inward reloid


Funcoidal products inside an inward reloid ★★

Author(s): Porton

\begin{conjecture} (solved) If $a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f$ then $a \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} b \subseteq f$ for every funcoid $f$ and atomic f.o. $a$ and $b$ on the source and destination of $f$ correspondingly. \end{conjecture}

A stronger conjecture:

\begin{conjecture} If $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f$ then $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} \mathcal{B} \subseteq f$ for every funcoid $f$ and $\mathcal{A} \in \mathfrak{F} \left( \ensuremath{\operatorname{Src}}f \right)$, $\mathcal{B} \in \mathfrak{F} \left( \ensuremath{\operatorname{Dst}}f \right)$. \end{conjecture}

Keywords: inward reloid

Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

\begin{conjecture} $( \mathsf{\tmop{RLD}})_{\tmop{in}} (g \circ f) = ( \mathsf{\tmop{RLD}})_{\tmop{in}} g \circ ( \mathsf{\tmop{RLD}})_{\tmop{in}} f$ for any composable \href[funcoids]{http://www.wikinfo.org/index.php/Funcoid} $f$ and $g$. \end{conjecture}

Keywords: distributive; distributivity; funcoid; functor; inward reloid; reloid

Reloid corresponding to funcoid is between outward and inward reloid ★★

Author(s): Porton

\begin{conjecture} For any \href[funcoid]{http://www.wikinfo.org/index.php/Funcoid} $f$ and \href[reloid]{http://www.wikinfo.org/index.php/Reloid} $g$ having the same source and destination \[ ( \mathsf{\tmop{RLD}})_{\tmop{out}} f \subseteq g \subseteq ( \mathsf{\tmop{RLD}})_{\tmop{in}} f \Leftrightarrow ( \mathsf{\tmop{FCD}}) g = f. \] \end{conjecture}

Keywords: funcoid; inward reloid; outward reloid; reloid

Inward reloid corresponding to a funcoid corresponding to convex reloid ★★

Author(s): Porton

\begin{conjecture} $( \mathsf{\tmop{RLD}})_{\tmop{in}} ( \mathsf{\tmop{FCD}}) f = f$ for any \href[convex reloid]{http://www.wikinfo.org/index.php/Convex_reloid} $f$. \end{conjecture}

Keywords: convex reloid; funcoid; functor; inward reloid; reloid

Funcoid corresponding to inward reloid ★★

Author(s): Porton

\begin{conjecture} $( \mathsf{\tmop{FCD}}) ( \mathsf{\tmop{RLD}})_{\tmop{in}} f = f$ for any \href[funcoid]{http://www.wikinfo.org/index.php/Funcoid} $f$. \end{conjecture}

Keywords: funcoid; inward reloid; reloid

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