In
mathematics, a cyclically ordered group is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
with both a
group structure and a
cyclic order
In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Ins ...
, 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 group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s: the
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
and the
finite cyclic groups . Since a
linear order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
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 ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G'', ≤) is a:
* le ...
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 (e.g. ). The set of all rat ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. 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
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Chris ...
'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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
to a certain subcategory of
MV-algebras, the "projectable MV-algebras".
Notes
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{{DEFAULTSORT:Cyclically Ordered Group
Ordered groups
Circles