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Intersection of complete funcoids (Solved)
Conjecture If
,
are complete funcoids (generalized closures) then
is a complete funcoid (generalized closure).
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
![$ f \cap^{\mathsf{\tmop{FCD}}} g $](/files/tex/41cbe85698929e50b9b99f1ce7f0e4212f18737d.png)
See Algebraic General Topology for definitions of used concepts.
Below is also a weaker conjecture:
Conjecture If
,
are binary relations then
is a binary relation; or equivalently,
for any binary relations
and
.
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
![$ f\cap^{\mathsf{\tmop{FCD}}} g $](/files/tex/866dbaca78b1cb59d21cb0ff85d584323b7c004e.png)
![$ f\cap^{\mathsf{\tmop{FCD}}} g=f\cap g $](/files/tex/22b042386cd914c6ff7eb6c559c6cfcf30d13f0c.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
The author has found a counterexample against this weaker conjecture and thus against the main conjecture. The example is and
. It is simple to show that
where
is the Fréchet filter and thus
.
See the section "Some counter-examples" in the online article "Funcoids and Reloids" for details.
Bibliography
*Victor Porton. Algebraic General Topology
* indicates original appearance(s) of problem.
Please improve presentation!
Please, provide
1) definitions of the used concepts (to make the statement self-contained)
2) motivation (why this is important, examples, ...)
At the present state, this text is unfortunately not very useful for someone not acquainted with your manuscripts.