Archimedean group

Last updated February 27, 2024

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.^[1]

Definition

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 cyclic subgroups: 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.

Conversely, as Otto Hölder showed, every Archimedean group is isomorphic (as an ordered group) to a subgroup of the real numbers.^[5]^[6]^[7]^[8] It follows from this that every Archimedean group is necessarily an abelian group: its addition operation must be commutative.^[5]

Examples of non-Archimedean groups

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, $(x,y)=x\epsilon +y,$ where $\epsilon$ is a unit infinitesimal: $\epsilon >0$ but $\epsilon <y$ for any positive real number $y>0$ . 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 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.

Related Research Articles

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being $0$ .

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.

In mathematics, the concept of an inverse element generalises the concepts of opposite and reciprocal of numbers.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups $such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading .$

In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A, and a finite set of identities, known as axioms, that these operations must satisfy.

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free $-modules,$ the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.

<span class="mw-page-title-main">Archimedean property</span> Mathematical property of algebraic structures

In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typically construed, states that given two positive numbers $and, there is an integer such that . It also means that the set of natural numbers is not bounded above. Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder .$

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 property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it ; such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

In algebra, a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.

In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups. It is named after Hans Hahn.

In mathematics, the Grothendieck group, or group of differences, of a commutative monoid $M$ is a certain abelian group. This abelian group is constructed from $M$ in the most universal way, in the sense that any abelian group containing a homomorphic image of $M$ will also contain a homomorphic image of the Grothendieck group of $M$ . The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.

In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order.

References

↑ Marvin, Stephen (2012), Dictionary of Scientific Principles, John Wiley & Sons, p. 17, ISBN 9781118582244 .
↑ 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.
↑ Alajbegovic, J.; Mockor, J. (1992), Approximation Theorems in Commutative Algebra: Classical and Categorical Methods, NATO ASI Series. Series D, Behavioural and Social Sciences, vol. 59, Springer, p. 5, ISBN 9780792319481 .
↑ Belegradek, Oleg (2002), "Poly-regular ordered abelian groups", Logic and algebra, Contemp. Math., vol. 302, Amer. Math. Soc., Providence, RI, pp. 101–111, doi:10.1090/conm/302/05049, MR 1928386 .
1 2 Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.: American Mathematical Society, p. 61, ISBN 978-0-8218-1963-0, MR 1794715
↑ Fuchs, László (2011) [1963]. Partially ordered algebraic systems. Mineola, New York: Dover Publications. pp. 45–46. ISBN 978-0-486-48387-0.
↑ Kopytov, V. M.; Medvedev, N. Ya. (1996), Right-Ordered Groups, Siberian School of Algebra and Logic, Springer, pp. 33–34, ISBN 9780306110603 .
↑ For a proof for abelian groups, see Ribenboim, Paulo (1999), The Theory of Classical Valuations, Monographs in Mathematics, Springer, p. 60, ISBN 9780387985251 .
↑ Krupka, Demeter (2000), Introduction to Global Variational Geometry, North-Holland Mathematical Library, vol. 13, Elsevier, p. 8, ISBN 9780080954202 .
↑ 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, MR 0215761 . 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, ISBN 9780821896662, MR 0268000 .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Marvin, Stephen (2012), Dictionary of Scientific Principles, John Wiley & Sons, p. 17, ISBN 9781118582244 .

[2] 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.

[3] Alajbegovic, J.; Mockor, J. (1992), Approximation Theorems in Commutative Algebra: Classical and Categorical Methods, NATO ASI Series. Series D, Behavioural and Social Sciences, vol. 59, Springer, p. 5, ISBN 9780792319481 .

[4] Belegradek, Oleg (2002), "Poly-regular ordered abelian groups", Logic and algebra, Contemp. Math., vol. 302, Amer. Math. Soc., Providence, RI, pp. 101–111, doi:10.1090/conm/302/05049, MR 1928386 .

[monnd-5] 1 2 Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.: American Mathematical Society, p. 61, ISBN 978-0-8218-1963-0, MR 1794715

[6] Fuchs, László (2011) [1963]. Partially ordered algebraic systems. Mineola, New York: Dover Publications. pp. 45–46. ISBN 978-0-486-48387-0.

[7] Kopytov, V. M.; Medvedev, N. Ya. (1996), Right-Ordered Groups, Siberian School of Algebra and Logic, Springer, pp. 33–34, ISBN 9780306110603 .

[8] For a proof for abelian groups, see Ribenboim, Paulo (1999), The Theory of Classical Valuations, Monographs in Mathematics, Springer, p. 60, ISBN 9780387985251 .

[9] Krupka, Demeter (2000), Introduction to Global Variational Geometry, North-Holland Mathematical Library, vol. 13, Elsevier, p. 8, ISBN 9780080954202 .

[10] 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, MR 0215761 . 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, ISBN 9780821896662, MR 0268000 .

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]