Алгебрадағы Архимедтік топтар

Type of classification in algebra
In abstract algebra, a branch of mathematics, an Archimedean group is a linearly ordered group for which the Archimedean property holds: every two positive group elements are bounded by integer multiples of each other. The set R of real numbers 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 to a subgroup of this group. The name "Archimedean" comes from Otto Stolz, who named the Archimedean property after its appearance in the works of Archimedes.

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. Conversely, as Otto Hölder showed, every Archimedean group is isomorphic (as an ordered group) to a subgroup of the real numbers. It follows from this that every Archimedean group is necessarily an abelian group: its addition operation must be commutative. 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.

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.

Archimedean groups can be generalised to Archimedean monoids, linearly ordered monoids 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.