Finiteness conditions for subdirectly irreducible $S$-acts and modules

It is proved that, for every semigroup $S$ of $n$ elements, the cardinalities of the subdirectly irreducible $S$-acts are less or equal to $2^{n+1}$. If the cardinalities of the subdirectly irreducible $S$-acts are bounded by a natural number then $S$ is a periodic semigroup. It is obtained a combinatorial proof of the fact that there exist only finitely many of unitary subdirect irreducible modules over a finite ring.