# Compatible condition

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

## Example

Let's talk about a function $\displaystyle{ k }$ over the number field $\displaystyle{ F }$:

 $\displaystyle{ k: F \to F }$ (1)

and

 $\displaystyle{ k: x \mapsto y }$ (2)

We can introduce a transformation $\displaystyle{ l }$ over arithmetical expression space $\displaystyle{ H }$, i.e.

 $\displaystyle{ l: H \to H }$ (3)

and

 $\displaystyle{ l: u \mapsto v }$ (4)

Under this setting, we have below condition

 $\displaystyle{ \begin{cases} x = \nu(u) \\ y = \nu(v) \end{cases} }$ (5)

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

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

The above diagram commute.