A new mapping from funcoids to reloids

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}.

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