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April 20, 2010 / porton

A theorem generalized

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.

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