  Assume that $X$ is an affine toric variety of characteristic $p > 0$. Let
$\Delta$ be an effective toric $Q$-divisor such that $K_X+\Delta$ is
$Q$-Cartier with index not divisible by $p$ and let $\phi_{\Delta}:F^e_* O_X
\to O_X$ be the toric map corresponding to $\Delta$. We identify all ideals $I$
of $O_X$ with $\phi_{\Delta}(F^e_* I)=I$ combinatorially and also in terms of a
log resolution (giving us a version of these ideals which can be defined in
characteristic zero). Moreover, given a toric ideal $\ba$, we identify all
ideals $I$ fixed by the Cartier algebra generated by $\phi_{\Delta}$ and $\ba$;
this answers a question by Manuel Blickle in the toric setting.
