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 […]
“Some (example) values” in my book
I’ve moved the section “Some (example) values” to my main book file (instead of the draft file addons.pdf where it was previously).
The math book rewritten with implicit arguments
I have rewritten my math book (volume 1) with implicit arguments (that is I sometimes write $latex \bot$ instead of $latex \bot^{\mathfrak{A}}$ to denote the least element of the lattice $latex \mathfrak{A}$). It considerably simplifies the formulas. If you want to be on this topic, learn what is called “dependent lambda calculus”. (Sadly, I do […]
Values of some concrete funcoids and reloids
I’ve calculated values of some concrete funcoids and reloids. The calculations are currently presented in the chapter 3 “Some (example) values” of addons.pdf.
New sections in my math book
I have added the sections “5.25 Bases on filtrators” (some easy theory generalizing filter bases) and “16.8 Funcoid bases” (mainly a counter-example against my former conjecture) to my math book.
The mystery of meet of funcoids solved?
It is not difficult to prove (see “Counter-examples about funcoids and reloids” in the book) that $latex 1^{\mathsf{FCD}} \sqcap^{\mathsf{FCD}} (\top\setminus 1^{\mathsf{FCD}}) = \mathrm{id}^{\mathsf{FCD}}_{\Omega}$ (where $latex \Omega$ is the cofinite filter). But the result is counterintuitive: meet of two binary relations is not a binary relation. After proving this I always felt that there is some […]
Three (seemingly not so difficult) new conjectures
I’ve noticed the following three conjectures (I expect not very difficult) for finite binary relations $latex X$ and $latex Y$ between some sets and am going to solve them: $latex X\sqcap^{\mathsf{FCD}} Y = X\sqcap Y$; $latex (\top \setminus X)\sqcap^{\mathsf{FCD}} (\top \setminus Y) = (\top \setminus X)\sqcap (\top \setminus Y)$; $latex (\top \setminus X)\sqcap^{\mathsf{FCD}} Y = […]
A counter-example to my conjecture
I’ve found the following counter-example, to this conjecture: Example For a set $latex S$ of binary relations $latex \forall X_0,\dots,X_n\in S:\mathrm{up}(X_0\sqcap^{\mathsf{FCD}}\dots\sqcap^{\mathsf{FCD}} X_n)\subseteq S$ does not imply that there exists funcoid $latex f$ such that $latex S=\mathrm{up}\, f$. The proof is currently available at this PDF file and this wiki page.
A new research project (a conjecture about funcoids)
I start the “research-in-the middle” project (an outlaw offspring of Polymath Project) introducing to your attention the following conjecture: Conjecture The following are equivalent (for every lattice $latex \mathsf{FCD}$ of funcoids between some sets and a set $latex S$ of principal funcoids (=binary relations)): $latex \forall X, Y \in S : \mathrm{up} (X \sqcap^{\mathsf{FCD}} Y) […]
A surprisingly easy proof of yesterday conjecture
I have found a surprisingly easy proof of this conjecture which I proposed yesterday. Theorem Let $latex S$ be a set of binary relations. If for every $latex X, Y \in S$ we have $latex \mathrm{up} (X \sqcap^{\mathsf{FCD}} Y) \subseteq S$ then there exists a funcoid $latex f$ such that $latex S = \mathrm{up}\, f$. […]