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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:

  1. every filters x and y;
  2. every principal filters x and y;
  3. every atomic filters x and y.
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