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This paper completes the classification of the set $S$ of all real parameterpairs $(\alpha,\beta)$ such that the dilated floor functions $f_\alpha(x) =\lfloor{\alpha x}\rfloor$, $f_\beta(x) = \lfloor{\beta x}\rfloor$ have anonnegative commutator, i.e. $ [ f_{\alpha}, f_{\beta}](x) = \lfloor{\alpha\lfloor{\beta x}\rfloor}\rfloor - \lfloor{\beta \lfloor{\alphax}\rfloor}\rfloor \geq 0$ for all real $x$. This paper treats the case whereboth dilation parameters $\alpha, \beta$ are negative. This result isequivalent to classifying all positive $\alpha, \beta$ satisfying $\lfloor{\alpha \lceil{\beta x}\rceil}\rfloor - \lfloor{\beta \lceil{\alphax}\rceil}\rfloor \geq 0$ for all real $x$. The classification analysis isconnected with the theory of Beatty sequences and with the DiophantineFrobenius problem in two generators.

Dilated floor functions having nonnegative commutator II. Negative dilations