Quasi injectivity and θ-internal order sum in partially ordered acts

Injectivity (weakly injectivity) of objects is an important concept which category theory inherited from homological and commutative algebra. One of the useful kinds of weakly injectivity is quasi injectivity. In this paper, we study the relation between different kinds of quasi injectivity and the concept of θ-internal order sum in the category of actions of an ordered monoid on ordered sets and some of its important subcategories. From the results obtained, we investigated the relations between these types of quasi injectivity.