Cyclically ordered group
{{Short description|Group with a cyclic order respected by the group operation}}
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.{{sfn|Pecinová-Kozáková|2005|p=194}} They are a generalization of cyclic groups: the infinite cyclic group {{math|Z}} and the finite cyclic groups {{math|Z/n}}. Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers {{math|Q}}, the real numbers {{math|R}}, and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group {{math|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 {{math|1=Zn = Z/nZ}} and {{math|1=T = R/Z}}. Even a once-linear group like {{math|Z}}, when bent into a circle, can be thought of as {{math|Z2 / Z}}. {{Harvs|last=Rieger|year=1946|year2=1947|year3=1948|txt}} showed that this picture is a generic phenomenon. For any ordered group {{mvar|L}} and any central element {{mvar|z}} that generates a cofinal subgroup {{mvar|Z}} of {{mvar|L}}, the quotient group {{math|L / Z}} is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as such a quotient group.{{sfn|Świerczkowski|1959a|p=162}}
The circle group
{{Harvtxt|Świerczkowski|1959a}} built upon Rieger's results in another direction. Given a cyclically ordered group {{mvar|K}} and an ordered group {{mvar|L}}, the product {{math|K × L}} is a cyclically ordered group. In particular, if {{math|T}} is the circle group and {{mvar|L}} is an ordered group, then any subgroup of {{math|T × L}} is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as a subgroup of such a product with {{math|T}}.{{sfn|Świerczkowski|1959a|pp=161–162}}
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 {{math|x, y}} such that {{math|[e, xn, y]}} for every positive integer {{mvar|n}}.{{sfn|Świerczkowski|1959a|pp=161–162}} Since only positive {{mvar|n}} are considered, this is a stronger condition than its linear counterpart. For example, {{math|Z}} no longer qualifies, since one has {{math|[0, n, −1]}} for every {{mvar|n}}.
As a corollary to Świerczkowski's proof, every Archimedean cyclically ordered group is a subgroup of {{math|T}} itself.{{sfn|Świerczkowski|1959a|pp=161–162}} This result is analogous to Otto Hölder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of {{math|R}}.{{Harvnb|Hölder|1901}}, cited after {{Harvnb|Hofmann|Lawson|1996|pp=19, 21, 37}}
Topology
Every compact cyclically ordered group is a subgroup of {{math|T}}.
Related structures
{{Harvtxt|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".{{sfn|Gluschankof|1993|p=261}}
Notes
{{Reflist|22em}}
References
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Further reading
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- {{Citation |last=Černák |first=Štefan |year=1989a |chapter=Completion and Cantor extension of cyclically ordered groups |editor-first=Katarzyna |editor-last=Hałkowska |editor2-first=Boguslaw |editor2-last=Stawski |title=Universal and Applied Algebra (Turawa, 1988) |publisher=World Scientific |pages=13–22 |isbn=978-9971-5-0837-1 |mr=1084391}}
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- {{Citation |last=Černák |first=Štefan |year=1991 |title=On the completion of cyclically ordered groups |journal=Mathematica Slovaca |volume=41 |issue=1 |pages=41–49 |hdl=10338.dmlcz/131783 |url=http://www.dml.cz/bitstream/handle/10338.dmlcz/131783/MathSlov_41-1991-1_7.pdf |access-date=22 May 2011}}
- {{Citation |last=Černák |first=Štefan |year=1995 |title=Lexicographic products of cyclically ordered groups |journal=Mathematica Slovaca |volume=45 |issue=1 |pages=29–38 |hdl=10338.dmlcz/130473 |url=http://dml.cz/bitstream/handle/10338.dmlcz/130473/MathSlov_45-1995-1_4.pdf |access-date=21 May 2011}}
- {{Citation |last=Černák |first=Štefan |year=2001 |title=Cantor extension of a half linearly cyclically ordered group |journal=Discussiones Mathematicae - General Algebra and Applications |volume=21 |issue=1 |pages=31–46 |doi=10.7151/dmgaa.1025 }}
- {{Citation |last=Černák |first=Štefan |year=2002 |title=Completion of a half linearly cyclically ordered group |journal=Discussiones Mathematicae - General Algebra and Applications |volume=22 |issue=1 |pages=5–23 |doi=10.7151/dmgaa.1043 |doi-access=free }}
- {{Citation |last1=Černák |first1=Štefan |last2=Jakubík |first2=Ján |year=1987 |title=Completion of a cyclically ordered group |journal=Czechoslovak Mathematical Journal |volume=37 |issue=1 |pages=157–174 |doi=10.21136/CMJ.1987.102144 |mr=875137 |zbl=0624.06021 |hdl=10338.dmlcz/102144 |doi-access=free }}
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{{DEFAULTSORT:Cyclically Ordered Group}}