Definition:Join Irreducible Element/Lattice

Definition

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

• Join Prime Element is Join Irreducible

• Join Prime Element Iff Join Irreducible In Distributive Lattice

• Definition:Join Prime Element

• Definition:Meet Irreducible Element (Lattice)

Sources

• 1971: George A. Grätzer: Lattice Theory: Chapter $1$: First Concepts, $\S 6$: Special Elements

• 2002: B.A. Davey and H.A. Priestley: Introduction to Lattices and Order (2nd ed.): Definition $2.42$