  We extend Yosida's 1941 version of Stone-Gelfand duality to metrically
complete unital lattice-ordered groups that are no longer required to be real
vector spaces. This calls for a generalised notion of compact Hausdorff space
whose points carry an arithmetic character to be preserved by continuous maps.
The arithmetic character of a point is (the complete isomorphism invariant of)
a metrically complete additive subgroup of the real numbers containing $1$,
namely, either $\frac{1}{n}\mathbb{Z}$ for an integer $n = 1, 2, \dots$, or the
whole of $\mathbb{R}$. The main result needed to establish the extended duality
theorem is a substantial generalisation of Urysohn's Lemma to such "arithmetic"
compact Hausdorff spaces. The original duality is obtained by considering the
full subcategory of spaces whose each point is assigned the entire group of
real numbers. In the introduction we indicate motivations from and connections
with the theory of dimension groups.
