I’ve noticed that the statement “Micronization is always reflexive.” in my math book is erroneous. It led also to some further errors in the section “Micronization”.

I am going to correct the errors in near time.

I have almost finished developing theory of filters on posets (not including cardinality issues, maps between filters, and maybe specifics of ultrafilters).

Yeah, it is finished! I have completely developed a field of math.

Well, there remains yet some informal problems, see the attached image:

Note that as it seems nobody before me researched filters in details. Let us congratulate this scientific achievement.

I have proved this recently formulated conjecture.

See my book. Currently it is theorem number 598.

Let $\mathfrak{F}(S)$ denotes the set of filters on a poset $S$, ordered reversely to set theoretic inclusion of filters. Let $Da$ for a lattice element $a$ denote its sublattice $\{ x \mid x \leq a \}$. Let $Z(X)$ denotes the set of complemented elements of the lattice $X$.

Conjecture $\mathfrak{F}(Z(D\mathcal{A}))$ is order-isomorphic to $D\mathcal{A}$ for every filter $\mathcal{A}$ on a set. If they are isomorphic, find an isomorphism.

Like a complete idiot, this took me a few years to disprove my conjecture, despite the proof is quite trivial.

Here is the complete solution:

Example $[S]\ne\{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\}$, where $[S]$ is the complete lattice generated by a strong partition $S$ of filter on a set.

Proof Consider any infinite set $U$ and its strong partition $\{\uparrow^U\{x\} \mid x\in U\}$.

$\{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\}$ consists only of principal filters.

But $[S]$ obviously contains some nonprincipal filters.

I noticed that there are two different things in mathematics both referred as “generalization”.

The first is like replacing real numbers with complex numbers, that is replacing a set in consideration with its superset.

The second is like replacing a metric space with its topology, that is abstracting away some properties.

Why are both called with the same word “generalization”? What is common in these two? Please comment.

I’ve found a counterexample to the following conjecture:

Statement For every composable funcoids $f$ and $g$ we have

$H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$

The counterexample is $f=a\times^{\mathsf{FCD}} \{p\}$ and $g=\{p\}\times^{\mathsf{FCD}}a$, $H=1$ where $a$ is an arbitrary nontrivial ultrafilter and $p$ is an arbitrary point.

I leave the proof that it is a counterexample as an easy exercise for the reader (however I am going to add the proof to my book soon).

That the conjecture failed invalidates my new proof of Urysohn’s lemma which was based on this conjecture.