Axiomatisability Problems for S-acts, Revisited

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