Axiomatizability of the class of subdirectly irreducible $S$-acts over a commutative monoid

An axiomatizability criterion is found for the class of subdirectly irreducible $S$-acts over a commutative monoid. As a corollary, a number of properties are presented which a commutative monoid should satisfy provided that the class of subdirectly irreducible acts over it is axiomatizable. The question about a complete description of monoids over which the class of subdirectly irreducible acts is axiomatizable remains open even for the case of a commutative monoid.