\[ \newcommand{abs}[1]{\left|#1\right|} \newcommand{\covers}{\mathrel{\triangleleft}} \]
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 \(v' \leq v\).