is closed. Moreover in [12], Shah and Bano proved that the varieties of structurally (0, n)-left regular bands are saturated in the varieties of structurally (0, k) left regular bands for any k and n with 1 ≤ k ≤ n. In this section, we generalize the above results by proving that the variety of structurally (0, n)-left seminormal bands is saturated in the variety of structurally (0, k)-bands for any k and n with 1 ≤ k ≤ n. In particular, we show that, in the category of structurally (0, k)-bands, any epi from a structurally (0, n)-left seminormal band is onto.…”

“…Higgins [10,11] had shown that epimorphisms are onto for generalised inverse semigroups and epimorphisms are onto for locally inverse semigroups, respectively. Recently, Shah et al [12] have shown that epis from a structurally (n, m) generalised inverse semigroup is surjective.…”

Saturated Varieties of Semigroups

“…Alam [8] has demonstrated that certain classes of permutative semigroups, which satisfy specific homotypical identities, are also saturated. Moreover, Shah et al [9] have shown that classes of structurally (n, m)-generalized inverse semigroups are saturated as well. Furthermore, Alam et al [10] have extended Howie's and Isbell's result to show that any H-commutative semigroup satisfying the minimum condition on principal ideals is saturated.…”

Epimorphisms, Dominions, and Various Classes of Saturated Semigroups

“…e following semigroups are not saturated: commutative cancellative semigroups, subsemigroups of finite inverse semigroups [17], commutative periodic semigroups [14], and bands, since Trotter [24] has produced a band with a correctly epimorphically embedded subband. In this direction, a very recent significant and remarkable work have been made by Ahanger and Shah on partially ordered semigroups (posemigroups), and commutative posemigroups (see [1][2][3], [23]). Now, we begin with the class of H-commutative semigroups whose concept was first developed by Tully [25].…”