I have added to my book, section “Equivalent filters and rebase of filters” some new results. Particularly I added “Rebase of unfixed filters” subsection. It remains to research the properties of the lattice of unfixed filters.
read moreEquivalence of filters can be described in terms of the lattice of filters on the poset of small sets. See new subsection “Embedding into the lattice of filters on small” in my book.
read moreI defined partial order on the set of unfixed filters and researched basic properties of this poset. Also some minor results about equivalence of filters. See my book.
read moreI added a few new propositions into “Equivalent filters and rebase of filters” section (now located in the chapter “Filters and filtrators”) of my book. I also define “unfixed filters” as the equivalence classes of (small) filters on sets. This is a…
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 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…
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 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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