Down closed-quasi-injectivity of partially ordered acts

Abstract

Action of a pomonoid on partially ordered sets ([Formula: see text]-posets) has beautiful aspects in practical subjects such as automata theory, projection algebra and theoretical computer science which makes it always capture the interest of mathematicians. On the other hand, the study of different kinds of weakly injectivity (which category theory inherited from homological and commutative algebra) is an interesting subject for mathematicians. One of the important kinds of weakly injectivity is quasi-injectivity. In this paper, we study quasi-injectivity in the category of [Formula: see text]-posets with respect to special kind of order embeddings, namely, down-closed embeddings (dc-quasi-injectivity).