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.

\begin{conjecture} "Other" is $\lambda f\in\mathsf{FCD}: \top$. \end{conjecture}

\begin{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? \end{question}

See \href [Algebraic General Topology]{} for definitions of used concepts.

The known part of the diagram is considered in \href[this file]{}.

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