reloid

Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

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 $g = \langle f \times f \rangle^{\ast} \Delta$ .

The main purpose here is to find a 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 this draft article.

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

Conjecture Let $f$ be a $T_1$ -separable compact reflexive symmetric funcoid and $g$ be a reloid such that

$( \mathsf{\tmop{FCD}}) g = f$

$g \circ g^{- 1} \sqsubseteq g$

Then $g = \langle f \times f \rangle^{\ast} \Delta$ .

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

Posted by porton
updated February 8th, 2014

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Topology

Decomposition of completions of reloids ★★

Author(s): Porton

Conjecture For composable reloids $f$ and $g$ it holds

$\operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f$

$f$

$\operatorname{CoCompl} ( f \circ g) = f \circ \operatorname{CoCompl} g$

$f$

$\operatorname{CoCompl} ( ( \operatorname{Compl} g) \circ f) = \operatorname{Compl} ( g \circ ( \operatorname{CoCompl} f)) = ( \operatorname{Compl} g) \circ ( \operatorname{CoCompl} f)$

$\operatorname{Compl} ( g \circ ( \operatorname{Compl} f)) = \operatorname{Compl} ( g \circ f)$

$\operatorname{CoCompl} ( ( \operatorname{CoCompl} g) \circ f) = \operatorname{CoCompl} ( g \circ f)$

Keywords: co-completion; completion; reloid

Posted by porton
updated September 2nd, 2013

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Topology

Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

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 funcoids $f$ and $g$ .

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

Posted by porton
updated December 31st, 2011

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Topology

Atomic reloids are monovalued ★★

Author(s): Porton

Conjecture Atomic reloids are monovalued.

Keywords: atomic reloid; monovalued reloid; reloid

Posted by porton
updated February 4th, 2009

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Topology

Composition of atomic reloids ★★

Author(s): Porton

Conjecture Composition of two atomic reloids is atomic or empty.

Keywords: atomic reloid; reloid

Posted by porton
updated February 4th, 2009

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Topology

S(S(f)) = S(f) for reloids ★★

Author(s): Porton

Question $S(S(f)) = S(f)$ for every endo-reloid $f$ ?

Keywords: reloid

Posted by porton
updated May 8th, 2008

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Topology

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

Author(s): Porton

Conjecture For any funcoid $f$ and reloid $g$ having the same source and destination $( \mathsf{\tmop{RLD}})_{\tmop{out}} f \subseteq g \subseteq ( \mathsf{\tmop{RLD}})_{\tmop{in}} f \Leftrightarrow ( \mathsf{\tmop{FCD}}) g = f.$

Keywords: funcoid; inward reloid; outward reloid; reloid

Posted by porton
updated September 2nd, 2007

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Topology

Distributivity of union of funcoids corresponding to reloids ★★

Author(s): Porton

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 reloids from a set $A$ to a set $B$ .