Axiomatisability problems for S-acts, revisited

This paper discusses necessary and sufficient conditions on a monoid S, such that a class of left S-acts is first order axiomatisable. Such questions have previously been considered by Bulman-Fleming, Gould, Stepanova and others. Let $mathcal{C}$ be a class of embeddings of right S-acts. A left S-act B is $mathcal{C}$ -flat if tensoring with B preserves the embeddings in $mathcal{C}$ . We find two sets (depending on a property of $mathcal{C}$ ) of necessary and sufficient conditions on S such that the class of all $mathcal{C}$ -flat left S-acts is axiomatisable. These results are similar to the ??replacement tossings?? results of Gould and Shaheen for S-posets. Further, we show how to axiomatise some classes using both replacement tossings and interpolation conditions, thus throwing some light on the former technique.