Less than a hour ago I discovered a new mapping from funcoids to reloids:

Definition $(\mathsf{RLD})_X f = \bigsqcap \left\{ g \in \mathsf{RLD} \mid (\mathsf{FCD}) g \sqsupseteq f \right\}$ for every funcoid $f$.

Now I am going to work on the following conjectures:

Conjecture $(\mathsf{RLD})_X f = \min \left\{ g \in \mathsf{RLD} \mid (\mathsf{FCD}) g \sqsupseteq f \right\}$, that is $(\mathsf{RLD})_X$ is the lower adjoint of $(\mathsf{FCD})$.

Conjecture $(\mathsf{RLD})_X f = f$ if $f$ is a principal funcoid.

Conjecture $(\mathsf{RLD})_X (f|_\mathcal{A}) = ((\mathsf{RLD})_X f)|_\mathcal{A}$.

Note that from the two last conjectures it follows that $(\mathsf{RLD})_X \mathrm{id}^{\mathsf{FCD}}_\mathcal{A} = \mathrm{id}^{\mathsf{RLD}}_\mathcal{A}$.

#### One Comment

1. porton / Jul 25 2016 23:09

$(\mathsf{RLD})_X$ is not a lower adjoint of $(\mathsf{FCD})$ because $(\mathsf{FCD})$ does not preserve binary meets.

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