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July 26, 2016 / porton

A vaguely formulated problem

Consider funcoid \mathrm{id}^{\mathsf{FCD}}_{\Omega} (restricted identity funcoids on Frechet filter on some infinite set).

Naturally 1\in\mathrm{up}\, \mathrm{id}^{\mathsf{FCD}}_{\Omega} (where 1 is the identity morphism).

But it also holds \top^{\mathsf{FCD}}\setminus 1\in\mathrm{up}\, \mathrm{id}^{\mathsf{FCD}}_{\Omega} (where 1 is the identity morphism). This result is not hard to prove but quite counter-intuitive (that is is a paradox).

I think that we should find modified \mathrm{up} (let’s denote it \mathrm{up}') such that 1\in\mathrm{up}'\, \mathrm{id}^{\mathsf{FCD}}_{\Omega} but \top^{\mathsf{FCD}}\setminus 1\notin\mathrm{up}'\, \mathrm{id}^{\mathsf{FCD}}_{\Omega}.

Currently I cannot formulate this problem exactly (what is \mathrm{up}'?) but I think (if you read my book) you can understand what I want.

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