Definition:Prime Element (Order Theory)

Definition
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.

Let $p \in S$.

Then $p$ is a prime element (of $\struct {S, \wedge, \preceq}$) :
 * $\forall x, y \in S: \paren {x \wedge y \preceq p \implies x \preceq p \text { or } y \preceq p}$

Also known as
A prime element of $\struct {S, \wedge, \preceq}$ can also be described as prime in $\struct {S, \wedge, \preceq}$.