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:

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

Let $L = \struct{S, \vee, \wedge, \preceq}$ be a lattice.

Let $z \in S$.

Then $z$ is said to be meet prime element in L if and only if

$z$ is a prime element in the meet semilattice $\struct{S, \wedge, \preceq}$

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

A prime element (of $\struct {S, \wedge, \preceq}$) is also referred to as a meet prime element or meet-prime element of $\struct {S, \wedge, \preceq}$ in some sources.

This is to distinguish a prime element from a join prime element in a lattice.