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.
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 […]
A conjecture about filters proved
I have proved this recently formulated conjecture. See my book. Currently it is theorem number 598.
A new conjecture about filters
Let $latex \mathfrak{F}(S)$ denotes the set of filters on a poset $latex S$, ordered reversely to set theoretic inclusion of filters. Let $latex Da$ for a lattice element $latex a$ denote its sublattice $latex \{ x \mid x \leq a \}$. Let $latex Z(X)$ denotes the set of complemented elements of the lattice $latex X$. […]
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 $latex [S]\ne\{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\}$, where $latex [S]$ is the complete lattice generated by a strong partition $latex S$ of filter on a set. Proof Consider any infinite set $latex U$ and its […]
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 […]
A counterexample to my conjecture
I’ve found a counterexample to the following conjecture: Statement For every composable funcoids $latex f$ and $latex g$ we have $latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$ The counterexample is $latex f=a\times^{\mathsf{FCD}} \{p\}$ and $latex g=\{p\}\times^{\mathsf{FCD}}a$, $latex H=1$ where $latex […]
Peer review for open access math books
We have a new kind of math publishing: Free books distributed through Internet. It is a new kind of mathematical culture. Some books of this kind appeared with daunting success. It has great advantages. It is how things should be done in modern times. But it miss an essential part of math tradition, peer review. For […]
Offtopic: A religious reason to do scientific research
See this my post in other blog for a religious reason to do scientific research.
A new funcoid discovered
It is easy to prove that the equation $latex \langle \mathscr{A} \rangle X = \mathrm{atoms}^{\mathfrak{A}}\, X$ (for principal filters $latex X$) defines a (unique) funcoid $latex \mathscr{A}$ which I call quasi-atoms funcoid. Note that as it is easy to prove $latex \langle \mathscr{A}^{-1} \rangle Y = \bigsqcup Y$ for every set $latex Y$ of ultrafilters. […]