Dedekind sigma-complete l-groups and Riesz spaces as varieties

TL;DR

This paper shows that Dedekind sigma-complete Riesz spaces form an infinitary variety, providing explicit axioms and demonstrating that the real numbers generate this category as a variety, with similar results for related structures.

Contribution

It establishes that Dedekind sigma-complete Riesz spaces and related categories are varieties with explicit axiomatizations, extending classical algebraic frameworks.

Findings

Dedekind sigma-complete Riesz spaces form an infinitary variety.

The real numbers generate this category as a variety.

Similar results hold for related ordered algebraic structures.

Abstract

We prove that the category of Dedekind $\sigma$-complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely many axioms suffice over the usual equational axiomatization of Riesz spaces. Our main result is that $\mathbb{R}$, regarded as a Dedekind $\sigma$-complete Riesz space, generates this category as a quasi-variety, and therefore as a variety. Analogous results are established for the categories of (i) Dedekind $\sigma$-complete Riesz spaces with a weak order unit, (ii) Dedekind $\sigma$-complete lattice-ordered groups, and (iii) Dedekind $\sigma$-complete lattice-ordered groups with a weak order unit.