= Integer-linear hinge functions have infinite independent rank
{title2=$h_n(x,y)=\max(0,x-ny)$}
The <piecewise linear functions> $h_n$, for distinct integer indices, are linearly independent over the integers. Restrict a finite integer relation to $y=1$. At $x=n$, only $h_n$ has a jump of one in its slope; hence the coefficient of that function must be zero. This proves that the underlying group of the two-generated <free Abelian lattice-ordered group> has infinite <rank of an abelian group>.
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