  A unital $\ell$-group is an abelian group equipped with a translation
invariant lattice-order and with a distinguished strong unit, i.e. an element
whose positive integer multiples eventually dominate every element of $G$.If
$X$ is a compact subset of $R^n$, the set $M(X)$ of real-valued piecewise
linear maps with integer coefficients, whose addition and lattice operations
defined pointwise and whose distinguished element is the constant map $1$, is a
unital $\ell$-group.
  In this paper we provide a geometric decription of finitely generated
(regular) projective unital $\ell$-groups. We prove that a finitely unital
$\ell$-group is projective if and only if it is isomorphic to $M(P)$ for some
polyhedron $P$ which is rational, contractible, contains an integer point, and
satisfies an elementary arithmetical-topological property.
