Join-irreducible elements of a nonempty finite lattice

Let $L$ be a nonempty finite lattice. Its least element, whose existence is proved in A finite lattice has a bottom and a top, and every element is the join of the join-irreducible elements below it ↗, is denoted $0_L$. An element $j\in L$ is join-irreducible if $j\ne0_L$ and

$j=a\vee b\quad\Longrightarrow\quad j=a\text{ or }j=b$

for all $a,b\in L$. The set of join-irreducible elements, with the order inherited from $L$, is denoted $\operatorname{JI}(L)$.

An element $j$ is join-prime if

$j\le a\vee b\quad\Longrightarrow\quad j\le a\text{ or }j\le b.$

Join-prime implies join-irreducible whenever $j\ne0_L$. In a distributive lattice the converse holds for join-irreducible elements.