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 $f^0$.