Splittability over semigroups. (Q862385)

Let \(S\) be a semigroup and \(\mathcal P\) be a class of semigroups. We say that \(S\) is splittable (or pointwise splittable) over \(\mathcal P\), if for every subset \(A\subseteq S\) (or every element \(s\in S\)) there exists a semigroup homomorphism \(f\colon S\to T\in\mathcal P\) such that \(f^{-1}(f(A))=A\) (or \(f^{-1}(f(s))=\{s\}\), resp.). If, moreover, \(f\) is surjective then \(S\) is onto-splittable over \(\mathcal P\). It is proved that if \(S\) is a semigroup with zero onto-splittable over \(0\)-simple semigroups then \(S\) is a \(0\)-simple semigroup, if \(S\) is pointwise onto-splittable over an infinite monogenic semigroup then \(S\) is an infinite monogenic semigroup, if \(S\) is a finite semigroup onto-splittable over a class \(\mathcal P\) consisting of finite monogenic semigroups with index distinct from \(2\) then \(S\in\mathcal P\). Open problems are given.