Document Type: Original Manuscript
Department of Mathematics, University of Maragheh, P.O. Box 55181-83111, Maragheh, Iran.
In this paper, we study the categorical and algebraic properties, such as limits and colimits of the category Pos-S with respect to order dense embeddings. Injectivity with respect to this class of monomorphisms has been studied by the author and used to obtain information about injectivity relative to regular monomorphisms. Then, we study three different kinds of essentiality, usually used in literature, with respect to the class of all order dense embed-dings of S-posets, and investigate their relations to order dense injectivity. We will see, among other things, that although all of these essential extensions
are not necessarily equivalent, they behave equivalently with respect to order dense injectivity. More precisely, it is proved that order dense injectivity well behaves regarding these essentialities. Finally, a characterization of these
essentialities over pogroups is given.