A diagram about funcoids and reloids

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Posted	by:	porton
	on:	November 26th, 2016

Define for posets with order $\sqsubseteq$ :

$\Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$ ;
$\Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigcap \{ x \in \mathfrak{B} \mid f x \sqsupseteq b \}$ .

Note that the above is a generalization of monotone Galois connections (with $\max$ and $\min$ replaced with suprema and infima).

Then we have the following diagram:

What is at the node "other" in the diagram is unknown.

Conjecture "Other" is $\lambda f\in\mathsf{FCD}: \top$ .

Question What repeated applying of $\Phi_{\ast}$ and $\Phi^{\ast}$ to "other" leads to? Particularly, does repeated applying $\Phi_{\ast}$ and/or $\Phi^{\ast}$ to the node "other" lead to finite or infinite sets?

See Algebraic General Topology for definitions of used concepts.

The known part of the diagram is considered in this file.

Bibliography

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* indicates original appearance(s) of problem.

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The value of node "other"

On November 29th, 2016 porton says:

It seems that the node "other" is not $\lambda f\in\mathsf{FCD}: \top$ .

I conjecture $\langle \Phi_{\ast} (\mathsf{RLD})_{\operatorname{out}} \rangle f = (\mathsf{FCD}) f$ where $f$ is the reloid defined by the cofinite filter on $A \times B$ and thus $\langle (\mathsf{FCD}) f \rangle \{ x \} = \bot$ for all singletons $\{ x \}$ and $\langle (\mathsf{FCD}) f \rangle p = \top$ for every nontrivial atomic filter $p$ .