In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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. They are a generalization of
cyclic groups: the
infinite cyclic group and the
finite cyclic groups . Since a
linear order induces a cyclic order, cyclically ordered groups are also a generalization of
linearly ordered group
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group (mathematics), group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G ...
s: the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s , the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s , and so on. Some of the most important cyclically ordered groups fall into neither previous category: the
circle group and its
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
s, such as the
subgroup of rational points.
Quotients of linear groups
It is natural to depict cyclically ordered groups as
quotients: one has and . Even a once-linear group like , when bent into a circle, can be thought of as . showed that this picture is a generic phenomenon. For any ordered group and any
central element that generates a
cofinal subgroup of , the quotient group is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as such a quotient group.
The circle group
built upon Rieger's results in another direction. Given a cyclically ordered group and an ordered group , the product is a cyclically ordered group. In particular, if is the circle group and is an ordered group, then any subgroup of is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as a subgroup of such a product with .
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 such that for every positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. Since only positive are considered, this is a stronger condition than its linear counterpart. For example, no longer qualifies, since one has for every .
As a corollary to Świerczkowski's proof, every Archimedean cyclically ordered group is a subgroup of itself. This result is analogous to
Otto Hölder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of .
[, cited after ]
Topology
Every
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
cyclically ordered group is a subgroup of .
Related structures
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".
Notes
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{{DEFAULTSORT:Cyclically Ordered Group
Ordered groups
Circles