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 […]
“What is physical reality?” in my other blog
I have published What is physical reality? blog post in my other blog. The post is philosophical.