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I have published What is physical reality? blog post in my other blog. The post is philosophical.

I proved that $\lvert \mathbb{R} \rvert_{\geq} \neq \lvert \mathbb{R} \rvert \sqcap \geq$ and so disproved one of my conjectures.

The proof is currently available in the section “Some inequalities” of this PDF file.

The proof isn’t yet thoroughly checked for errors.

Note that I have not yet proved $\lvert \mathbb{R} \rvert_{>} \neq \lvert \mathbb{R} \rvert \sqcap >$, but the proof is expected to be similar to the above.

I’ve moved the section “Some (example) values” to my main book file (instead of the draft file addons.pdf where it was previously).

I have rewritten my math book (volume 1) with implicit arguments (that is I sometimes write $\bot$ instead of $\bot^{\mathfrak{A}}$ to denote the least element of the lattice $\mathfrak{A}$).

It considerably simplifies the formulas.

If you want to be on this topic, learn what is called “dependent lambda calculus”. (Sadly, I do not use it in my book explicitly, in order to make my book easier to understand. But I weight the possibility to rewrite my book in a dependent lambda calculus proof-assistant language, that is in the language of an automatic proof verification software, to make it even greater.)

I’ve calculated values of some concrete funcoids and reloids.

The calculations are currently presented in the chapter 3 “Some (example) values” of addons.pdf.

I have added the sections “5.25 Bases on filtrators” (some easy theory generalizing filter bases) and “16.8 Funcoid bases” (mainly a counter-example against my former conjecture) to my math book.

It is not difficult to prove (see “Counter-examples about funcoids and reloids” in the book) that $1^{\mathsf{FCD}} \sqcap^{\mathsf{FCD}} (\top\setminus 1^{\mathsf{FCD}}) = \mathrm{id}^{\mathsf{FCD}}_{\Omega}$ (where $\Omega$ is the cofinite filter). But the result is counterintuitive: meet of two binary relations is not a binary relation.

After proving this I always felt that there is some “mystery” about meet of funcoids: It behaves in a weird way and what it is in general (not this one special counterexample case) is not known.

Today I noted a simple formula which decomposes $f \sqcap^{\mathsf{FCD}} g$: $f \sqcap^{\mathsf{FCD}} g = (\mathsf{FCD})((\mathsf{RLD})f \sqcap (\mathsf{RLD})g)$ for every funcoids $f$ and $g$ and more generally $\bigsqcap^{\mathsf{FCD}} S = (\mathsf{FCD}) \bigsqcap^{\mathsf{RLD}} \langle (\mathsf{RLD})_{\mathrm{in}} \rangle^{\ast} S$ for a set $S$ of funcoids. (It follows from that $(\mathsf{FCD})$ is an upper adjoint and that $(\mathsf{FCD})(\mathsf{RLD})_{\mathrm{in}} f=f$ for every funcoid $f$.) This way it looks much more clear and less counterintuitive.

So now it looks more clear, but I have not yet found particular implications of these formulas leading to interesting results.