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.
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 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 $latex S = \mathrm{up}\, f$. […]
A base of a funcoid which is not a filter base
The converse of this theorem does not hold. Counterexample: Take $latex S = \mathrm{up}\, \mathrm{id}^{\mathsf{FCD}}_{\Omega}$. We know that $latex S$ is not a filter base. But it is trivial to prove that $latex S$ is a base of the funcoid $latex \mathrm{id}^{\mathsf{FCD}}_{\Omega}$.
A new theorem proved
Definition A set $latex S$ of binary relations is a base of a funcoid $latex f$ when all elements of $latex S$ are above $latex f$ and $latex \forall X \in \mathrm{up}\, f \exists T \in S : T \sqsubseteq X$. It was easy to show: Proposition A set $latex S$ of binary relations is […]
A new proposition proved
I’ve proved the following lemma: Lemma Let for every $latex X, Y \in S$ and $latex Z \in \mathrm{up} (X \sqcap^{\mathsf{FCD}} Y)$ there is a $latex T \in S$ such that $latex T \sqsubseteq Z$. Then for every $latex X_0, \ldots, X_n \in S$ and $latex Z \in \mathrm{up} (X_0 \sqcap^{\mathsf{FCD}} \ldots \sqcap^{\mathsf{FCD}} X_n)$ there […]
I proved a conjecture
After prayer in tongues and going down anointment of Holy Spirit I proved this conjecture about funcoids. The proof is currently located in this PDF file. Well, the proof is for special cases of distributive lattices, but more general case seems not necessary (at least now). It seems easy to generalize it for more general […]
New conjecture about funcoids
New conjecture: Conjecture $latex \mathrm{up} (f \sqcap^{\mathsf{FCD}} g) \subseteq \{ F \sqcap G \mid F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \}$ for all funcoids $latex f$, $latex g$ (with corresponding sources and destinations). Looks trivial? But how to (dis)prove it?