in this paper $s$ is a monoid with a left zero and $a_s$ (or $a$) is a unitary right $s$-act. it is shown that a monoid $s$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $s$-act is quasi-projective. also it is shown that if every right $s$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that ...

A notion of metric invariance entropy is constructed with respect to a conditionally invariant measure for control systems in discrete time. It is shown that the metric invariance entropy is invariant under conjugacies, the power rule holds, and the (topological) invariance entropy provides an upper bound. November 20, 2014 Key words. invariance entropy, conditionally invariant measures, quasi-...