I’ve noticed the following three conjectures (I expect not very difficult) for finite binary relations $latex X$ and $latex Y$ between some sets and am going to solve them:

- $latex X\sqcap^{\mathsf{FCD}} Y = X\sqcap Y$;
- $latex (\top \setminus X)\sqcap^{\mathsf{FCD}} (\top \setminus Y) = (\top \setminus X)\sqcap (\top \setminus Y)$;
- $latex (\top \setminus X)\sqcap^{\mathsf{FCD}} Y = (\top \setminus X)\sqcap Y$.

I’ve proved the first one. I am going to publish the (easy) proof soon.

All three conjectures follow from the fact that $latex \Gamma$ is a sublattice of $latex \mathsf{FCD}$.