  In this paper and its sequel we classify the set $S$ of all real parameter
pairs $(\alpha,\beta)$ such that the dilated floor functions $f_\alpha(x) =
\lfloor{\alpha x}\rfloor$ and $f_\beta(x) = \lfloor{\beta x}\rfloor$ have a
nonnegative commutator, i.e. $ [ f_{\alpha}, f_{\beta}](x) = \lfloor{\alpha
\lfloor{\beta x}\rfloor}\rfloor - \lfloor{\beta \lfloor{\alpha
x}\rfloor}\rfloor \geq 0$ for all real $x$. The relation
$[f_\alpha,f_\beta]\geq 0$ induces a preorder on the set of non-zero dilation
factors $\alpha, \beta$, which extends the divisibility partial order on
positive integers. This paper treats the cases where at least one of the
dilation parameters $\alpha$ or $\beta$ is nonnegative. The analysis of the
positive dilations case is related to the theory of Beatty sequences and to the
Diophantine Frobenius problem in two generators.
