Open Problem Garden

Upgrading a completary multifuncoid ★★

Author(s): Porton

Let $\mho$ be a set, $\mathfrak{F}$ be the set of filters on $\mho$ ordered reverse to set-theoretic inclusion, $\mathfrak{P}$ be the set of principal filters on $\mho$ , let $n$ be an index set. Consider the filtrator $\left( \mathfrak{F}^n ; \mathfrak{P}^n \right)$ .

Conjecture If $f$ is a completary multifuncoid of the form $\mathfrak{P}^n$ , then $E^{\ast} f$ is a completary multifuncoid of the form $\mathfrak{F}^n$ .

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords:

Posted by porton
updated February 4th, 2012

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pseudodifference

graph minors

Joret, Gwenaël

Barát ,János

Graph Theory » Basic G.T. » Minors

Forcing a $K_6$-minor ★★

Author(s): Barát ; Joret; Wood

Conjecture Every graph with minimum degree at least 7 contains a $K_6$ -minor.

Conjecture Every 7-connected graph contains a $K_6$ -minor.

Keywords: connectivity; graph minors

Posted by David Wood
updated January 16th, 2012

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Topology

Funcoidal products inside an inward reloid ★★

Author(s): Porton

Conjecture (solved) If $a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f$ then $a \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} b \subseteq f$ for every funcoid $f$ and atomic f.o. $a$ and $b$ on the source and destination of $f$ correspondingly.

A stronger conjecture:

Conjecture If $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f$ then $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} \mathcal{B} \subseteq f$ for every funcoid $f$ and $\mathcal{A} \in \mathfrak{F} \left( \ensuremath{\operatorname{Src}}f \right)$ , $\mathcal{B} \in \mathfrak{F} \left( \ensuremath{\operatorname{Dst}}f \right)$ .

Keywords: inward reloid

Posted by porton
updated January 1st, 2012

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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 $g$ .