= Solution
A concrete model of the two-generated <free Abelian lattice-ordered group> is the lattice subgroup of real-valued functions on $\mathbb R^2$ generated by the coordinate projections $X(x,y)=x$, $Y(x,y)=y$. Its elements are finite <joins> of finite <meets> of integer linear forms $ax+by$. Thus they are continuous <positively homogeneous> <piecewise linear functions>, with pointwise addition and lattice operations. This is the <piecewise integer-linear representation of a free Abelian lattice-ordered group>.
For the construction, first form the term algebra on two symbols using addition, negation, zero, <join> and <meet>, then identify terms that agree in every <Abelian lattice-ordered group>. Evaluation under any assignment of the two generators factors through this quotient, proving its <universal property>. To obtain the concrete function model, use the identity-detection theorem that the real line generates the variety of Abelian lattice-ordered groups. Its proof passes a failed identity to a prime ordered quotient, chooses its finitely many active linear pieces, and uses the rational-inequality alternative underlying <Farkas' lemma> to find a real witness. This makes evaluation on $\mathbb R^2$ injective as well as surjective onto the generated function group.
\b[Its underlying group is not $\mathbb Z^n$ for any finite $n$.] The <integer-linear hinge functions have infinite independent rank>: $h_k(x,y)=(x-ky)\vee0$ are integer-linearly independent for distinct integers $k$. On $y=1$, only $h_k$ has a slope jump at $x=k$, so each coefficient in a finite relation must vanish. The underlying group therefore has infinite <rank of an abelian group> despite having two lattice generators.
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