%0 Journal Article %T Categorical Properties of Down Closed Embeddings %J Sahand Communications in Mathematical Analysis %I University of Maragheh %Z 2322-5807 %A Shahbaz, Leila %D 2022 %\ 02/01/2022 %V 19 %N 1 %P 89-99 %! Categorical Properties of Down Closed Embeddings %K S-poset %K Down closed embeddings %K dc-embeddings %K Limit %K Colimit %R 10.22130/scma.2022.545343.1038 %X 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. %U https://scma.maragheh.ac.ir/article_249406_3d57569204846263835eb8b5bf190414.pdf