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February 1, 2018 / porton

Error in my math book

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.

January 5, 2018 / porton

Theory of filters is FINISHED!

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.

January 5, 2018 / porton

A conjecture about filters proved

I have proved this recently formulated conjecture.

See my book. Currently it is theorem number 598.

January 3, 2018 / porton

A new conjecture about filters

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.

January 2, 2018 / porton

Very easy solution of my old conjecture

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.

October 25, 2017 / porton

Two kinds of generalization

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.

October 15, 2017 / porton

A counterexample to my conjecture

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.