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 […]
Pointfree funcoid induced by a locale or frame?
I have shown in my research monograph that topological (even pre-topological) spaces are essentially (via an isomorphism) a special case of endo-funcoids. It was natural to suppose that locales or frames induce pointfree funcoids, in a similar way. But I just spent a few minutes on defining the pointfree funcoid corresponding to a locale or […]
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, […]
A new math problem about funcoids
Just a few seconds ago I realized that I have never considered and and even never formulated the following problem: Explicitly describe the set of complemented funcoids. Note that not all principal funcoids are complemented. For example see my book for a proof that the identity funcoid on some set is not complemented.
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 […]
A partial proof of “Partitioning a filter into ultrafilters” conjecture
I’ve put a partial partial proof of “Every filter on a set can be strongly partitioned into ultrafilters” conjecture at PlanetMath. Please collaborate in solving this conjecture.
Rough idea: Operations on singularities and their application to general relativity
I discovered a math theory which (among other things) gives an alternate interpretation of the equations of general relativity (something like to replacing real numbers with complex numbers in a quadratic equation). This theory is a theory of limits in points of singularities and properties of singularities based on my theory of funcoids (a new […]
Welcome to collaborative research on PlanetMath.org
Background information: I’ve written a math book and submitted it to a publisher. But afterward I found a serious error (a wrong proof of the conjecture below) and have withdrawn it from the publisher. Now the draft book is available online. (You will enjoy reading it if you are interested in general topology.) I welcome […]
A conjecture related with subatomic product
With subatomic products first mentioned here and described in this article are related the following conjecture (or being precise three conjectures): Conjecture For every funcoid $latex f: \prod A\rightarrow\prod B$ (where $latex A$ and $latex B$ are indexed families of sets) there exists a funcoid $latex \Pr^{\left( A \right)}_k f$ defined by the formula $latex […]
New math research wiki
I’ve created a new wiki site for math research. The motto of this wiki is “a research in the middle”. The site is intended to discuss research ideas, aspiring ways of research, usage of open problems and ways to prove open problems, etc. The exact rules are not yet defined, but I published several example […]