Characterization of Join Irreducible Element

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Theorem

Let $\struct{S, \vee, \preceq}$ be a join semilattice.

Let $z \in S$.

Then:

$z$ is join-irreducible

if and only if

$\forall x, y \in S : \leftparen{x \prec z}$ and $\rightparen{y \prec z} \implies \paren{x \vee y \prec z}$

where $x \prec z$ denotes that $x \preceq z$ and $x \neq z$.

Proof

By Dual Pairs (Order Theory):

• join semilattice is dual to meet semilattice.
• join irreducible element is dual to meet irreducible element.
• join is dual to meet.
• succeeds is dual to precedes.

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

$\blacksquare$