Strongly alg-universal semigroup varieties (Q5950782)

It is known that every monoid is isomorphic to the endomorphism monoid of some semigroup. The authors deal with varieties of semigroups having the same property. They prove that if \(\mathcal V\) is a semigroup variety such that every monoid is isomorphic to the endomorphism monoid of a semigroup from \(\mathcal V\), then any full category \(\mathcal A\) of algebras has a full embedding into \(\mathcal V\) carried by an endofunctor of the category \({\mathcal S}{\mathcal E}{\mathcal T}\) of all sets and mappings.