1. Prime/irreducible elements

In    Z$$, two concepts coincide:

(i) $p$ is irreducible if $p = ab$ implies $p = a$ or $p = b$, and $p > 1$;

(ii) $p$ is prime if $p \vert ab$ implies $p \vert a$ or $p \vert b$, and $p > 1$.

Here $a$ and $b$ are positive integers.

In a lattice, one can make similar definitions:

(i) $p$ is join-irreducible if $p = a \vee b$ implies $p = a$ or $p = b$, and $p > 0$;

(ii) $p$ is join-prime if $p \leq a \vee b$ implies $p \leq a$ or $p \leq b$, and $p > 0$.

Examples: (a) In $M _ 3$ (which is not distributive), each atom is join-irreducible, but not join-prime. (b) In    R$$, every element is both.

Some basic facts for lattices in general:

Observation 1. In any finite lattice, an element is join-irreducible if and only if it covers exactly one other element.

Observation 2. In any lattice, $p$ is join-prime $\Rightarrow$ $p$ is join-irreducible.

Observation 3. In any finite lattice, every element is a join of join-irreducible elements.

(Here $0$ is considered to be the join of no elements.)

Lemma. In a distributive lattice, $p$ is join-prime $\Leftrightarrow$ $p$ is join-irreducible.