# Natural unwound of a cyclically ordered group

The Rieger's unwound (https://arxiv.org/pdf/1311.0499.pdf) of a cyclically ordered group $$G(+)$$ is the Cartesian product $$\mathbb Z \times G$$ with a binary operation defined as:

$$(m, a) + (n, b) =$$

• $$(m + n, a + b)$$ if $$a = 0$$ or $$b = 0$$ or $$[0, a, a + b]$$;
• $$(m + n + 1, a + b)$$ otherwise.

The Rieger's unwound uses $$0$$ as the cutting element for the cyclic order of $$G$$.

Using the natural cut (Natural cut of a cyclically ordered group) we can introduce the natural unwound:

the Cartesian product $$\mathbb Z \times G$$ with a binary operation defined on the natural cut of $$G$$:

$$(m, a) + (n, b) =$$

• $$(m + n - 1, a + b)$$ if $$a < 0$$ and $$b < 0$$ and $$a + b > 0$$;
• $$(m + n + 1, a + b)$$ if $$a > 0$$ and $$b > 0$$ and $$a + b \le 0$$;
• $$(m + n, a + b)$$ otherwise.

The natural unwound is still a group:

• it is associative (checking all the combinations);
• $$(0, 0)$$ is the identity element;
• $$(-n, -a)$$ is the inverse element of $$(n, a)$$ if $$a$$ is not an apex;
• $$(-n - 1, a)$$ is the inverse element of $$(n, a)$$ if $$a$$ is an apex.

Is this correct?