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August 22, 2014 / porton

Funcoids are filters?

I am not doing math research this month (because a bug in TeXmacs software which I use for writing my book and articles). I instead do writing some free software not to waste my time.

But today (this hour) I unexpectedly had a new interesting idea about my math research:

Let denote Q the set of finite joins of funcoidal products of two principal filters.

Conjecture The poset of funcoids is order-isomorphic to the set of filters on the set Q (moreover the isomorphism is (possibly infinite) meet of the filter).

If proved positively, this may reveal new properties of funcoids and probably solve some of my open problems.

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3 Comments

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  1. porton / Aug 23 2014 09:10

    I’ve provided a counter-example for this conjecture (or rather an equivalent conjecture presented in elementary terms).

    The counter-example is available here:
    http://math.stackexchange.com/questions/906543/a-conjecture-about-filters-and-finite-unions-of-cartesian-products/906626

  2. porton / Aug 25 2014 02:55

    Clarification to my previous comment. To disprove the conjecture it is enough to prove that \bigsqcap^{\mathsf{FCD}} \mathcal{A} = \bigsqcap^{\mathsf{FCD}} \mathcal{B} (that is \left\langle \bigsqcap^{\mathsf{FCD}} \mathcal{A} \right\rangle a = \left\langle \bigsqcap^{\mathsf{FCD}} \mathcal{B} \right\rangle a for every ultrafilter a) does not imply \mathcal{A} = \mathcal{B}.

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