A new function which is a counter-example to a conjecture found
For this conjecture there was found a counter-example, see this online article. The counter-example states that $latex (\mathsf{RLD})_{\Gamma} f \sqsupset (\mathsf{RLD})_{\mathrm{in}} f \sqsupset (\mathsf{RLD})_{\mathrm{out}} f$ for funcoid $latex f=(=)|_{\mathbb{R}}$. This way I discovered a new function $latex (\mathsf{RLD})_{\Gamma}$ defined by the formula $latex (\mathsf{RLD})_{\Gamma} f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\mathrm{Src}\,f ; \mathrm{Dst}\,f)} f$. While $latex (\mathsf{RLD})_{\mathrm{in}} […]
I’ve proved one more conjecture
I’ve proved yet one conjecture. The proof is presented in this online article. Theorem For every funcoid $latex f$ and filters $latex \mathcal{X}\in\mathfrak{F}(\mathrm{Src}\,f)$, $latex \mathcal{Y}\in\mathfrak{F}(\mathrm{Dst}\,f)$: $latex \mathcal{X} \mathrel{[(\mathsf{FCD}) f]} \mathcal{Y} \Leftrightarrow \forall F \in \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)} f : \mathcal{X} \mathrel{[F]} \mathcal{Y}$; $latex \langle (\mathsf{FCD}) f \rangle \mathcal{X} = \bigsqcap_{F \in \mathrm{up}^{\Gamma […]
Restricting a reloid to Gamma before converting it into a funcoid formula
I have just proved this my conjecture. The proof is presented in this online article. Theorem $latex (\mathsf{FCD}) f = \bigsqcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \mathrm{GR}\, f)$ for every reloid $latex f \in \mathsf{RLD} (A ; B)$.
Todd Trimble’s commentary analyzed
I have written a short article with my response on Todd Trimble’s commentary on my book. In this response I present these of Todd Trimble’s results which are new for me. Note that I skipped specifically category-theoretic results (such as that the category of endofuncoids is topological). I am going to amend my article with […]
Todd Trimble’s commentary on my math research
Todd Trimble has notified me that he has written a “commentary” (notes) on my theory of funcoids presented in my monograph. His commentary is available at this nLab wiki page. I’ve started to read his notes. First I needed to lookup into Wikipedia to know what Chu space is. He uses category theory however as […]
Funcoid corresponding to reloid through lattice Gamma
Conjecture For every funcoid $latex f$ and filter $latex \mathcal{X}\in\mathfrak{F}(\mathrm{Src}\,f)$, $latex \mathcal{Y}\in\mathfrak{F}(\mathrm{Dst}\,f)$: $latex \mathcal{X} \mathrel{[(\mathsf{FCD}) f]} \mathcal{Y} \Leftrightarrow \forall F \in \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)} f : \mathcal{X} \mathrel{[F]} \mathcal{Y}$; $latex \langle (\mathsf{FCD}) f \rangle \mathcal{X} = \bigsqcap_{F \in \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)} f} \langle F \rangle \mathcal{X}$.
Conjecture: Restricting a reloid to Gamma before converting it into a funcoid
Conjecture $latex (\mathsf{FCD}) f = \bigsqcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \mathrm{GR}\, f)$ for every reloid $latex f \in \mathsf{RLD} (A ; B)$.
A new conjecture about relationships of funcoids and reloids
Conjecture $latex (\mathsf{RLD})_{\mathrm{in}} f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)}\, f$ for every funcoid $latex f$. (I use notation from this note and this draft article.)
Funcoids as filters and composition
I 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.
Two new theorems about complete reloids
I 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 this note for the proofs.