Axiomatizability of the class of weakly injective $S$-acts

The concepts of weakly injective, fg-weakly injective, and p-weakly injective $S$-acts generalize that of injective $S$-act. We study the monoids $S$ over which the classes of weakly injective, fg-weakly injective, and p-weakly injective $S$-acts are axiomatizable. We prove that the class of p-weakly injective $S$-acts over a regular monoid is axiomatizable.