Yet three conjectures
Conjecture $latex \mathrm{dom}\, (\mathsf{RLD})_{\Gamma} f = \mathrm{dom}\, f$ and $latex \mathrm{im}\, (\mathsf{RLD})_{\Gamma} f = \mathrm{im}\, f$ for every funcoid $latex f$. Conjecture $latex (\mathsf{RLD})_{\Gamma} g \circ (\mathsf{RLD})_{\Gamma} f = (\mathsf{RLD})_{\Gamma} (g \circ f)$ for every composable funcoids $latex f$ and $latex g$. Conjecture For every funcoid $latex g$ we have $latex \mathrm{Cor}\, (\mathsf{RLD})_{\Gamma} g = […]
A new theorem and a conjecture
I’ve just proved the following: Theorem $latex (\mathsf{FCD}) (\mathsf{RLD})_{\Gamma} f = f$ for every funcoid $latex f$. For a proof see this online article. I’ve also posed the conjecture: Conjecture $latex (\mathsf{FCD}) : \mathsf{RLD} (A ; B) \rightarrow \mathsf{FCD} (A ; B)$ is the upper adjoint of $latex (\mathsf{RLD})_{\Gamma} : \mathsf{FCD} (A ; B) \rightarrow […]
Some new definitions, propositions, and conjectures
I added to this online article the following definitions, propositions, and conjectures: Definition $latex \boxbox f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)} \, f$ for reloid $latex f$. Obvious $latex \boxbox f \sqsupseteq f$ for every reloid $latex f$. Example $latex (\mathsf{RLD})_{\Gamma} f \neq \boxbox (\mathsf{RLD})_{\mathrm{out}} f$ for some funcoid $latex f$. Conjecture […]
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 $latex f \in \mathsf{RLD} (A ; B)$.