\begin{definition}[Definition:Prime Element (Order Theory)]
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $p \in S$.
Then $p$ is '''a prime element (of $\struct {S, \wedge, \preceq}$)''' {{iff}}:
:$\forall x, y \in S: \paren {x \wedge y \preceq p \implies x \preceq p \text { or } y \preceq p}$
\end{definition}