Algebraically equipped posets

TL;DR

This paper introduces algebraically equipped posets, a new structure combining posets with algebraic subspace data, and explores their representation theory, including matrix classification.

Contribution

It defines algebraically equipped posets and establishes their equivalences with module categories and differential tensor algebra representations, providing new tools for their analysis.

Findings

Established equivalences with module categories

Developed matrix representations and classification methods

Connected algebraically equipped posets to differential tensor algebra

Abstract

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.