Transfer of properties between a truncated Riesz space and its unitization

Abstract

We study the transfer of fundamental order and completeness properties between a truncated Riesz space and its Alexandroff unitization. In particular, we distinguish genuine completeness notions, including relatively uniform completeness, Dedekind completeness, lateral completeness, universal completeness, and the (principal) projection property, from order-theoretic properties such as the Archimedean property. We establish characterizations and equivalences for several of these notions and provide counterexamples to highlight the necessity of certain assumptions and the independence of different completeness properties.