I have added a new easy (but unnoticed before) theorem to my book:

Proposition $(\mathsf{RLD})_{\mathrm{out}} f\sqcup (\mathsf{RLD})_{\mathrm{out}} g = (\mathsf{RLD})_{\mathrm{out}}(f\sqcup g)$ for funcoids $f$, $g$.

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 $\mathrm{other} = \Phi_{\ast}(\mathsf{RLD})_{\mathrm{out}}$.

A few minutes ago I’ve proved $(\Phi_{\ast}(\mathsf{RLD})_{\mathrm{out}})\bot = \Omega^{\mathsf{FCD}}$, that is found the value of the function “other” at $\bot$. It is yet a “mistery” what values it has at different arguments.

I’ve added a new short chapter “Generalized Cofinite Filters” to my book.

I have proved the conjecture that $S^{\ast}(\mu)\circ S^{\ast}(\mu)=S^{\ast}(\mu)$ for every endoreloid $\mu$. The easy proof is currently available in this file.

I have proved (the proof is currently available in this file) that $((\mathsf{FCD}), (\mathsf{RLD})_{\mathrm{in}})$ are components of a pointfree funcoid between boolean lattices.

See my book for definitions.

I have updated my math book with new (easy but) general theorem similar to this (but in other notation):

Theorem If $\mathfrak{Z}$ is an ideal base, then the set of filters on $\mathfrak{Z}$ is a join-semilattice and the binary join of filters is described by the formula $\mathcal{A}\sqcup\mathcal{B} = \mathcal{A}\cap\mathcal{B}$.

I have updated some other theorems to use this general result and so themselves to become a little more general.

In the course of rewriting my book I found and corrected several small errors.

The latest changes of the book are not yet as thoroughly checked for errors as the rest of the book.

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 introduced new errors. The current version is not 100% stable. However, I am sure the errors (if any) are small and don’t break the exposition.