Another star-category of funcoids
I’ve introduced another version of cross-composition of funcoids. This forms a category with star-morphisms. It is conjectured that this category is quasi-invertible, because I have failed to prove it. This should be included in the next version of my book.
What are hyperfuncoids isomorphic to?
Let $latex \mathfrak{A}$ be an indexed family of sets. Products are $latex \prod A$ for $latex A \in \prod \mathfrak{A}$. Hyperfuncoids are filters $latex \mathfrak{F} \Gamma$ on the lattice $latex \Gamma$ of all finite unions of products. Problem Is $latex \bigsqcap^{\mathsf{FCD}}$ a bijection from hyperfuncoids $latex \mathfrak{F} \Gamma$ to: prestaroids on $latex \mathfrak{A}$; staroids on […]
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 […]
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 $latex f \in \mathsf{RLD} (A ; B)$.
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.)