A new little theorem (Galois connections)
I’ve added the following to my research book: Definition Galois surjection is the special case of Galois connection such that $latex f^{\ast} \circ f_{\ast} $ is identity. Proposition For Galois surjection $latex \mathfrak{A} \rightarrow \mathfrak{B}$ such that $latex \mathfrak{A}$ is a join-semilattice we have (for every $latex y \in \mathfrak{B}$) $latex f_{\ast} y = \max […]
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Two new kinds of product of funcoids
I have defined two new kinds of products of funcoids: $latex \prod^{\mathrm{in}}_{i \in \mathrm{dom}\, f} f = \prod^{(C)}_{i \in \mathrm{dom}\, f} (\mathsf{RLD})_{\mathrm{in}} f_i$ (cross-inner product). $latex \prod^{\mathrm{out}}_{i \in \mathrm{dom}\, f} f = \prod^{(C)}_{i \in \mathrm{dom}\, f} (\mathsf{RLD})_{\mathrm{out}} f_i$ (cross-outer product). These products are notable that their values are also funcoids (not just pointfree funcoids). See […]
A conjecture about outward funcoids
I’ve added to my book the following conjecture: Conjecture For every composable funcoids $latex f$ and $latex g$ $latex (\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq (\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{RLD})_{\mathrm{out}} f.$
An error in my math book corrected
After noticing an error in my math book, I rewritten its section “Funcoids and filters” to reflect that $latex (\mathsf{RLD})_\Gamma = (\mathsf{RLD})_{\mathrm{in}}$. Previously I proved an example demonstrating that $latex (\mathsf{RLD})_\Gamma \ne (\mathsf{RLD})_{\mathrm{in}}$, but this example is believed by me to be wrong. The example was removed from the book. Thus I removed all references […]
An error in my book
I proved both $latex (\mathsf{RLD})_\Gamma \ne (\mathsf{RLD})_{\mathrm{in}}$ and $latex (\mathsf{RLD})_\Gamma = (\mathsf{RLD})_{\mathrm{in}}$. So there is an error in my math research book. I will post the details of the resolution as soon as I will locate and correct the error. While the error is not yet corrected I have added a red font note in […]
I’ve solved a conjecture about pseudodifference of filters
I claimed earlier that I partially solved this open problem. Today I solved it completely. The proof is available in this PDF file.
Two new chapters in my math draft
I’ve added chapters “Cartesian closedness” and “Singularities” (from the site http://tiddlyspace.com which will be closed soon) to volume 2 draft. Both chapters are very rough draft and present not rigorous proofs but rough ideas.
My article is ignored
The journal European Journal of Pure and Applied Mathematics has accepted my article after a peer review and asked me to send it in their LaTeX format. I had a hyperref trouble with my LaTeX file. So I’ve said them that I withdraw my article. But later I realized that the best thing I can […]
Common generalizations of convergences and funcoids
I am reading the book “Convergence Foundations of Topology” by Szymon Dolecki and Frédéric Mynard. (Well, I am more skimming than reading, I may read it more carefully in the future.) After reading about a half of the book, I tried to integrate my theory of funcoids with their theory of convergences. And I noticed, […]