Archimedean group
Encyclopedia
In abstract algebra
, a branch of mathematics
, an Archimedean group is an algebraic structure
consisting of a set together with a binary operation
and binary relation
satisfying certain axioms detailed below. We can also say that an Archimedean group is a linearly ordered group
for which the Archimedean property
holds. For example, the set R of real number
s together with the operation of addition and usual ordering relation (≤) is an Archimedean group. The concept is named after Archimedes
.
(where 
is in the set N of natural number
s) for the sum of a with itself n times.
An Archimedean group (G, +, ≤) is a linearly ordered group
subject to the following condition:
for any a and b in G which are greater than 0, the inequality na ≤ b for any n in N implies a = 0.
s, the rational number
s, the real number
s, together with the operation of addition and the usual ordering (≤), are Archimedean groups.
Proof: Consider the elements (1, 0) and (0, 1). For all n in N one evidently has n (1, 0) < (0, 1).
For another example, see p-adic number
.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an Archimedean group is an algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
consisting of a set together with a binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
and binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
satisfying certain axioms detailed below. We can also say that an Archimedean group is a linearly ordered group
Linearly ordered group
In abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation "≤" is total...
for which the Archimedean property
Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...
holds. For example, the set R of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s together with the operation of addition and usual ordering relation (≤) is an Archimedean group. The concept is named after Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
.
Definition
In the subsequent, we use the notation (where is in the set N of natural numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s) for the sum of a with itself n times.
An Archimedean group (G, +, ≤) is a linearly ordered group
Linearly ordered group
In abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation "≤" is total...
subject to the following condition:
for any a and b in G which are greater than 0, the inequality na ≤ b for any n in N implies a = 0.
Examples of Archimedean groups
The sets of the integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, together with the operation of addition and the usual ordering (≤), are Archimedean groups.
Examples of non-Archimedean groups
An ordered group (G, +, ≤) defined as follows is not Archimedean:- G = R × R.
- Let a = (u, v) and b = (x, y) then a + b = (u + x, v + y)
- a ≤ b iffIFFIFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
v < y or (v = y and u ≤ x) (lexicographical orderLexicographical orderIn mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...
with the least-significant number on the left).
Proof: Consider the elements (1, 0) and (0, 1). For all n in N one evidently has n (1, 0) < (0, 1).
For another example, see p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
.