I’ve proved the conjecture about composition of reloids through atomic reloids
From a new version of preprint of my book: Corollary 7.18 If $latex f$ and $latex g$ are composable reloids, then $latex g \circ f = \bigsqcup \left\{ G \circ F \, | \, F \in \mathrm{atoms}\, f, G \in \mathrm{atoms}\, g \right\}$.
Equalizers in certain categories
In this my rough draft article I construct equalizers for certain categories (such as the category of continuous maps between endofuncoids). Products and co-products were already proved to exist in my categories, so these categories are complete. In the above mentioned article I also claim co-equalizers (however have not yet proved that they are really […]
A draft proof of distributivity of composition with a principal reloid over join of reloids
Recently I’ve announced that I have an elegant proof idea of this conjecture, but have a trouble to fill in details of the proof: Statement Composition with a principal reloid is distributive over join of reloids. Now I have almost complete draft proof of the above statement (Well, it is yet to be checked for […]
A conjecture about atomic funcoids
A new conjecture: Conjecture $latex \langle f \rangle \mathcal{X} = \bigsqcup_{F \in \mathrm{atoms}\, f} \langle F \rangle \mathcal{X}$ for every funcoid $latex f$ and $latex \mathcal{X} \in \mathfrak{F} (\mathrm{Src}\, f)$. This conjecture seems important for the notion of exponential object in the category of continuous maps between endofuncoids, which I am investigating now.
One more conjecture about provability without axiom of choice
I addition to this conjecture I formulate one more similar conjecture: Conjecture $latex a\setminus^{\ast} b = a\#b$ for arbitrary filters $latex a$ and $latex b$ on a powerset cannot be proved in ZF (without axiom of choice). Notation (where $latex \mathfrak{F}$ is the set of filters on a powerset ordered reverse to set-theoretic inclusion): $latex […]
Conjecture: Distributivity of a lattice of funcoids is not provable without axiom of choice
Conjecture Distributivity of the lattice $latex \mathsf{FCD}(A;B)$ of funcoids (for arbitrary sets $latex A$ and $latex B$) is not provable in ZF (without axiom of choice). It is a remarkable conjecture, because it establishes connection between logic and a purely algebraic equation. I have come to this conjecture in the following way: My proof that […]
My further study plans
I remind that I am not a professional mathematician. Nevertheless I have written research monograph “Algebraic General Topology. Volume 1”. Yesterday I have asked on MathOverflow how to characterize a poset of all filters on a set. From the answer: the posets isomorphic to lattices of filters on a set are precisely the atomic compact […]
A (possibly open) problem about filters on a set
http://mathoverflow.net/questions/139608/a-characterization-of-the-poset-of-filters-on-a-set For the lattices of all subsets of a given set it is known an axiomatic characterization: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra. The question: How to characterize the sets of filters on a set? That is having a poset, […]
My conjecture partially solved
I’ve partially solved my conjecture, proposed Polymath problem described at this page. The problem asks which of certain four expressions about filters on a set are always pairwise equal. I have proved that the first three of them are equal, equality with the fourth remains an open problem. For the (partial) solution see this online […]
The history of discovery of funcoids
In my book I introduce funcoids as a generalization of proximity spaces. This is the most natural way to introduce funcoids, but it was not the actual way I’ve discovered them. The first thing discovered equivalent to funcoids was a function $latex \Delta$ (generalizing a topological space) which I defined to get a set as […]