inward reloid


Funcoidal products inside an inward reloid ★★

Author(s): Porton

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.

A stronger conjecture:

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) $.

Keywords: inward reloid

Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

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 funcoids $ f $ and $ g $.

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

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

Author(s): Porton

Conjecture   For any funcoid $ f $ and reloid $ g $ having the same source and destination \[ ( \mathsf{\tmop{RLD}})_{\tmop{out}} f \subseteq g \subseteq (    \mathsf{\tmop{RLD}})_{\tmop{in}} f \Leftrightarrow ( \mathsf{\tmop{FCD}}) g    = f. \]

Keywords: funcoid; inward reloid; outward reloid; reloid

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

Author(s): Porton

Conjecture   $ ( \mathsf{\tmop{RLD}})_{\tmop{in}} ( \mathsf{\tmop{FCD}}) f = f $ for any convex reloid $ f $.

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

Funcoid corresponding to inward reloid ★★

Author(s): Porton

Conjecture   $ ( \mathsf{\tmop{FCD}}) ( \mathsf{\tmop{RLD}})_{\tmop{in}} f = f $ for any funcoid $ f $.

Keywords: funcoid; inward reloid; reloid

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