A new theorem about funcoids and generalizated filter bases
I proved the following theorem: Theorem If $latex S$ is a generalized filter base then $latex \left\langle f \right\rangle \bigcap{\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} \left\langle\left\langle f \right\rangle \right\rangle S$ for every funcoid $latex f$. The proof (presented in updated version of this online article) is short but not quite trivial. It was originally formulated as […]
A surprisingly hard problem
I am now trying to prove or disprove this innocently looking but somehow surprisingly hard conjecture: Conjecture If $latex S$ is a generalized filter base then $latex \left\langle f \right\rangle \bigcap{\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} \left\langle\left\langle f \right\rangle \right\rangle S$ for every funcoid $latex f$.
Orderings of filters in terms of reloids – in terms of category theory
My draft article “Orderings of filters in terms of reloids. Extensions of Rudin-Keisler ordering” is updated. Read here. Now it uses the language of basic category theory. It is yet not carefully checked for errors.
My motivation to do math research
In the past I considered my purpose to exactly and directly follow commandments of Bible. I had some purposes hardly set as the aim of my life. My life was driven by these purposes not by my wish or my heart. I understood that it was wrong, Bible is more subtle than just a list […]
Orderings of filters in terms of reloids – online draft
I put a preliminary draft (not yet checked for errors) of the article “Orderings of filters in terms of reloids” at Algebraic General Topology site. The abstract Orderings of filters which extend Rudin-Keisler preorder of ultrafilters are defined in terms of reloids that (roughly speaking) is filters on sets of binary relations between some sets. […]
Two elementary theorems
I proved the following two elementary but useful theorems: Theorem For every funcoids $latex f$, $latex g$: If $latex \mathrm{im}\, f \supseteq \mathrm{im}\, g$ then $latex \mathrm{im}\, (g\circ f) = \mathrm{im}\, g$. If $latex \mathrm{im}\, f \subseteq \mathrm{im}\, g$ then $latex \mathrm{dom}\, (g\circ f) = \mathrm{dom}\, g$. Theorem For every reloids $latex f$, $latex g$: […]
Funcoid corresponding to a monovalued reloid is monovalued
I proved the following simple theorem: 1. $latex (\mathsf{FCD}) f$ is a monovalued funcoid if $latex f$ is a monovalued reloid. 2. $latex (\mathsf{FCD}) f$ is an injective funcoid if $latex f$ is an injective reloid. See online article “Funcoids and Reloids” and this Web page.
New corollary
To the theorem “Every monovalued reloid is a restricted function.” I added a new corollary “Every monovalued injective reloid is a restricted injection.” See the online article “Funcoids and Reloids” and this Web page.
“Pointfree Funcoids” is now a good draft
I updated online article “Pointfree Funcoids” from “preliminary draft” to just “draft”. This means that it was somehow checked for errors and ready for you to read. (No 100% warranty against errors however.) Having finished with that draft the way is now free for my further Algebraic General Topology research, such as n-ary (multidimensional) funcoids, […]
Draft: Pointfree funcoids
My draft article Pointfree Funcoids was not yet thoroughly checked for errors. However at this stage of the draft I expect that there are no big errors there only possible little errors. Familiarize yourself with Algebraic General Topology.