I’ve added a new section “Double filtrators” to the book “Algebraic General Topology. Volume 1”. I show that it’s possible to describe $latex (\mathsf{FCD})$, $latex (\mathsf{RLD})_{\mathrm{out}}$, and $latex (\mathsf{RLD})_{\mathrm{in}}$ entirely in terms of filtrators (order). This seems not to lead to really interesting…
read moreI have added a new easy (but unnoticed before) theorem to my book: Proposition $latex (\mathsf{RLD})_{\mathrm{out}} f\sqcup (\mathsf{RLD})_{\mathrm{out}} g = (\mathsf{RLD})_{\mathrm{out}}(f\sqcup g)$ for funcoids $latex f$, $latex g$.
read moreI am attempting to find the value of the node “other” in a diagram currently located at this file, chapter “Extending Galois connections between funcoids and reloids”. By definition $latex \mathrm{other} = \Phi_{\ast}(\mathsf{RLD})_{\mathrm{out}}$. A few minutes ago I’ve proved $latex (\Phi_{\ast}(\mathsf{RLD})_{\mathrm{out}})\bot = \Omega^{\mathsf{FCD}}$, that…
read moreI’ve added a new short chapter “Generalized Cofinite Filters” to my book.
read moreI have proved the conjecture that $latex S^{\ast}(\mu)\circ S^{\ast}(\mu)=S^{\ast}(\mu)$ for every endoreloid $latex \mu$. The easy proof is currently available in this file.
read moreI have proved (the proof is currently available in this file) that $latex ((\mathsf{FCD}), (\mathsf{RLD})_{\mathrm{in}})$ are components of a pointfree funcoid between boolean lattices. See my book for definitions.
read moreI updated my math research book to use “weakly down-aligned” and “weakly up-aligned” instead of “down-aligned” and “up-aligned” (see the book for the definitions) where appropriate to make theorems slightly more general. During this I also corrected an error. (One theorem referred to complement of…
read moreI proved: Theorem $latex T$ is a left adjoint of both $latex F_{\star}$ and $latex F^{\star}$, with bijection which preserves the “function” part of the morphism. The details and the proof is available in the draft of second volume of my online book. The…
read moreAfter proposing this conjecture I quickly found a counterexample: $latex S = \left\{ (- a ; a) \mid a \in \mathbb{R}, 0 < a < 1 \right\}$, $latex f$ is the usual Kuratowski closure for $latex \mathbb{R}$.
read moreConjecture $latex \langle f \rangle \bigsqcup S = \bigsqcup_{\mathcal{X} \in S} \langle f \rangle \mathcal{X}$ if $latex S$ is a totally ordered (generalize for a filter base) set of filters (or at least set of sets).
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