On the Transfer of Completeness and Projection Properties in Truncated Vector Lattices

In this work we investigate the transfer of fundamental order and completeness properties between truncated Riesz spaces and their unitizations. Specifically, we provide characterizations and equivalences for several notions of completeness: the Archimedean property, relatively uniform completeness, Dedekind completeness, lateral completeness, universal completeness, and the projection property. Counterexamples are presented to illustrate the necessity of assumptions and the independence of various completeness notions.