In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a branch of
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 ...
, an Archimedean group is a
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 ...
for which the
Archimedean property
In abstract algebra and mathematical analysis, analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, Italy, Syracuse, is a property held by some algebraic structures, such as ordered or normed g ...
holds: every two positive group elements are bounded by
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 ...
multiples of each other. The set R of
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 together with the
operation of addition and the usual ordering relation between pairs of numbers is an Archimedean group. By a result of
Otto Hölder, every Archimedean group is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to a
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  ...
of this group. The name "Archimedean" comes from
Otto Stolz, who named the Archimedean property after its appearance in the works of
Archimedes
Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
.
Definition
An
additive group consists of a set of elements, an
associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
addition operation that combines pairs of elements and returns a single element,
an
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
(or zero element) whose sum with any other element is the other element, and an
additive inverse
In mathematics, the additive inverse of an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero el ...
operation such that the sum of any element and its inverse is zero.
A group is a
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 ...
when, in addition, its elements can be
linearly ordered in a way that is compatible with the group operation: for all elements ''x'', ''y'', and ''z'', if ''x'' ≤ ''y'' then ''x'' + ''z'' ≤ ''y'' + ''z'' and ''z'' + ''x'' ≤ ''z'' + ''y''.
The notation ''na'' (where ''n'' is a
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
) stands for the group sum of ''n'' copies of ''a''.
An Archimedean group (''G'', +, ≤) is a linearly ordered group subject to the following additional condition, the Archimedean property: For every ''a'' and ''b'' in ''G'' which are greater than 0, it is possible to find a natural number ''n'' for which the inequality ''b'' ≤ ''na'' holds.
An equivalent definition is that an Archimedean group is a linearly ordered group without any bounded
cyclic 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: there does not exist a cyclic subgroup ''S'' and an element ''x'' with ''x'' greater than all elements in ''S''. It is straightforward to see that this is equivalent to the other definition: the Archimedean property for a pair of elements ''a'' and ''b'' is just the statement that the cyclic subgroup generated by ''a'' is not bounded by ''b''.
Examples of Archimedean groups
The sets of the integers, 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, and the real numbers, together with the operation of addition and the usual ordering (≤), are Archimedean groups. Every subgroup of an Archimedean group is itself Archimedean, so it follows that every subgroup of these groups, such as the additive group of the
even numbers or of the
dyadic rationals, also forms an Archimedean group.
Conversely, as
Otto Hölder showed, every Archimedean group is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
(as an ordered group) to a
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  ...
of the real numbers.
It follows from this that every Archimedean group is necessarily an
abelian group: its addition operation must be
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
.
Examples of non-Archimedean groups
Groups that cannot be linearly ordered, such as the
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s, are not Archimedean. For another example, see the
''p''-adic numbers, a system of numbers generalizing the rational numbers in a different way to the real numbers.
Non-Archimedean ordered groups also exist; the ordered group (''G'', +, ≤) defined as follows is not Archimedean. Let the elements of ''G'' be the points of the
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
, given by their
Cartesian coordinates: pairs (''x'', ''y'') of real numbers. Let the group addition operation be
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some Function (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...
(vector) addition, and order these points in
lexicographic order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
: if ''a'' = (''u'', ''v'') and ''b'' = (''x'', ''y''), then ''a'' + ''b'' = (''u'' + ''x'', ''v'' + ''y''), and
''a'' ≤ ''b'' exactly when either ''v'' < ''y'' or ''v'' = ''y'' and ''u'' ≤ ''x''. Then this gives an ordered group, but one that is not Archimedean. To see this, consider the elements (1, 0) and (0, 1), both of which are greater than the zero element of the group (the
origin). For every natural number ''n'', it follows from these definitions that ''n'' (1, 0) = (''n'', 0) < (0, 1), so there is no ''n'' that satisfies the Archimedean property. This group can be thought of as the additive group of pairs of a real number and an
infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
,
where
is a unit infinitesimal:
but
for any positive real number
.
Non-Archimedean ordered field
In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Such fields will contain infinitesimal and infinitely large elements, suitably defined.
Definition
Suppose is an ordered field. ...
s can be defined similarly, and their additive groups are non-Archimedean ordered groups. These are used in
non-standard analysis, and include the
hyperreal numbers and
surreal numbers.
While non-Archimedean ordered groups cannot be
embedded in the real numbers, they can be embedded in a power of the real numbers, with lexicographic order, by the
Hahn embedding theorem; the example above is the 2-dimensional case.
Additional properties
Every Archimedean group has the property that, for every
Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, ...
of the group, and every group element ε > 0, there exists another group element ''x'' with ''x'' on the lower side of the cut and ''x'' + ε on the upper side of the cut. However, there exist non-Archimedean ordered groups with the same property. The fact that Archimedean groups are abelian can be generalized: every ordered group with this property is abelian.
[. Translated into English in .]
Generalisations
Archimedean groups can be generalised to Archimedean monoids,
linearly ordered monoids that obey the
Archimedean property
In abstract algebra and mathematical analysis, analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, Italy, Syracuse, is a property held by some algebraic structures, such as ordered or normed g ...
. Examples include the natural numbers, the non-negative rational numbers, and the non-negative real numbers, with the usual
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation ...
and order
. Through a similar
proof as for Archimedean groups, Archimedean monoids can be shown to be
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
.
See also
*
Archimedean equivalence
References
{{DEFAULTSORT:Archimedean Group
Ordered groups