Intersection of complete funcoids (Solved)

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
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 $.

See the section "Some counter-examples" in the online article "Funcoids and Reloids" for details.


*Victor Porton. Algebraic General Topology

* indicates original appearance(s) of problem.

Please improve presentation!

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