  Combining tools from category theory, model theory, and non-standard analysis
we extend Baker-Beynon dualities to the classes of all Abelian $\ell$-groups
and all Riesz spaces (also known as vector lattices). The extended dualities
have a strong geometrical flavor, as they involve a non-standard version of the
category of polyhedral cones and piecewise (homogeneous) linear maps between
them. We further show that our dualities are induced by the functor
$\mathtt{Spec}$, once it is understood how to endow it with "coordinates" in
some ultrapower $\mathcal{U}$ of $\mathbb{R}$. This also allows us to
characterize the topological spaces arising as spectra of Abelian $\ell$-groups
and Riesz spaces as certain subspaces of $\mathcal{U}^{\kappa}$ endowed with
the Zariski topology given by definable functions in their respective
languages. Furthermore, we provide some applications of the extended duality by
characterizing, in geometrical terms, semisimplicity, Archimedeanity, and the
existence of weak and strong order-units. Finally, we show that our dualities
afford a neat and simpler proof of Panti's celebrated characterization of the
prime ideals in free Abelian $\ell$-groups and Riesz spaces.
