  In this paper, we introduce (Rees) artinian S-acts as those acts that satisfy
the descending chain condition on their (Rees) congruences. Then, we show that
artinian S-acts are those acts whose all factor acts are finitely (Rees)
cogenerated. In addition, we continue the study of (Rees) noetherian S-acts,
namely, those acts whose (Rees) congruences are finitely generated. Moreover,
as a useful tool for the investigation of chain conditions, we prove that the
properties of being (Rees) noetherian and artinian are inherited in Rees short
exact sequences. Next, we specifically consider the chain conditions on
monoids. Finally, we prove that every right Rees artinian, commutative monoid
is right Rees noetherian.
