Define for posets with order $latex \sqsubseteq$:

  1. $latex \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigsqcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$;
  2. $latex \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigsqcap \{ x \in \mathfrak{B} \mid f x \sqsupseteq b \}$.

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

Then we get the following diagram (see this PDF file for a proof):


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

1 thought on “A new diagram about funcoids and reloids

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