  This paper completes the classification of the set $S$ of all real parameter
pairs $(\alpha,\beta)$ such that the dilated floor functions $f_\alpha(x) =
\lfloor{\alpha x}\rfloor$, $f_\beta(x) = \lfloor{\beta x}\rfloor$ 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 $x$. This paper treats the case where
both dilation parameters $\alpha, \beta$ are negative. This result is
equivalent to classifying all positive $\alpha, \beta$ satisfying $
\lfloor{\alpha \lceil{\beta x}\rceil}\rfloor - \lfloor{\beta \lceil{\alpha
x}\rceil}\rfloor \geq 0$ for all real $x$. The classification analysis is
connected with the theory of Beatty sequences and with the Diophantine
Frobenius problem in two generators.
