This paper is a continuation of recent researches concerning generalization of
injectivity of acts over moniods, namely, C-injectivity and InD-injectivity. We introduce new
kinds of injectivity, such as, LC-injectivity and CQ-injectivity. Classications of monoids
by properties of these kinds of injective acts are presented. It is proved that a monoid S
is completely (cyclic) injective if and only if it is completely quasi (CQ-) injective. Some
results on quasi-injective acts are proved. Also new characterizations for right hereditary
monoids and right PP-monoids are given.