Algebraically equipped posets

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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.