Unitization of Ball truncated ${\ell}$-groups

Abstract

Recently, Ball defined a truncated ${\ell}$-group to be an ${\ell}$-group G along with a truncation. We constructively prove that if G is a truncated ${\ell}$-group, then the direct sum ${G \oplus \mathbb{Q}}$ is equipped with a structure of an ${\ell}$-group with weak unit the rational number 1. As a simple consequence, we get a description of the truncated ${\ell}$- group obtained by Ball via representation theory. On the other hand, we derive some characterizations of truncation morphisms as defined by Ball himself. In particular, we show that the group homomorphism ${f : G \rightarrow H}$ is a truncation morphism if and only its natural extension ${f^*}$ from ${G \oplus \mathbb{Q}}$ into ${H \oplus \mathbb{Q}}$ is an ${\ell}$-homomorphism.