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May 14, 2017 / porton

The mystery of meet of funcoids solved?

It is not difficult to prove (see “Counter-examples about funcoids and reloids” in the book) that 1^{\mathsf{FCD}} \sqcap^{\mathsf{FCD}} (\top\setminus 1^{\mathsf{FCD}}) = \mathrm{id}^{\mathsf{FCD}}_{\Omega} (where \Omega is the cofinite filter). But the result is counterintuitive: meet of two binary relations is not a binary relation.

After proving this I always felt that there is some “mystery” about meet of funcoids: It behaves in a weird way and what it is in general (not this one special counterexample case) is not known.

Today I noted a simple formula which decomposes f \sqcap^{\mathsf{FCD}} g: f \sqcap^{\mathsf{FCD}} g = (\mathsf{FCD})((\mathsf{RLD})f \sqcap (\mathsf{RLD})g) for every funcoids f and g and more generally \bigsqcap^{\mathsf{FCD}} S = (\mathsf{FCD}) \bigsqcap^{\mathsf{RLD}} \langle (\mathsf{RLD})_{\mathrm{in}} \rangle^{\ast} S for a set S of funcoids. (It follows from that (\mathsf{FCD}) is an upper adjoint and that (\mathsf{FCD})(\mathsf{RLD})_{\mathrm{in}} f=f for every funcoid f.) This way it looks much more clear and less counterintuitive.

So now it looks more clear, but I have not yet found particular implications of these formulas leading to interesting results.

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