\[ \newcommand{abs}[1]{\left|#1\right|} \newcommand{\covers}{\mathrel{\triangleleft}} \]
Definition 1 A posite \((P, \leq, \covers)\) is a poset \((P, \leq)\) with a binary relation \({\covers} \subseteq P \times 2^{\abs{P}}\) such that:
if \(u \covers V\) and \(v \in V\), then \(v \leq u\);
if \(u \covers V\) and \(u' \leq u\), then there exists \(V'\) such that \(u' \covers V'\), and for all \(v' \in V'\), there exists \(v \in V\) such that \(v' \leq v\).
Given posites \(P\) and \(Q\), a morphism of posites \(f : P \to Q\) is a monotone function such that:
given any \(u \in Q\), for some \(v \in P\), we have \(u \leq f(v)\);
if \(w \leq f(v)\), \(f(v) \in Q\), then for some \(x \leq u\), \(v \in P\), we have \(w \leq f(x)\);
if \(u \covers V\), then \(f(u) \covers \{f(v) : v \in V\}\).
Citation
@online{choudhury2026,
author = {Choudhury, Vikraman},
title = {Normalization of Sums},
date = {2026-05-15},
url = {https://vikraman.org/notes/sums.html},
langid = {en}
}