\begin{definition}[Definition:Completely Irreducible]
Let $\struct {S, \preceq}$ be an ordered set.
Let $p \in S$.
An element $p$ is '''completely irreducible''' {{iff}}
:$p^\succeq \setminus \set p$ admits a minimum element
where $p^\succeq$ denotes the upper closure of $p$.
\end{definition}