I’ve proved the theorem: Theorem $latex f \mapsto \bigsqcap^{\mathsf{RLD}} f$ and $latex \mathcal{A} \mapsto \Gamma (A ; B) \cap \mathcal{A}$ are mutually inverse bijections between $latex \mathfrak{F} (\Gamma (A ; B))$ and funcoidal reloids. These bijections preserve composition. (The second items is…
read moreTheorem $latex (\mathsf{RLD})_{\mathrm{in}} (g \circ f) = (\mathsf{RLD})_{\mathrm{in}} g \circ (\mathsf{RLD})_{\mathrm{in}} f$ for every composable funcoids $latex f$ and $latex g$. See proof in this online article.
read moreIn this online article I’ve proved: Theorem $latex \mathrm{dom}\, (\mathsf{RLD})_{\mathrm{in}} f = \mathrm{dom}\, f$ and $latex \mathrm{im}\, (\mathsf{RLD})_{\mathrm{in}} f = \mathrm{im}\, f$ for every funcoid $latex f$. and its easy consequence: Proposition $latex \mathrm{dom}\, (\mathsf{RLD})_{\Gamma} f = \mathrm{dom}\, f$ and $latex \mathrm{im}\,…
read moreConjecture $latex \mathrm{dom}\, (\mathsf{RLD})_{\Gamma} f = \mathrm{dom}\, f$ 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 $latex g$….
read moreI’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 $latex (\mathsf{FCD}) : \mathsf{RLD} (A ; B) \rightarrow \mathsf{FCD} (A ; B)$…
read moreI 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…
read moreFor 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…
read moreI’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 :…
read moreI 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 $latex f \in \mathsf{RLD} (A ; B)$.
read more