# Strict inequalities for products of filters

 Importance: Low ✭
 Author(s): Porton, Victor
 Subject: Topology
 Keywords: filter products
 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]{http://www.mathematics21.org/algebraic-general-topology.html} 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.

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

## Bibliography

*Victor Porton. \href [Algebraic General Topology]{http://www.mathematics21.org/algebraic-general-topology.html} % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.