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 $\struct {S, \wedge, \preceq}$) if and only if:

$\forall x, y \in S: \paren {x \wedge y \preceq p \implies x \preceq p \text { or } y \preceq p}$

Also known as

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

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_6:def 6