Alexandroff unitization of a truncated vector lattice

Abstract

Recently Ball introduced the remarkable concept of truncated vector lattices and studied their representations by continuous functions. Weakening the original Ball definition, we introduce the notion of an unitization of a truncated vector lattice T and we prove that T has a smallest unitization $T^{*}$, called the Alexandroff unitization of T. We show that $T^{*}$ has universal properties and we prove that T is dense in $T^{*}$ if and only if T is not unital. Otherwise, T and its polar are cardinal summands for $T^{*}$. All these facts are also interpreted in category-theoretic terms.