Entourages of a composition of funcoids ★★

Author(s): Porton

Conjecture   $ \forall H \in \operatorname{up} (g \circ f) \exists F \in \operatorname{up} f, G \in \operatorname{up} g : H \sqsupseteq G \circ F $ for every composable funcoids $ f $ and $ g $.

Keywords: composition of funcoids; funcoids

Restricting a reloid to lattice Gamma before converting it into a funcoid ★★

Author(s): Porton

Conjecture   $ (\mathsf{FCD}) f = \bigcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \operatorname{GR} f) $ for every reloid $ f \in \mathsf{RLD} (A ; B) $.

Keywords: funcoid corresponding to reloid; funcoids; reloids

Inner reloid through the lattice Gamma ★★

Author(s): Porton

Conjecture   $ (\mathsf{RLD})_{\operatorname{in}} f = \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f $ for every funcoid $ f $.

Counter-example: $ (\mathsf{RLD})_{\operatorname{in}} f \sqsubset \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f $ for the funcoid $ f = (=)|_\mathbb{R} $ is proved in this online article.

Keywords: filters; funcoids; inner reloid; reloids

Coatoms of the lattice of funcoids

Author(s): Porton

Problem   Let $ A $ and $ B $ be infinite sets. Characterize the set of all coatoms of the lattice $ \mathsf{FCD}(A;B) $ of funcoids from $ A $ to $ B $. Particularly, is this set empty? Is $ \mathsf{FCD}(A;B) $ a coatomic lattice? coatomistic lattice?

Keywords: atoms; coatoms; funcoids

Domain and image of inner reloid ★★

Author(s): Porton

Conjecture   $ \ensuremath{\operatorname{dom}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{in}}} f =\ensuremath{\operatorname{dom}}f $ and $ \ensuremath{\operatorname{im}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{in}}} f =\ensuremath{\operatorname{im}}f $ for every funcoid $ f $.

Keywords: domain; funcoids; image; reloids

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