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 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 (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.

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

Clarification to my previous comment. To disprove the conjecture it is enough to prove that (that is for every ultrafilter ) does not imply .