Strict inequalities for products of filters

Importance: Low ✭
Author(s): Porton, Victor
Subject: Topology
Keywords: filter products
Recomm. for undergrads: no
Posted by: porton
on: August 9th, 2011

\begin{conjecture} $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B}$ for some filter objects $\mathcal{A}$, $\mathcal{B}$. Particularly, is this formula true for $\mathcal{A} = \mathcal{B} = \Delta \cap \uparrow^{\mathbb{R}} \left( 0 ; + \infty \right)$? \end{conjecture}

A weaker conjecture:

\begin{conjecture} $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B}$ for some filter objects $\mathcal{A}$, $\mathcal{B}$. \end{conjecture}

See \href [Algebraic General Topology]{} for definitions of used concepts.

The first conjecture probably has no use by itself but proving it may be somehow challenging, just like Fermat Last Theorem.

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*Victor Porton. \href [Algebraic General Topology]{} % (Put an empty line between individual entries)

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