Intersection of complete funcoids (Solved)

Importance: Medium ✭✭

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Keywords:	complete funcoid
	funcoid
	generalized closure

Recomm. for undergrads: no

Posted	by:	porton
	on:	August 9th, 2007

Solved by:

Porton, Victor

Conjecture If $f$ , $g$ are complete funcoids (generalized closures) then $f \cap^{\mathsf{\tmop{FCD}}} g$ is a complete funcoid (generalized closure).

See Algebraic General Topology for definitions of used concepts.

Below is also a weaker conjecture:

Conjecture If $f$ , $g$ are binary relations then $f\cap^{\mathsf{\tmop{FCD}}} g$ is a binary relation; or equivalently, $f\cap^{\mathsf{\tmop{FCD}}} g=f\cap g$ for any binary relations $f$ and $g$ .

The author has found a counterexample against this weaker conjecture and thus against the main conjecture. The example is $f = {(=)} |_{\mho}$ and $g = \mho\times\mho \setminus f$ . It is simple to show that $f\cap^{\mathsf{\tmop{FCD}}} g = {(=)} |_{\Omega}$ where $\Omega$ is the Fréchet filter and thus $f\cap^{\mathsf{\tmop{FCD}}} g \ne \emptyset = f\cap g$ .