Unitization of a lattice ordered ring with a truncation

TL;DR

This paper investigates conditions under which the unitization of a lattice ordered ring with a truncation remains a lattice ordered ring, focusing especially on the unique truncation in the Archimedean case.

Contribution

It provides a necessary and sufficient condition for the unitization to be a lattice ordered ring and identifies the unique truncation in the Archimedean setting.

Findings

Unitization preserves lattice order under specific conditions.

There is at most one truncation making the unitization a lattice ordered ring.

The unique truncation in the Archimedean case is characterized.

Abstract

Let $R$ be a lattice ordered ring along with a truncation in the sense of Ball. We give a necessary and sufficient condition on $R$ for its unitization $R\oplus\mathbb{Q}$ to be again a lattice ordered ring. Also, we shall see that $R\oplus\mathbb{Q}$ is a lattice ordered ring for at most one truncation. Particular attention will be paid to the Archimedean case. More precisely, we shall identify the unique truncation on an Archimedean $\ell$-ring $R$ which makes $R\oplus\mathbb{Q}$ into a lattice ordered ring.