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| In [[abstract algebra]], a branch of [[mathematics]], an '''Archimedean group''' is an [[algebraic structure]] consisting of a [[Set (mathematics)|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]].
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| ==Definition==
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| In the subsequent, we use the notation <math>na</math> (where <math>n</math> is in the set '''N''' of [[natural number]]s) for the sum of ''a'' with itself ''n'' times.
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| An '''Archimedean group''' (''G'', +, ≤) is a [[linearly ordered group]] subject to the following condition:
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| for any ''a'' and ''b'' in ''G'' which are greater than ''0'', the inequality ''na'' ≤ ''b'' holding for every ''n'' in '''N''' implies ''a'' = 0.
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| ==Examples of Archimedean groups==
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| The sets of the [[integer]]s, the [[rational number]]s, the [[real number]]s, together with the operation of addition and the usual ordering (≤), are Archimedean groups.
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| ==Examples of non-Archimedean groups==
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| An ordered group (''G'', +, ≤) defined as follows is not Archimedean:
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| * ''G'' = '''R''' × '''R'''.
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| * Let ''a'' = (''u'', ''v'') and ''b'' = (''x'', ''y'') then ''a'' + ''b'' = (''u'' + ''x'', ''v'' + ''y'')
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| * ''a'' ≤ ''b'' [[iff]] ''v'' < ''y'' or (''v'' = ''y'' and ''u'' ≤ ''x'') ([[lexicographical order]] with the least-significant number on the left).
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| '''Proof:''' Consider the elements (1, 0) and (0, 1). For all ''n'' in '''N''' one evidently has ''n'' (1, 0) < (0, 1).
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| For another example, see [[p-adic number]].
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| ==Theorems==
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| For each ''a'', ''b'' in ''G'' there exist ''m'', ''n'' in '''N''' such that ''ma'' ≤ ''b'' and ''a'' ≤ ''nb''.
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| {{DEFAULTSORT:Archimedean Group}}
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| [[Category:Ordered groups]] | |
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