# On the linearity of order-isomorphisms

## TL;DR

This paper characterizes conditions under which order-isomorphisms in certain partially ordered vector spaces are linear, extending previous results to infinite dimensions and broader classes of cones.

## Contribution

It introduces a milder condition for linearity of order-isomorphisms, extending known results to infinite-dimensional spaces and general cones.

## Key findings

- Order-isomorphisms are linear for every Archimedean cone satisfying the new condition.
- Provides a general form of order-isomorphisms on specific cones.
- Proves linearity of homogeneous order-isomorphisms in new settings.

## Abstract

A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein-Avidan and Slomka to infinite dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.