funcoids


Entourages of a composition of funcoids ★★

Author(s): Porton

\begin{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 $g$. \end{conjecture}

Keywords: composition of funcoids; funcoids

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

Author(s): Porton

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

Keywords: funcoid corresponding to reloid; funcoids; reloids

Inner reloid through the lattice Gamma ★★

Author(s): Porton

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

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 \href[this online article]{http://www.mathematics21.org/binaries/funcoids-are-filters.pdf}.

Keywords: filters; funcoids; inner reloid; reloids

Coatoms of the lattice of funcoids

Author(s): Porton

\begin{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? \end{problem}

Keywords: atoms; coatoms; funcoids

Domain and image of inner reloid ★★

Author(s): Porton

\begin{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 $f$. \end{conjecture}

Keywords: domain; funcoids; image; reloids

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