Dear Quentin,
An element $x$ in a (multiplicatively written) Dedekind-finite monoid $H$ is a non-unit $a \in H$ such that $a \ne xy$ for all non-units $x, y \in H$ such that $x$ and $y$ are proper divisors of $a$ in $H$ (meaning that each of the two-sided principal
ideals generated by $x$ and $y$ properly contains the two-sided principal ideal generated by $a$). Considering that every commutative monoid is Dedekind-finite:
- A bounded semilattice with identity element $e$ is a commutative monoid whose only unit is $e$, hence a join-irreducible element in a join-semilattice with a bottom element is precisely an irreducible element as per the above definition.
- Regarding "colimit-irreducible elements" in a category $C$ with all finite colimits, the (equivalence) classes made up of the isomorphic objects of $C$ form a commutative monoid $V(C)$ with a trivial group of units when endowed with the binary operation
$([A], [B]) \mapsto [A \coprod B]$, where $[ \cdot ]$ denotes the isomorphism class of an object. An object of $C$ is then
colimit-irreducible (in the sense that it is isomorphic neither to the initial object nor to the coproduct of two non-initial objects) if and only if its isomorphism class is irreducible in the monoid $V(C)$.
Best,
Salvo