= Piecewise integer-linear representation of a free Abelian lattice-ordered group
{title2=$F_n=\langle x_1,\ldots,x_n\rangle_\ell\subseteq\mathbb R^{\mathbb R^n}$}
Evaluate the generators as coordinate projections on $\mathbb R^n$. The <lattice-ordered group> they generate consists of continuous <positively homogeneous> <piecewise linear functions> with finitely many integer-linear pieces. Equivalently its elements are finite <joins> of finite <meets> of integer linear forms. The <universal property> follows because the real line detects every identity of <Abelian lattice-ordered groups>: if an identity fails, pass to a prime ordered quotient and select the finitely many active linear pieces. The resulting rational linear inequalities have a real solution by the alternative underlying <Farkas' lemma>, giving a real witness to failure. This is a statement about integer coefficients and positive homogeneity, not arbitrary real coefficients or affine constant terms.
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