  Let $R$ be a commutative Noetherian $F$-finite ring of prime characteristic
and let $\mathcal{D}$ be a Cartier algebra. We define a self-map on the
Frobenius split locus of the pair $(R,\mathcal{D})$ by sending a point $P$ to
the splitting prime of $(R_P, \mathcal{D}_P)$. We prove this map is continuous,
containment preserving, and fixes the $\mathcal{D}$-compatible ideals. We show
this map can be extended to arbitrary ideals $J$, where in the Frobenius split
case it gives the largest $\mathcal{D}$-compatible ideal contained in $J$.
Finally, we apply Glassbrenner's criterion to prove that the prime uniformly
$F$-compatible ideals of a Stanley-Reisner rings are the sums of its minimal
primes.
