Join-prime element

Summary

An element $a$ of a lattice is join-prime if it is not the bottom element and has the following property. Whenever $a$ lies below a join $b \vee c$, it lies below $b$ or below $c$. The definition copies that of a prime element of a ring, with "divides" replaced by "lies below" and the product replaced by the join. Every join-prime element is join-irreducible, and in a distributive lattice the two notions coincide.