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April 23, 2016 / porton

An important conjecture about funcoids

Just a few minutes ago I’ve formulated a new important conjecture about funcoids:

Let A, B be sets.

Conjecture Funcoids f from A to B bijectively corresponds to the sets R of pairs
(\mathcal{X}; \mathcal{Y}) of filters (on A and B correspondingly) that

  1. R is nonempty.
  2. R is a lower set.
  3. R (ordered pointwise) is a dcpo

by the mutually inverse formulas:
(\mathcal{X} ; \mathcal{Y}) \in R \Leftrightarrow \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \sqsubseteq f \quad \mathrm{and} \quad f = \bigsqcup^{\mathsf{FCD}} \left\{ \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}.

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4 Comments

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  1. porton / Apr 23 2016 01:16

    Oh, a trivial counter-example: f = ([0 ; 2] \times [0 ; 1]) \cup ([0 ; 1] \times [0 ; 2]). I will try to figure another axioms.

  2. porton / Apr 23 2016 02:04

    Oh, sorry, the counter-example is wrong. However it inspired me with another similar conjecture.

  3. porton / Apr 23 2016 20:30

    No, it is wrong. For a counter-example take R=\{\uparrow\emptyset,\uparrow\{0\},\uparrow\{1\}\}.

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