An additive group consists of a set of elements, an associative addition operation that combines pairs of elements and returns a single element,
an identity element (or zero element) whose sum with any other element is the other element, and an additive inverse operation such that the sum of any element and its inverse is zero.[2]
A group is a linearly ordered group 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) 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.[3]
An equivalent definition is that an Archimedean group is a linearly ordered group without any bounded cyclicsubgroups: there does not exist a cyclic subgroup S and an element x with x greater than all elements in S.[4] 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 numbers, 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.
Groups that cannot be linearly ordered, such as the finite groups, 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, given by their Cartesian coordinates: pairs (x, y) of real numbers. Let the group addition operation be pointwise (vector) addition, and order these points in lexicographic order: 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.[9] This group can be thought of as the additive group of pairs of a real number and an infinitesimal, where is a unit infinitesimal: but for any positive real number . Non-Archimedean ordered fields 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 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.[10]
Generalisations
Archimedean groups can be generalised to Archimedean monoids, linearly orderedmonoids that obey the Archimedean property. Examples include the natural numbers, the non-negative rational numbers, and the non-negative real numbers, with the usual binary operation and order . Through a similar proof as for Archimedean groups, Archimedean monoids can be shown to be commutative.
^Additive notation for groups is usually only used for abelian groups, in which the addition operation is commutative. The definition here does not assume commutativity, but it will turn out to follow from the Archimedean property.
^Vinogradov, A. A. (1967), "Ordered algebraic systems", Algebra, Topology, Geometry, 1965 (Russian) (in Russian), Akad. Nauk SSSR Inst. Naučn. Tehn. Informacii, Moscow, pp. 83–131, MR0215761. Translated into English in Filippov, N. D., ed. (1970), Ten papers on algebra and functional analysis, American Mathematical Society Translations, Series 2, vol. 96, American Mathematical Society, Providence, R.I., pp. 69–118, ISBN9780821896662, MR0268000.