Join Prime Element Iff Join Irreducible In Distributive Lattice

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Theorem

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

Let $z \in S$.

Then:

$z$ is join-irreducible

if and only if

$z$ is join-prime

Proof

By Dual Pairs (Order Theory):

• join prime element is dual to meet prime element.
• join irreducible element is dual to meet irreducible element.

Thus the theorem holds by the duality principle applied to Characterization of Meet Irreducible Element.

$\blacksquare$