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July 26, 2012 / porton

A conjecture about direct product of funcoids

I am attempting to define direct products in the category cont(mepfFcd) (the category of monovalued, entirely defined continuous pointfree funcoids), see this draft article for a definition of this category.

A direct product of objects may possibly be defined as the cross-composition product (see this article).

A candidate for product of morphisms f_1:\mathfrak{A}\rightarrow\mathfrak{B} and f_2:\mathfrak{A}\rightarrow\mathfrak{C} of this category is the pointfree funcoid f_1\times^{(D)} f_2 defined by the formula:

\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X  \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} |  \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x  \right\}.

The trouble is that I failed to prove that such a pointfree funcoid exists. So a conjecture:

Conjecture If f_1 and f_2 are monovalued, entirely defined funcoids then the pointfree funcoid defined by the above formula exists.

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