A step forward to solve an open problem
I 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 is found the value of the function “other” at $latex \bot$. It is […]
New short chapter
I’ve added a new short chapter “Generalized Cofinite Filters” to my book.
A conjecture proved
I 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.
New theorem about relationships between funcoids and reloids
I 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.
My math book updated
I have updated my math book with new (easy but) general theorem similar to this (but in other notation): Theorem If $latex \mathfrak{Z}$ is an ideal base, then the set of filters on $latex \mathfrak{Z}$ is a join-semilattice and the binary join of filters is described by the formula $latex \mathcal{A}\sqcup\mathcal{B} = \mathcal{A}\cap\mathcal{B}$. I have updated some […]
My math book updated
I 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 a lattice element without stating that the lattice is boolean.) Well, maybe I […]
I proved that certain functors between topological spaces and endofuncoid are adjoint
I 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 proof is not yet enough checked for errors.
A counterexample to my recent conjecture
After 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 $latex \mathbb{R}$.
New conjecture about funcoids
Conjecture $latex \langle f \rangle \bigsqcup S = \bigsqcup_{\mathcal{X} \in S} \langle f \rangle \mathcal{X}$ if $latex S$ is a totally ordered (generalize for a filter base) set of filters (or at least set of sets).
Mappings between endofuncoids and topological spaces
I started research of mappings between endofuncoids and topological spaces. Currently the draft is located in volume 2 draft of my online book. I define mappings back and forth between endofuncoids and topologies. The main result is a representation of an endofuncoid induced by a topological space. The formula is $latex f\mapsto 1\sqcup\mathrm{Compl}\, f\sqcup(\mathrm{Compl}\, f)^2\sqcup […]