Conjecture For every funcoid $latex f$ and filter $latex \mathcal{X}\in\mathfrak{F}(\mathrm{Src}\,f)$, $latex \mathcal{Y}\in\mathfrak{F}(\mathrm{Dst}\,f)$: $latex \mathcal{X} \mathrel{[(\mathsf{FCD}) f]} \mathcal{Y} \Leftrightarrow \forall F \in \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)} f : \mathcal{X} \mathrel{[F]} \mathcal{Y}$; $latex \langle (\mathsf{FCD}) f \rangle \mathcal{X} = \bigsqcap_{F \in \mathrm{up}^{\Gamma…

read moreConjecture $latex (\mathsf{FCD}) f = \bigsqcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \mathrm{GR}\, f)$ for every reloid $latex f \in \mathsf{RLD} (A ; B)$.

read moreConjecture $latex (\mathsf{RLD})_{\mathrm{in}} f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)}\, f$ for every funcoid $latex f$. (I use notation from this note and this draft article.)

read moreI have recently proved that there is an order isomorphism between funcoids and filters on the lattice of finite unions of Cartesian products of sets. Today I’ve proved that this bijection preserves composition. See this note (updated) for the proofs.

read moreI have just proven the following two new theorems: Theorem Composition of complete reloids is complete. Theorem $latex (\mathsf{RLD})_{\mathrm{out}} g \circ (\mathsf{RLD})_{\mathrm{out}} f = (\mathsf{RLD})_{\mathrm{out}} (g \circ f)$ if $latex f$ and $latex g$ are both complete funcoids (or both co-complete). See…

read moreI’ve proved this conjecture (not a long standing conjecture, it took just one day to solve it) and found a stronger theorem than these propositions. So my new theorem: Theorem $latex (\mathsf{FCD})$ and $latex (\mathsf{RLD})_{\mathrm{out}}$ form mutually inverse bijections between complete reloids…

read moreIn this recent blog post I have formulated the conjecture: Conjecture A funcoid $latex f$ is complete iff $latex f=(\mathsf{FCD}) g$ for a complete reloid $latex g$. This conjecture has not been living long, I have quickly proved it in this note.

read moreAbout new theorems in in this my blog post: I’ve simplified this theorem: Theorem A reloid $latex f$ is complete iff $latex f = \bigsqcap^{\mathsf{RLD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, |…

read moreI’ve proved some new theorems. The proofs are currently available in this PDF file. Theorem The set of funcoids is with separable core. Theorem The set of funcoids is with co-separable core. Theorem A funcoid $latex f$ is complete iff $latex f…

read moreI have published online a short article saying that the set of funcoids is isomorphic to the set of filters on a certain lattice. Then I found a counter-example and decided that my theorem was wrong. I was somehow sad about this….

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