I generalized a theorem in the preprint article “Filters on posets and generalizations” on my Algebraic General Topology site.

The new theorem is formulated as following:

Theorem If $(\mathfrak{A}; \mathfrak{Z})$ is a join-closed filtrator and $\mathfrak{A}$ is a meet-semilattice and $\mathfrak{Z}$ is a complete lattice, then $\mathrm{Cor}' (a \cap^{\mathfrak{A}} b) = \mathrm{Cor}' a \cap^{\mathfrak{Z}} \mathrm{Cor}' b.$

From this theorem also follow the following formulas about funcoids:

• $\mathrm{Compl}(f\cap^{\mathsf{FCD}}g) = \mathrm{Compl}f\cap^{\mathsf{FCD}}\mathrm{Compl}g$.
• $\mathrm{CoCompl}(f\cap^{\mathsf{FCD}}g) = \mathrm{CoCompl}f\cap^{\mathsf{FCD}}\mathrm{CoCompl}g$.

The problem, whether meet of any two discrete funcoids is the same on the lattice of all funcoids and on the lattice of discrete funcoids, remains empty.