Dedekind $\sigma$-complete $\ell$-groups and Riesz spaces as varieties

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 variety; further, we use this fact to obtain the even stronger result that ${\mathbb {R}}$ generates this category as a quasi-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.