  We introduce partially ordered sets (posets) with an additional structure
given by a collection of vector subspaces of an algebra $A$. We call them
algebraically equipped posets. Some particular cases of these, are generalized
equipped posets and $p$-equipped posets, for a prime number $p$. We study their
categories of representations and establish equivalences with some module
categories, categories of morphisms and a subcategory of representations of a
differential tensor algebra. Through this, we obtain matrix representations and
its corresponding matrix classification problem.
