Pseudofinite $S$-acts

The work has begun to study the structure of pseudofinite acts over a monoid. A theorem on the finiteness of an arbitrary cyclic subacts of $S$-act is proved under the condition that this $S$-act is pseudofinite and the number of types of isomorphisms of finite cyclic $S$-acts is finite. It is shown that a coproduct of finite $S$-acts is pseudofinite. As a consequence, it is shown that any $S$-act, where $S$ is a finite group, is pseudofinite.