  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.
