Stability of the class of divisible $S$-acts

We describe monoids $S$ such that the theory of the class of all divisible $S$-acts is stable, superstable or, for commutative monoid, $\omega$-stable. More precisely, we prove that the theory of the class of all divisible $S$-acts is stable (superstable) iff $S$ is a linearly ordered (well ordered) monoid. We also prove that for a commutative monoid $S$ the theory of the class of all divisible $S$-acts is $\omega$-stable iff $S$ is either an abelian group with at most countable number of subgroups or is finite and has only one proper ideal. Classes of regular, projective and strongly flat $S$-acts were considered in [1, 2]. Using results from [3] we obtain necessary and sufficient conditions for stability, superstability and $\omega$-stability of theories of classes of all divisible $S$-acts.