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May 23, 2012 / porton

Candidate formulas for product of reloids

First, we can define product of reloids as a trivial generalization of the alternative definition of product of uniform spaces.

There are no trivial simplification of this relatively inelegant definition, it is not algebraic as I would want.

I (without any evidence or intuition) propose two open questions one of which may be true despite of no evidence. (Having no evidence I expect that the answers to these questions are false, but I may happily mistake about this.)

The displacement of a pointfree funcoid f\in \mathsf{FCD}( \mathsf{FCD}(\mathrm{Src}\,f;\mathrm{Src}\,g) ; \mathsf{FCD}(\mathrm{Dst}\,f;\mathrm{Dst}\,g) ) is the upgrading of downgrading the pointfree funcoid f where downgrading is taken for the filtrator (\mathsf{FCD}( \mathsf{FCD}(\mathrm{Src}\,f;\mathrm{Src}\,g) ; \mathsf{FCD}(\mathrm{Dst}\,f;\mathrm{Dst}\,g) ); \mathsf{FCD}(\mathrm{Src}\,f\times\mathrm{Src}\,g; \mathrm{Dst}\,f\times\mathrm{Dst}\,g)) and upgrading for the filtrator (\mathsf{FCD}( \mathsf{RLD}(\mathrm{Src}\,f;\mathrm{Src}\,g) ; \mathsf{RLD}(\mathrm{Dst}\,f;\mathrm{Dst}\,g) ); \mathsf{FCD}(\mathrm{Src}\,f\times\mathrm{Src}\,g; \mathrm{Dst}\,f\times\mathrm{Dst}\,g)).

The displaced product f\times^\mathrm{(DP)} g of reloids f and g is the displacement of the cross-composition product f\times^\mathrm{(C)} g of reloids f and g.

Question The product of reloids f and g is the reloid (\mathsf{RLD})_{\mathrm{in}}(f\times^\mathrm{(DP)} g).

Question The product of reloids f and g is the reloid (\mathsf{RLD})_{\mathrm{out}}(f\times^\mathrm{(DP)} g).

Read about funcoids and reloids here.

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