Funcoidal groups and a curious theorem
I 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 I work on another projects and may be not very active in researching […]
New simple theorem in my book
I 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 quasi of Math.SE has helped me with the proof.
A wrong result
I’ve published in my blog a new theorem. The proof was with an error (see the previous edited post)!
A new unexpected result (ERROR!)
The 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 complete it’s enough $latex f$ to be complete. The proof which I missed […]
A new easy proposition about funcoids
I 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 $latex X \in \mathscr{P} A$ and a function $latex t \in \mathscr{F} (B)^X$ there exists (obviously unique) $latex […]
Error in my theorem – found
I found the exact error noticed in Error in my theorem post. The error was that I claimed that infimum of a greater set is greater (while in reality it’s lesser). I will delete the erroneous theorem from my book soon.
Error in my theorem
It seems that there is an error in proof of this theorem. Alleged counter-example: $latex f=\bot$ and $latex z(p)=\top$ for infinite sets $latex A$ and $latex B$. I am now attempting to locate the error in the proof.
New theorem about funcoids (ERROR!)
I have proved (and added to my online book) the following theorem: Theorem Let $latex f \in \mathsf{FCD} (A ; B)$ and $latex z \in \mathscr{F} (B)^A$. Then there is an (obviously unique) funcoid $latex g \in \mathsf{FCD} (A ; B)$ such that $latex \langle g\rangle x = \langle f\rangle x$ for nontrivial ultrafilters $latex […]
An open problem solved
I proved the following (in)equalities, solving my open problem which stood for a few months: $latex \lvert \mathbb{R} \rvert_{>} \sqsubset \lvert \mathbb{R} \rvert_{\geq} \sqcap \mathord{>}$ $latex \lvert \mathbb{R} \rvert_{>} = \lvert \mathbb{R} \rvert_{>} \sqcap \mathord{>}$ The proof is currently available in the section “Some inequalities” of this PDF file. Note that earlier I put online […]
A conjecture about funcoids on real numbers disproved
I proved that $latex \lvert \mathbb{R} \rvert_{\geq} \neq \lvert \mathbb{R} \rvert \sqcap \geq$ 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 $latex \lvert \mathbb{R} \rvert_{>} \neq \lvert […]