Cross-composition product of reloids is a quasi-cartesian function

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
Recomm. for undergrads: no
Posted by: porton
on: July 5th, 2012
Conjecture   Cross-composition product (for small indexed families of reloids) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation $ \mathfrak{S}_0 $ of reloids to the quasi-cartesian situation $ \mathfrak{S}_1 $ of pointfree funcoids over posets with least elements.

This conjecture is unsolved even for product of two multipliers.

An obviously equivalent reformulation of this conjecture for the special case of two multipliers:

Conjecture   Provided that reloids $ f $ and $ g $ on some set $ \mho $ are proper filters, we can restore the values of $ f $ and $ g $ knowing only $ g\circ a\circ f^{-1} $ for every reloid $ a $ on $ \mho $.

Reloids are defined simply as filters on a Cartesian product of two sets. The reverse reloid $ f^{-1} $ of a reloids $ f $ is defined by the formula: $ f^{-1} = \{F^{-1} \,|\, F\in f \} $. Composition $ g\circ f $ of reloids $ f $ and $ g $ is defined as the reloid whose base is $$\{ G\circ F \,|\, F\in f, G\in g\}.$$

See Algebraic General Topology for definitions of used concepts.

Bibliography

*Victor Porton. Algebraic General Topology


* indicates original appearance(s) of problem.