Semigroups over which all acts are residually finite

The properties ($*$) and ($**$) of semigroup $S$ are investigated, namely: ($*$) every subdirectly irreducible right $S$-act is finite; ($**$) the cardinalities of subdirectly irreducible right $S$ acts are bounded by a natural number. We prove that if $S$ is a nilsemigroup then then these conditions are equivalent to each other and to finiteness of $S$. We characterize the commutative semigroups satisfying ($**$).