A new diagram about funcoids and reloids

Define for posets with order \sqsubseteq:

  1. \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigsqcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \};
  2. \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigsqcap \{ 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 get the following diagram (see this PDF file for a proof):

diagram2

It is yet unknown what will happen if we keep apply \Phi_{\ast} and/or \Phi^{\ast} to the node “other”. Will this lead to a finite or infinite set?

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