Subatomic products – a new kind of product of funcoids

I’ve discovered a new kind of product of funcoids, which I call subatomic product.

Definition Let f : A_0 \rightarrow A_1 and g : B_0 \rightarrow B_1 are funcoids. Then f \times^{\left( A \right)} g (subatomic product) is a funcoid A_0 \times B_0 \rightarrow A_1 \times B_1 such that for every a \in \mathrm{atoms}\,1^{\mathfrak{F} \left( A_0 \times B_0 \right)}, b \in \mathrm{atoms}\,1^{\mathfrak{F} \left( A_1 \times B_1 \right)}

a \mathrel{\left[ f \times^{\left( A \right)} g \right]} b \Leftrightarrow \mathrm{dom}\,a \mathrel{\left[ f \right]} \mathrm{dom}\,b \wedge \mathrm{im}\,a \mathrel{\left[ g \right]} \mathrm{im}\,b.

This (subatomic) composition has the merit that for funcoids f : A \rightarrow B and g : A \rightarrow C the destination of product is B \times C is the same as for categorical product in the category \boldsymbol{\mathrm{Set}}.

See This online draft article for details. There it is also proved that subatomic product exists.

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