Consider the map
$$f:\mathbb C^2\to\mathbb C^2$$
$$(x,y)\mapsto(x^2y,xy^2)$$
We can view $f$ as induced by the map of monoids $g:\mathbb Z^2\_{\geq 0}\to\mathbb Z^2\_{\geq 0}$ given by the matrix $(\begin{smallmatrix}2&1\cr 1&2\end{smallmatrix})$. Thus $f$ is a map of log schemes.



> 
> Is $f$ log smooth?
> 
> 
> 


If I understand Proposition 6.1 from <https://arxiv.org/abs/alg-geom/9406004> correctly, the answer is supposed to be yes, since $\ker g$ and $(\operatorname{coker} g)\_{\mathrm{tors}}$ are both finite.


On the other hand, log smooth maps are supposed to be flat, and flat maps are open. Thus if $f$ were log smooth it would have to be open. But it is not open: $(0,0)$ is in the image of $f$, but $(0,a)$ is not whenever $a\ne 0$.



Is $(x^2y,xy^2)$ log smooth?
It is smooth. For a log map to have good "fiber bundle properties", one needs more than smoothness. Rather, one needs the relevant maps of monoids to be **exact** in the sense of Kato. A reference is Nakayama--Ogus "Relative rounding in toric and logarithmic geometry" [Mathscinet](https://mathscinet.ams.org/mathscinet-getitem?mr=2740645) [Journal](https://msp.org/gt/2010/14-4/p09.xhtml).


