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Galois connections


Topology

A diagram about funcoids and reloids ★★

Author(s): Porton

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?

Keywords: Galois connections

Posted by porton
updated November 29th, 2016
2 comments | 2 attachments
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