F.Kato has a statement said that a log smooth curve $f:(X, M,\alpha)\to (k,N,\beta)$ where $k$ is an algebraic closed field and $N$ is some fine log structure on $k$, is equvalent to pointed node curve in a paper "Log sooth deformation and moduli of log smooth curves". I am quite confused with the log structure $N$ on k, when one associate a log smooth curve to a pointed node curve.


Let's consider the following example, $X$ has only one double point $x$, and only one marked point $s$, and $x\neq s$. Let $A=k[t\_1, t\_2]/(t\_1t\_2)$, $U=Spec (A)$ be an etale neighborhood of $x$ , with $x= (0,0)$, we see that the log structure over $U$ is defined by the following $\alpha:M\_U:=k^\*\oplus N^2 \to A $ by $\alpha(k\_1,n\_1,n\_2)= k\_1t\_1^{n\_1}t\_2^{n\_2}$, and the log structure $N:=k^\*\oplus \mathbb{N}$ and the map $\beta:N\to k $ defined by $\beta(k\_1,0)=k\_1$, and $\beta(k\_1,n)=0$ for $n>0$.
The map $f$ on the monoid is defined by the following $f^0:N\to M\_U$, $f^0(k\_1,n)=(k\_1,n,n)$.


On the other hand Let $B=k[T]$ , $V=Spec(B)$, be an etale neighborhood of $s$, with $s=(0)$. Then the log structure on V is $M\_V:=k^\*\oplus \mathbb{N}$ with $\alpha(k\_1,n)=k\_1T^n$.


Now F. Kato claimed one extend $M\_U$, $M\_V$ to $X$, and define $M:=M\_U\oplus\_{\mathcal{O}^{\times}\_{X}} M\_V$, then $(X, M, \alpha)$ is log smooth over $(k, N, \beta)$. My question is how to define the map $f^0 : N\to M$ on the monoid? Over $U$, it is already given above. But what about over V, it seem that it is impossible to define such a map, for one need to satisfy the following relation:
$ \alpha\circ f^0= i\circ \beta$, where $i: k\to k[T]$ is the natural map. 
And one fine that $\alpha\circ f^0(k\_1, n)=i(0)=0 $ for $n>0$, however $0\notin Imge( \alpha)$, so it is impossible to define $f^0$. 



log smooth curve vs pointed node curve
You haven't really explained your notation, but it seems that you are covering the log curve $X$ with two neighborhoods $U$ and $V$, where $U$ contains a node, and $V$ contains a log point. I think the problem with your definition of $M\_V$ is that you forgot about the nontrivial log structure on the base log scheme. If I'm not mistaken, the neighborhood a log point on the curve should have log structure associated to $k^\* \times \mathbb{N}^2$ instead of $k^\* \times \mathbb{N}$, with $\alpha(k\_1,m,n) = k\_1 0^m T^n$ and $f^0(k\_1,m) = (k\_1,m,0)$.


