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 […]
Definition of order of pointfree funcoids changed in my book
In this blog post I announced that I am going to change the definition of order of pointfree funcoids in my book. Now in the last preprint the changes are done.
Changed the definition of order of pointfree funcoids
In my preprint I defined pre-order of pointfree funcoids by the formula $latex f\sqsubseteq g \Leftrightarrow [f]\subseteq[g]$. Sadly this does not define a poset, but only a pre-order. Recently I’ve found an other (non-equivalent) definition of an order on pointfree funcoids, this time this is a partial order not just a pre-order: $latex f \sqsubseteq […]
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 […]
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, […]
(Not math) My spiritual experience related with the theory of funcoids
This post is not about mathematics. It is about spirituality. In the very beginning of my research, when I was formulating the definition of funcoids I felt certain spiritual experience. While thinking about it, I felt myself in a kind of virtual reality, at the same time not only sitting in a chair but also […]
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 […]