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
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) $?

A weaker conjecture:

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} $.

See 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.


*Victor Porton. Algebraic General Topology

* indicates original appearance(s) of problem.