August 29, 2012 / porton

A conjecture related with subatomic product

With subatomic products first mentioned here and described in this article are related the following conjecture (or being precise three conjectures):

Conjecture For every funcoid $f: \prod A\rightarrow\prod B$ (where $A$ and $B$ are indexed families of sets) there exists a funcoid $\Pr^{\left( A \right)}_k f$ defined by the formula

$x \mathrel{\left[ \Pr^{\left( A \right)}_k f \right]} y \Leftrightarrow \prod^{\mathsf{\mathrm{RLD}}} \left( \left\{ \begin{array}{ll} 1^{\mathfrak{F} \left( \mathrm{Base} \left( x \right) \right)} & \mathrm{if} i \neq k ;\\ x & \mathrm{if} i = k \end{array} \right. \right) \mathrel{\left[ f \right]} \prod^{\mathsf{\mathrm{RLD}}} \left( \left\{ \begin{array}{ll} 1^{\mathfrak{F} \left( \mathrm{Base} \left( y \right) \right)} & \mathrm{if} i \neq k ;\\ y & \mathrm{if} i = k \end{array} \right. \right)$

for: