Some Model-Theoretic Properties of the Class of T-Pseudofinite S-Acts

Source abstract

The structure M in a language L is called pseudofinite if every sentence in a language L true in M has a finite model. A model M of a theory T in a language L is called T-pseudofinite if every sentence in a language L true in M has a finite model, which is a model of the theory T. In this work we prove that for any theory T the class of all pseudofinite (T-pseudofinite) models of this theory is axiomatizable. A (left) S-act over monoid S is a set A upon which S acts unitarily on the left. If T is the theory of all S-acts, then completeness, model completeness, and totally categoricalness of the class of all T-pseudofinite S-acts are equivalent to the following condition: every T-pseudofinite S-act is a coproduct of one-element S-acts. If S is a finite monoid or a commutative Noetherian and Artinian monoid then the theory of every S-act (T-pseudofinite S-act) is stable iff S is a linearly ordered monoid, where T is the theory of all S-acts.