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Importance: Medium ✭✭

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Keywords:	cross-composition product
	quasi-cartesian function

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 $\{ G\circ F \,|\, F\in f, G\in g\}.$