Prime Element iff Meet Irreducible in Distributive Lattice

Theorem

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

Let $p \in S$.

Then $p$ is a prime element if and only if $p$ is meet irreducible.

Proof

By Prime Element is Meet Irreducible:

if $p$ is a prime element, then $p$ is meet irreducible.

Assume that:

$p$ is meet irreducible.

Let $x, y \in S$ such that:

$x \wedge y \preceq p$

\(\ds p\)

\(=\)

\(\ds p \vee \paren {x \wedge y}\)

Preceding iff Join equals Larger Operand

\(\ds \)

\(=\)

\(\ds \paren {p \vee x} \wedge \paren {p \vee y}\)

Definition of Distributive Lattice

By definition of meet irreducible:

$p = p \vee x$ or $p = p \vee y$

Thus by Preceding iff Join equals Larger Operand:

$x \preceq p$ or $y \preceq p$

$\blacksquare$

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_6:27