Circle plus

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Let $h>0$ and $z_1,z_2 \in \mathbb{C}_h$, the Hilger complex plane. Then we define the $\oplus_h$ operation by $$z_1 \oplus_h z_2 = z_1+z_2+z_1 z_2h.$$

Properties

Theorem: The structure $(\mathbb{C}_h,\oplus_h)$ is an Abelian group.

Proof:

Circle minus inverse of circle plus
Hilger real part oplus Hilger imaginary part equals z