Cancellation Property for Acts Over Monoids

The cancellation property in the theory of actions over a monoid is introduced and examined in this paper. We will find some significant classes of acts which are cancellable. In addition, we give a characterization of cancellable acts. Then, we prove that every act is cancellable if and only if it is internally cancellable. Finally, using significant properties of refinement monoids, we show that any Dedekind-finite S-act is cancellable and it has also multiplicative cancellation in the category of Dedekind-finite S-acts with a unique zero.