Compatible condition

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Compatible condition is a condition on which a mathematical construction on a number field [math]\displaystyle{ F }[/math] can be introduced or borrowed into an arithmetic expression space.


Let's talk about a function [math]\displaystyle{ k }[/math] over the number field [math]\displaystyle{ F }[/math]:

[math]\displaystyle{ k: F \to F }[/math]



[math]\displaystyle{ k: x \mapsto y }[/math]


We can introduce a transformation [math]\displaystyle{ l }[/math] over arithmetical expression space [math]\displaystyle{ H }[/math], i.e.

[math]\displaystyle{ l: H \to H }[/math]



[math]\displaystyle{ l: u \mapsto v }[/math]


Under this setting, we have below condition

[math]\displaystyle{ \begin{cases} x = \nu(u) \\ y = \nu(v) \end{cases} }[/math]


i.e, [math]\displaystyle{ k }[/math] is a shadow or projection of [math]\displaystyle{ l }[/math] from [math]\displaystyle{ H }[/math] to [math]\displaystyle{ F }[/math] by evaluation function [math]\displaystyle{ \nu }[/math], and equation (5) is the compatible condition for this case.

And we say the compatible condition (5) projects the transformation [math]\displaystyle{ l }[/math] over [math]\displaystyle{ H }[/math] to a function [math]\displaystyle{ k }[/math] over [math]\displaystyle{ F }[/math].


The above diagram commute.

See also