Categorical Properties of Down Closed Embeddings

Let $mathcal M$ be a  class of (mono)morphisms in a category $mathcal A$. To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair (${mathcal A}$,${mathcal M}$). In this paper, we take $mathcal A$ to be the category {bf Pos}-$S$ of $S$-posets over a posemigroup $S$, and ${mathcal M}_{dc}$ to be the class of down closed embeddings and study the categorical properties, such as limits and colimits, of the  pair (${mathcal A}$,${mathcal M}$). Injectivity with respect to this class of monomorphisms have been studied by Shahbaz et al., who used it to obtain  information about regular injectivity.