funcoid


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}

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity

Distributivity of a lattice of funcoids is not provable without axiom of choice

Author(s): Porton

\begin{conjecture} Distributivity of the lattice $\mathsf{FCD}(A;B)$ of funcoids (for arbitrary sets $A$ and $B$) is not provable in ZF (without axiom of choice). \end{conjecture}

A similar conjecture:

\begin{conjecture} $a\setminus^{\ast} b = a\#b$ for arbitrary filters $a$ and $b$ on a powerset cannot be proved in ZF (without axiom of choice). \end{conjecture}

Keywords: axiom of choice; distributive lattice; distributivity; funcoid; reverse math; reverse mathematics; ZF; ZFC

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}

Keywords: distributive; distributivity; funcoid; functor; inward reloid; 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

Distributivity of outward reloid over composition of funcoids ★★

Author(s): Porton

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

Keywords: distributive; distributivity; funcoid; functor; outward reloid; reloid

Funcoid corresponding to inward reloid ★★

Author(s): Porton

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

Keywords: funcoid; inward reloid; reloid

Intersection of complete funcoids ★★

Author(s): Porton

\begin{conjecture} If $f$, $g$ are \href[complete funcoids]{http://www.wikinfo.org/index.php/Complete_funcoid} (generalized closures) then $f \cap^{\mathsf{\tmop{FCD}}} g$ is a complete funcoid (generalized closure). \end{conjecture}

Keywords: complete funcoid; funcoid; generalized closure

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