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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?

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