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…
read moreI’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)…
read moreI’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…
read moreI 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…
read moreI 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…
read moreConjecture 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 $latex S = \mathrm{up}\, f$.
read moreThe 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 $latex \mathrm{id}^{\mathsf{FCD}}_{\Omega}$.
read moreDefinition 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…
read moreI’ve proved the following (for every funcoids $latex f$ and $latex g$): Statement $latex \mathrm{up}\, (f \sqcap^{\mathsf{FCD}} g) \subseteq \bigcup \{ \mathrm{up}\, (F \sqcap^{\mathsf{FCD}} G) \mid F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \}$ or equivalently: If $latex Z\in\mathrm{up}\, (f \sqcap^{\mathsf{FCD}}…
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