# reloid

## Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

\begin{conjecture} Let $f$ is a $T_1$-separable (the same as $T_2$ for symmetric transitive) compact funcoid and $g$ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $( \mathsf{\tmop{FCD}}) g = f$. Then $g = \langle f \times f \rangle^{\ast} \Delta$. \end{conjecture}

The main purpose here is to find a \emph{direct} proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in \href[this draft article]{http://www.mathematics21.org/binaries/compact.pdf}.

Direct proof could be better because with it we would get a little more general statement like this:

\begin{conjecture} Let $f$ be a $T_1$-separable compact reflexive symmetric funcoid and $g$ be a reloid such that \begin{enumerate} \item $( \mathsf{\tmop{FCD}}) g = f$; \item $g \circ g^{- 1} \sqsubseteq g$. \end{enumerate} Then $g = \langle f \times f \rangle^{\ast} \Delta$. \end{conjecture}

## Decomposition of completions of reloids ★★

Author(s): Porton

\begin{conjecture} For composable reloids $f$ and $g$ it holds \begin{enumerate} \item $\operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f$ if $f$ is a co-complete reloid; \item $\operatorname{CoCompl} ( f \circ g) = f \circ \operatorname{CoCompl} g$ if $f$ is a complete reloid; \item $\operatorname{CoCompl} ( ( \operatorname{Compl} g) \circ f) = \operatorname{Compl} ( g \circ ( \operatorname{CoCompl} f)) = ( \operatorname{Compl} g) \circ ( \operatorname{CoCompl} f)$; \item $\operatorname{Compl} ( g \circ ( \operatorname{Compl} f)) = \operatorname{Compl} ( g \circ f)$; \item $\operatorname{CoCompl} ( ( \operatorname{CoCompl} g) \circ f) = \operatorname{CoCompl} ( g \circ f)$. \end{enumerate} \end{conjecture}

Keywords: co-completion; completion; reloid

## Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

\begin{conjecture} $( \mathsf{\tmop{RLD}})_{\tmop{in}} (g \circ f) = ( \mathsf{\tmop{RLD}})_{\tmop{in}} g \circ ( \mathsf{\tmop{RLD}})_{\tmop{in}} f$ for any composable \href[funcoids]{http://www.wikinfo.org/index.php/Funcoid} $f$ and $g$. \end{conjecture}

## Atomic reloids are monovalued ★★

Author(s): Porton

\begin{conjecture} Atomic \href[reloids]{http://www.wikinfo.org/index.php/Reloid} are monovalued. \end{conjecture}

Keywords: atomic reloid; monovalued reloid; reloid

## Composition of atomic reloids ★★

Author(s): Porton

\begin{conjecture} Composition of two atomic \href[reloids]{http://www.wikinfo.org/index.php/Reloid} is atomic or empty. \end{conjecture}

Keywords: atomic reloid; reloid

## S(S(f)) = S(f) for reloids ★★

Author(s): Porton

\begin{question} $S(S(f)) = S(f)$ for every endo-\href[reloid]{http://www.wikinfo.org/index.php/Reloid} $f$? \end{question}

Keywords: reloid

## Reloid corresponding to funcoid is between outward and inward reloid ★★

Author(s): Porton

\begin{conjecture} For any \href[funcoid]{http://www.wikinfo.org/index.php/Funcoid} $f$ and \href[reloid]{http://www.wikinfo.org/index.php/Reloid} $g$ having the same source and destination $( \mathsf{\tmop{RLD}})_{\tmop{out}} f \subseteq g \subseteq ( \mathsf{\tmop{RLD}})_{\tmop{in}} f \Leftrightarrow ( \mathsf{\tmop{FCD}}) g = f.$ \end{conjecture}

Keywords: funcoid; inward reloid; outward reloid; reloid

## Distributivity of union of funcoids corresponding to reloids ★★

Author(s): Porton

\begin{conjecture} $\bigcup \left\langle ( \mathsf{\tmop{FCD}}) \right\rangle S = ( \mathsf{\tmop{FCD}}) \bigcup S$ if $S\in\mathscr{P}\mathsf{RLD}(A;B)$ is a set of \href[reloids]{http://www.wikinfo.org/index.php/Reloid} from a set $A$ to a set $B$. \end{conjecture}

Keywords: funcoid; infinite distributivity; reloid

## Inward reloid corresponding to a funcoid corresponding to convex reloid ★★

Author(s): Porton

\begin{conjecture} $( \mathsf{\tmop{RLD}})_{\tmop{in}} ( \mathsf{\tmop{FCD}}) f = f$ for any \href[convex reloid]{http://www.wikinfo.org/index.php/Convex_reloid} $f$. \end{conjecture}

Keywords: convex reloid; funcoid; functor; inward reloid; reloid

## Outward reloid corresponding to a funcoid corresponding to convex reloid ★★

Author(s): Porton

\begin{conjecture} $( \mathsf{\tmop{RLD}})_{\tmop{out}} ( \mathsf{\tmop{FCD}}) f = f$ for any \href[convex reloid]{http://www.wikinfo.org/index.php/Convex_reloid} $f$. \end{conjecture}

Keywords: convex reloid; funcoid; functor; outward reloid; reloid