TITLE: Have arrows in a category with this property a special name?
QUESTION [5 upvotes]: Studying posets I encountered the notation $a\prec b$. It means
that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then
in words $a$ is covered by $b$. Looking at a poset $P$ as a category
you could say that for the arrow $f$ in $P\left(a,b\right)$ we have:
$$f=g\circ h\Rightarrow g=1\vee h=1$$ It reminds me of elements in
a ring that are irreducible. 
Suppose that more generally in a category $\mathcal{C}$ there is
an arrow $f$ - which is not an isomorphism itself - satisfying: $$f=g\circ h\Rightarrow g\text{ is isomorphism}\vee h\text{ is isomorphism}$$
Is there a name (p.e. irreducible) for arrows having that property?

REPLY [4 votes]: Auslander and Reiten (Representation theory of artin algebras) introduced the notion of an irreducible morphism, but it differs from your notion: A morphism $g$ is irreducible if $g$ is neither a split mono nor a split epi, and if, for any factorization $g=fh$, either $f$ is a split mono or $h$ is a split epi.
For monoids (regarded as categories with one object), this reads: $g$ is irreducible, if $g$ is neither left nor right invertible, and if, for any factorization $g=fh$, either $f$ is left invertible or $h$ is right invertible. But your notion becomes the usual notion of irreducibility (no unit, and in every factorization a unit occurs). In the commutatice case, there is no difference.
I think it really depends on the application which notion one uses.