Join-irreducible element

Remarks

Clause 2 says that $a$ is not the join of two elements strictly below it. Example: in the power set of a set $X$, each singleton $\set{x}$ is join-irreducible. It is not $\emptyset$, and if $\set{x} = B \cup C$ then $B$ or $C$ contains $x$, so it equals $\set{x}$. Non-example: $\set{1, 2} = \set{1} \cup \set{2}$ with $\set{1} \ne \set{1, 2}$ and $\set{2} \ne \set{1, 2}$, so clause 2 fails. In a chain, every element other than the least element is join-irreducible, since $b \vee c = \max(b, c)$. In a finite lattice, every element is the join of the join-irreducible elements below it.