A diagram about funcoids and reloids

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
Keywords: Galois connections
Recomm. for undergrads: no
Posted by: porton
on: November 26th, 2016

Define for posets with order $ \sqsubseteq $:

  1. $ \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \} $;
  2. $ \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.


Blog post

* indicates original appearance(s) of problem.

The value of node "other"

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 $.

This is my very recent thoughts and yet needs to be checked.

-- Victor Porton - http://www.mathematics21.org

The diagram was with an error

My diagram was with an error. I have uploaded a corrected version of the diagram.

Victor Porton - http://www.mathematics21.org

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