Decomposition of completions of reloids

 Importance: Medium ✭✭
 Author(s): Porton, Victor
 Subject: Topology
 Keywords: co-completion completion reloid
 Posted by: porton on: September 2nd, 2013

\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}

Well, in fact this is three separate problems (if we count dual formulas as one formula), but I am lazy to create three pages for them.

This conjecture is inspired by the proven fact that the above formulas hold for every composable funcoids $f$ and $g$ (instead of reloids). Properties of reloids are expected to be similar to properties of funcoids.

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Bibliography

*\href [Algebraic General Toplogy. Volume 1]{http://www.mathematics21.org/algebraic-general-topology.html}

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.