Cyclically ordered group

In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order.

Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947.^[1] They are a generalization of cyclic groups: the infinite cyclic group $Z$ and the finite cyclic groups $Z / n$ . Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers $Q$ , the real numbers $R$ , and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group $T$ and its subgroups, such as the subgroup of rational points.

Quotients of linear groups

It is natural to depict cyclically ordered groups as quotients: one has $Z n = Z / n Z$ and $T = R / Z$ . Even a once-linear group like $Z$ , when bent into a circle, can be thought of as $Z 2 / Z$ . Rieger (1946, 1947, 1948) showed that this picture is a generic phenomenon. For any ordered group $L$ and any central element $z$ that generates a cofinal subgroup $Z$ of $L$ , the quotient group $L / Z$ is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as such a quotient group.^[2]

The circle group

Świerczkowski (1959a) built upon Rieger's results in another direction. Given a cyclically ordered group $K$ and an ordered group $L$ , the product $K \times L$ is a cyclically ordered group. In particular, if $T$ is the circle group and $L$ is an ordered group, then any subgroup of $T \times L$ is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as a subgroup of such a product with $T$ .^[3]

By analogy with an Archimedean linearly ordered group, one can define an Archimedean cyclically ordered group as a group that does not contain any pair of elements $x, y$ such that $[e, x n, y]$ for every positive integer $n$ .^[3] Since only positive $n$ are considered, this is a stronger condition than its linear counterpart. For example, $Z$ no longer qualifies, since one has $[0, n, - 1]$ for every $n$ .

As a corollary to Świerczkowski's proof, every Archimedean cyclically ordered group is a subgroup of $T$ itself.^[3] This result is analogous to Otto Hölder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of $R$ .^[4]

Topology

Every compact cyclically ordered group is a subgroup of $T$ .

Related structures

Gluschankof (1993) showed that a certain subcategory of cyclically ordered groups, the "projectable Ic-groups with weak unit", is equivalent to a certain subcategory of MV-algebras, the "projectable MV-algebras".^[5]

Notes

↑ Pecinová-Kozáková 2005, p. 194.
↑ Świerczkowski 1959a, p. 162.
1 2 3 Świerczkowski 1959a, pp. 161–162.
↑ Hölder 1901, cited after Hofmann & Lawson 1996, pp. 19, 21, 37
↑ Gluschankof 1993, p. 261.

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[FOOTNOTEPecinová-Kozáková2005194-1] Pecinová-Kozáková 2005, p. 194.

[FOOTNOTEŚwierczkowski1959a162-2] Świerczkowski 1959a, p. 162.

[FOOTNOTEŚwierczkowski1959a161–162-3] 1 2 3 Świerczkowski 1959a, pp. 161–162.

[4] Hölder 1901, cited after Hofmann & Lawson 1996, pp. 19, 21, 37

[FOOTNOTEGluschankof1993261-5] Gluschankof 1993, p. 261.

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