Definition:Join Irreducible Element

Definition

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

Let $z \in S$.

Then $z$ is said to be join irreducible if and only if

$\forall x, y \in S : x \vee y = z \implies x = z$ or $y = z$

Lattice

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

Let $z \in S$.

Then $z$ is said to be join irreducible in L if and only if

$z$ is join irreducible in the join semilattice $\struct{S, \vee, \preceq}$

Also known as

The term join irreducible is sometimes hyphenated as join-irreducible in some sources.

Also see

• Characterization of Join Irreducible Element

• Join Prime Element Iff Join Irreducible In Distributive Lattice

• Definition:Join Prime Element

• Definition:Meet Irreducible Element

Sources

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