I’ve found a counterexample to the following conjecture: Statement For every composable funcoids $latex f$ and $latex g$ we have $latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$…
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 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 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 $latex X \in \mathscr{P} A$…
read moreI 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.
read moreIt 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.
read moreI 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…
read moreI 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…
read moreI 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….
read moreI’ve moved the section “Some (example) values” to my main book file (instead of the draft file addons.pdf where it was previously).
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