Attempt to generalize filter bases for more general filtrators
In this draft I present some definitions and conjectures on how to generalize filter bases for more general filtrators (such as the filtrator of funcoids). This is a work-in-progress. This seems an interesting research by itself, but I started to develop it as a way to prove this conjecture.
My proof was with an error
I claimed that I have proved the following conjecture: Conjecture $latex \forall H \in \mathrm{up} (g \circ f) \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \sqsupseteq G \circ F$ for every composable funcoids $latex f$ and $latex g$. The proof was with an error. So it remains a conjecture.
A conjecture about funcoids proved
WARNING: The proof was with an error! I have proved the following theorem: Theorem $latex \forall H \in \mathrm{up} (g \circ f) \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \sqsupseteq G \circ F$ for every composable funcoids $latex f$ and $latex g$. This theorem (being a conjecture at that time) was […]
A new counter-example
Example There is such a non-symmetric reloid $latex f$ that $latex (\mathsf{FCD})f$ is symmetric. Take $latex f=((\mathsf{RLD})_{\mathrm{in}}(\mathord{=})|_{\mathbb{R}})\sqcap (\mathord{\geq})_{\mathbb{R}}$. I have added this to my online book.
I replaced semicolons with commas
I’ve released my math research book and all supplementary materials free with semicolons replaced with commas to denote tuples: $latex (a;b)$ → $latex (a,b)$, in order to comply with usual math notation of other mathematicians.
An informal open problem in mathematics
Characterize the set $latex \{f\in\mathsf{FCD} \mid (\mathsf{RLD})_{\mathrm{in}} f=(\mathsf{RLD})_{\mathrm{out}} f\}$. (This seems a difficult problem.)
A new theorem proved
I have proved $latex (\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}} = \Omega^{\mathsf{RLD}}$ (where $latex \Omega^{\mathsf{FCD}}$ is a cofinite funcoid and $latex \Omega^{\mathsf{RLD}}$ is a cofinite reloid that is reloid defined by a cofinite filter). The proof is currently available in this draft. Note that in the previous draft there was a wrong formula for $latex (\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}}$.
A typo in my math book
I’ve found a typo in my math book. I confused existential quantifiers with universal quantifiers in the section “Second product. Oblique product” in the chapter “Counter-examples about funcoids and reloids”.
More on generalized cofinite filters
I added more properties of cofinite funcoids to this draft.
Generalized cofinite filters
I have described generalized cofinite filters (including the “cofinite funcoid”). See the draft at http://www.math.portonvictor.org/binaries/addons.pdf