Coproduct Cancellation on \textbf{Act}-$S$

Abstract: The themes of cancellation, internal cancellation, substitution have led to a lot of interesting research in the theory of modules over commutative and noncommutative rings. In this paper, we introduce and study cancellation problem in the theory of acts over monoids. We show that if $A$ is an $S$-act and $A={\dot\bigcup_{i\in I}}A_i$ is the unique decomposition of $A$ into indecomposable subacts $A_i, i\in I$ such that the set $P={{\rm Card} [i] \mid i\in I}$ is finite, then $A$ is cancellable if and only if the equivalence class $[i]={j\in I \mid A_i\cong A_j}$ is finite, for every $i\in I$. Likewise, we prove that every $S$-act is cancellable if and only if it is internally cancellable. Thus, the concepts cancellation and internal cancellation coincide here.