  There have been many parallel streams of research studying order isomorphisms
of some specific sets $\mathcal{G}$ of functions from a set $\mathcal{X}$ to
$\mathbb{R}\cup\{\pm\infty\}$, such as the sets of convex or Lipschitz
functions. We provide in this article a unified abstract approach inspired by
$c$-convex functions. Our results are obtained highlighting the role of inf and
sup-irreducible elements of $\mathcal{G}$ and the usefulness of characterizing
them, to subsequently derive the structure of order isomorphisms, and in
particular of those commuting with the addition of scalars. We show that in
many cases all these isomorphisms $J:\mathcal{G}\to\mathcal{G}$ are of the form
$Jf=g+f\circ \phi$ for a translation $g:\mathcal{X}\to\mathbb{R}$ and a
bijective reparametrization $\phi:\mathcal{X}\to \mathcal{X}$. We apply our
theory to the sets of $c$-convex functions on compact Hausdorff spaces, to the
set of lower semicontinuous (convex) functions on a Hausdorff topological
vector space and to Lipschitz and 1-Lipschitz functions of complete metric
spaces.
