Topology

Decomposition of completions of reloids ★★

Author(s): Porton

Conjecture   For composable reloids $ f $ and $ g $ it holds
    \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) $.

Keywords: co-completion; completion; reloid

Posted by porton
updated September 2nd, 2013
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Topology

A construction of direct product in the category of continuous maps between endo-funcoids ★★★

Author(s): Porton

Consider the category of (proximally) continuous maps (entirely defined monovalued functions) between endo-funcoids.

Remind from my book that morphisms $ f: A\rightarrow B $ of this category are defined by the formula $ f\circ A\sqsubseteq B\circ f $ (here and below by abuse of notation I equate functions with corresponding principal funcoids).

Let $ F_0, F_1 $ are endofuncoids,

We define $ F_0\times F_1 = \bigsqcup \left\{ \Phi \in \mathsf{FCD} \,|\, \pi_0 \circ \Phi \sqsubseteq F_0 \circ \pi_0 \wedge \pi_1 \circ \Phi \sqsubseteq F \circ \pi_1 \right\} $

(here $ \pi_0 $ and $ \pi_1 $ are cartesian projections).

Conjecture   The above defines categorical direct product (in the above mentioned category, with products of morphisms the same as in Set).

Keywords: categorical product; direct product

Posted by porton
updated August 31st, 2013
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Graph Theory » Coloring

List Total Colouring Conjecture ★★

Author(s): Borodin; Kostochka; Woodall

Conjecture   If $ G $ is the total graph of a multigraph, then $ \chi_\ell(G)=\chi(G) $.

Keywords: list coloring; Total coloring; total graphs

Posted by Jon Noel
updated August 29th, 2013
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Geometry

Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by projective plane we mean the set of all lines through the origin in $ \mathbb{R}^3 $.

Definition   Say that a subset $ S $ of the projective plane is octahedral if all lines in $ S $ pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition   Say that a subset $ S $ of the projective plane is weakly octahedral if every set $ S'\subseteq S $ such that $ |S'|=3 $ is octahedral.
Conjecture   Suppose that the projective plane can be partitioned into four sets, say $ S_1,S_2,S_3 $ and $ S_4 $ such that each set $ S_i $ is weakly octahedral. Then each $ S_i $ is octahedral.

Keywords: Partitioning; projective plane

Posted by Jon Noel
updated August 27th, 2013
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