$(\mathcal{A}\ltimes\mathcal{B})\sqcap(\mathcal{A}\rtimes\mathcal{B})=\mathcal{A}\times_{F}^{\mathsf{RLD}}\mathcal{B}$ for every filters $\mathcal{A}$, $\mathcal{B}$ on sets.
(Here $\ltimes$ and $\rtimes$ is what I call oblique products.)