Semigroups that are factors of subdirectly irreducible semigroups by their monolith. (Q2583584)

A universal algebra \(\mathbf A\) is called subdirectly irreducible if it possesses a smallest non-identity congruence \(\mu\). An algebra is called a monolith factor if it is isomorphic to \({\mathbf A}/\mu\) for some subdirectly irreducible \(\mathbf A\). Strengthening the results of \textit{J. Ježek} and \textit{T. Kepka} [Algebra Univers. 47, No. 3, 319-327 (2002; Zbl 1065.08006)] the authors prove that every semigroup possessing a Suschkewitsch kernel (that is, a smallest ideal) is the monolith factor of a subdirectly irreducible semigroup but there exist finite semigroups that are not monolith factors of any finite subdirectly irreducible semigroup (an example is provided by every finite right zero semigroup with more than one element).