Commutative semigroups satisfying the equation $2x+y=2x$ and having only two $G$-invariant congruences for an automorphism group $G$ are considered. Some classes of these semigroups are characterized and some other examples are constructed.
For every module $M$ we have a natural monomorphism \[ \Psi :\coprod _{i\in I}\mathop {\mathrm Hom}\nolimits _R(M,A_i)\rightarrow \mathop {\mathrm Hom}\nolimits _R\biggl (M,\coprod _{i\in I}A_i\biggr) \] and we focus our attention on the case when $\Psi $ is also an epimorphism. Some other colimits are also considered.
This short note is a continuation of and and its purpose is to show that every simple zeropotent paramedial groupoid containing at least three elements is strongly balanced in the sense of [4].