Join-prime element

Remarks

Every join-prime element is join-irreducible: if $a = b \vee c$, then $a \le b$ or $a \le c$, and since $b, c \le a$ this gives $a = b$ or $a = c$. In a distributive lattice the converse holds: if $a$ is join-irreducible and $a \le b \vee c$, then $a = a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$, so $a = a \wedge b$ or $a = a \wedge c$. Example: in the power set of a set, a singleton $\set{x}$ is join-prime, since $x \in B \cup C$ implies $x \in B$ or $x \in C$. Non-example: in the diamond lattice $M_3 = \set{0, a, b, c, 1}$, where $a$, $b$, $c$ are pairwise incomparable and the join of any two of them is $1$, the element $a$ is join-irreducible. It is not join-prime, because $a \le b \vee c = 1$ while $a \not\le b$ and $a \not\le c$. Meet-prime elements are defined dually.