Order isomorphisms of sup-stable function spaces: continuous, Lipschitz, c-convex, and beyond

Abstract: 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 $\phi:\mathcal{X}\to \mathcal{X}$. Given a reference anti-isomorphism, this characterization then allows to recover all the other anti-isomorphisms. 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 1-Lipschitz functions of complete metric spaces. The latter application is obtained using properties of the horoboundary of a metric space.