Just a few minutes ago I conceived a definition of generalized Fréchet filters with definition for every poset on which filters are considered (however, I have not yet calculated the class of posets for which generalized Fréchet filter is defined; it should be easy but I am busy with other business).

Generalized Fréchet filter on a poset $\mathfrak{A}$ is a filter $\Omega$ such that $\partial \Omega = \left\{ x \in \mathfrak{A} \hspace{0.5em} | \hspace{0.5em} \mathrm{atoms}\, x \text{ is infinite} \right\} .$

See my book for a definition of $\partial$.

I’ve done a little discovery today: Proximities are the same reflexive, symmetric, transitive funcoids.

For now I leave to prove this as an exercise for a reader. But later I am going to include this theorem into the book I am writing.

My article was accepted for publication in European Journal of Pure and Applied Mathematics, but it didn’t compile with their LaTeX templates. After waiting a reasonable time until they would tackle the problem, I have withdrawn my article and sent it to another journal.

I would search for the bug in their LaTeX template myself, but I was a great LaTeX expert in the past, I have forgotten much of LaTeX tricks, as now I use TeXmacs to write my manuscript, I convert them to simple LaTeX only to send it to a publisher. So I have chosen not to attempt to resolve the problem with LaTeX but just to send it to an other journal.

I’ve introduced another version of cross-composition of funcoids. This forms a category with star-morphisms. It is conjectured that this category is quasi-invertible, because I have failed to prove it.

This should be included in the next version of my book.

A mathematician named Todd Trimble has helped me to prove that the set of funcoids between two given sets (and more generally certain pointfree funcoids) is always a co-frame. (I knew this for funcoids but my proof required axiom of choice, while Todd’s does not require axiom of choice.)

He initially published his proof here but because his proof relies on advanced category theory I didn’t understood his proof. However Todd was so kind that he preserved me a longer more elementary version of the proof in email correspondence.

I wrote my own version of this proof in this short article which I am going to incorporate into my book.

This my proof needs some revision. Possibly I confused just join-semilattices and join-semilattices with least element are confused with each other.

I’ve found today earlier stated conjecture that lattices $\mathrm{Compl}\mathsf{FCD}(A;B)$ and $\mathrm{Compl}\mathsf{RLD}(A;B)$ are co-brouwerian.

Exercise: Prove this fact.

I’ve proved the following conjecture:

Theorem Let $f$ be a staroid such that $(\mathrm{form}\, f)_i$ is an atomic lattice for
each $i \in \mathrm{arity}\, f$. We have

$\displaystyle L \in \mathrm{GR}\, f \Leftrightarrow \mathrm{GR}\, f \cap \prod_{i \in \mathrm{dom}\, \mathfrak{A}} \mathrm{atoms}\, L_i \neq \emptyset$

for every $L \in \prod_{i \in \mathrm{arity}\, f} (\mathrm{form}\, f)_i$ (where upgrading is taken on the primary filtrator).

The proof is based on transfinite recursion. See this online article for the proof.

The above proof was with an error. Now there is a counter-example in the same article.