Restricting a reloid to lattice Gamma before converting it into a funcoid ★★

Author(s): Porton

\begin{conjecture} $(\mathsf{FCD}) f = \bigcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \operatorname{GR} f)$ for every reloid $f \in \mathsf{RLD} (A ; B)$. \end{conjecture}

Keywords: funcoid corresponding to reloid; funcoids; reloids

Inner reloid through the lattice Gamma ★★

Author(s): Porton

\begin{conjecture} $(\mathsf{RLD})_{\operatorname{in}} f = \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f$ for every funcoid $f$. \end{conjecture}

Counter-example: $(\mathsf{RLD})_{\operatorname{in}} f \sqsubset \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f$ for the funcoid $f = (=)|_\mathbb{R}$ is proved in \href[this online article]{}.

Keywords: filters; funcoids; inner reloid; reloids

Domain and image of inner reloid ★★

Author(s): Porton

\begin{conjecture} $\ensuremath{\operatorname{dom}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{in}}} f =\ensuremath{\operatorname{dom}}f$ and $\ensuremath{\operatorname{im}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{in}}} f =\ensuremath{\operatorname{im}}f$ for every funcoid $f$. \end{conjecture}

Keywords: domain; funcoids; image; reloids

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