Weak projectives of finite semigroups (Q1403862)

Let \(\mathcal P\) be a collection of finite bands of certain type, let \(\mathcal F\) be the category of finite semigroups and \(\mathcal C\) the category of compact right topological semigroups. An algebra is called `weak projective' if it is projective with respect to surjective epimorphisms. The main theorem proves equivalence of eight statements for a finite semigroup \(S\) including the following: (1) \(S\) is isomorphic to some semigroup of \(\mathcal P\); (2) \(S\) is a weak projective in \(\mathcal F\); (3) \(S\) is a weak projective in \(\mathcal C\).