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…

read moreI’ve found a counterexample to the following conjecture: Statement For every composable funcoids $latex f$ and $latex g$ we have $latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$…

read moreWe 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…

read moreSee this my post in other blog for a religious reason to do scientific research.

read moreIt 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…

read moreI started to work on funcoidal groups (a generalization of topological groups). I defined it and promptly found a curious theorem. Not sure if this theorem has use for anything. See the definition and the “curious” proposition in this draft. Note that…

read moreI added to my online research book the following theorem: Theorem Let $latex \mathfrak{A}$ be a distributive lattice with least element. Let $latex a,b\in\mathfrak{A}$. If $latex a\setminus b$ exists, then $latex a\setminus^* b$ also exists and $latex a\setminus^* b=a\setminus b$. The user…

read moreI’ve published in my blog a new theorem. The proof was with an error (see the previous edited post)!

read moreThe below is wrong! The proof requires $latex \langle g^{-1}\rangle J$ to be a principal filter what does not necessarily hold. I knew that composition of two complete funcoids is complete. But now I’ve found that for $latex g\circ f$ to be…

read moreI have proved (see new version of my book) the following proposition. (It is basically a special case of my erroneous theorem which I proposed earlier.) Proposition For $latex f \in \mathsf{FCD} (A, B)$, a finite set $latex X \in \mathscr{P} A$…

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