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</nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of classification in algebra</div> <p>In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>Archimedean group</b> is a <a href="/wiki/Linearly_ordered_group" title="Linearly ordered group">linearly ordered group</a> for which the <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a> holds: every two positive group elements are bounded by <a href="/wiki/Integer" title="Integer">integer</a> multiples of each other. The set <b>R</b> of <a href="/wiki/Real_number" title="Real number">real numbers</a> together with the <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> of addition and the usual ordering relation between pairs of numbers is an Archimedean group. By a result of <a href="/wiki/Otto_H%C3%B6lder" title="Otto Hölder">Otto Hölder</a>, every Archimedean group is <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of this group. The name "Archimedean" comes from <a href="/wiki/Otto_Stolz" title="Otto Stolz">Otto Stolz</a>, who named the Archimedean property after its appearance in the works of <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Archimedean_group&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">additive group</a> consists of a set of elements, an <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associative</a> addition operation that combines pairs of elements and returns a single element, an <a href="/wiki/Identity_element" title="Identity element">identity element</a> (or zero element) whose sum with any other element is the other element, and an <a href="/wiki/Inverse_element" title="Inverse element">additive inverse</a> operation such that the sum of any element and its inverse is zero.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A group is a <a href="/wiki/Linearly_ordered_group" title="Linearly ordered group">linearly ordered group</a> when, in addition, its elements can be <a href="/wiki/Linear_order" class="mw-redirect" title="Linear order">linearly ordered</a> in a way that is compatible with the group operation: for all elements <i>x</i>, <i>y</i>, and <i>z</i>, if <i>x</i> ≤ <i>y</i> then <i>x</i> + <i>z</i> ≤ <i>y</i> + <i>z</i> and <i>z</i> + <i>x</i> ≤ <i>z</i> + <i>y</i>. </p><p>The notation <i>na</i> (where <i>n</i> is a <a href="/wiki/Natural_number" title="Natural number">natural number</a>) stands for the group sum of <i>n</i> copies of <i>a</i>. An <b>Archimedean group</b> (<i>G</i>, +, ≤) is a linearly ordered group subject to the following additional condition, the Archimedean property: For every <i>a</i> and <i>b</i> in <i>G</i> which are greater than 0, it is possible to find a natural number <i>n</i> for which the inequality <i>b</i> ≤ <i>na</i> holds.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>An equivalent definition is that an Archimedean group is a linearly ordered group without any bounded <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a> <a href="/wiki/Subgroup" title="Subgroup">subgroups</a>: there does not exist a cyclic subgroup <i>S</i> and an element <i>x</i> with <i>x</i> greater than all elements in <i>S</i>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It is straightforward to see that this is equivalent to the other definition: the Archimedean property for a pair of elements <i>a</i> and <i>b</i> is just the statement that the cyclic subgroup generated by <i>a</i> is not bounded by <i>b</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_of_Archimedean_groups">Examples of Archimedean groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Archimedean_group&action=edit&section=2" title="Edit section: Examples of Archimedean groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sets of the integers, the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, 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 <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even numbers</a> or of the <a href="/wiki/Dyadic_rational" title="Dyadic rational">dyadic rationals</a>, also forms an Archimedean group. </p><p><a href="/wiki/Converse_(logic)" title="Converse (logic)">Conversely</a>, as <a href="/wiki/Otto_H%C3%B6lder" title="Otto Hölder">Otto Hölder</a> showed, every Archimedean group is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> (as an ordered group) to a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the real numbers.<sup id="cite_ref-monnd_5-0" class="reference"><a href="#cite_note-monnd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> It follows from this that every Archimedean group is necessarily an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>: its addition operation must be <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>.<sup id="cite_ref-monnd_5-1" class="reference"><a href="#cite_note-monnd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples_of_non-Archimedean_groups">Examples of non-Archimedean groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Archimedean_group&action=edit&section=3" title="Edit section: Examples of non-Archimedean groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Groups that cannot be linearly ordered, such as the <a href="/wiki/Finite_group" title="Finite group">finite groups</a>, are not Archimedean. For another example, see the <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>, a system of numbers generalizing the rational numbers in a different way to the real numbers. </p><p>Non-Archimedean ordered groups also exist; the ordered group (<i>G</i>, +, ≤) defined as follows is not Archimedean. Let the elements of <i>G</i> be the points of the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>, given by their <a href="/wiki/Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinates</a>: pairs (<i>x</i>, <i>y</i>) of real numbers. Let the group addition operation be <a href="/wiki/Pointwise" title="Pointwise">pointwise</a> (vector) addition, and order these points in <a href="/wiki/Lexicographic_order" title="Lexicographic order">lexicographic order</a>: if <i>a</i> = (<i>u</i>, <i>v</i>) and <i>b</i> = (<i>x</i>, <i>y</i>), then <i>a</i> + <i>b</i> = (<i>u</i> + <i>x</i>, <i>v</i> + <i>y</i>), and <i>a</i> ≤ <i>b</i> exactly when either <i>v</i> < <i>y</i> or <i>v</i> = <i>y</i> and <i>u</i> ≤ <i>x</i>. 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 <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>). For every natural number <i>n</i>, it follows from these definitions that <i>n</i> (1, 0) = (<i>n</i>, 0) < (0, 1), so there is no <i>n</i> that satisfies the Archimedean property.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This group can be thought of as the additive group of pairs of a real number and an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)=x\epsilon +y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>ϵ</mi> <mo>+</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)=x\epsilon +y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87dc5ad0c38811603e54fb6eee1583282b7db6dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.343ex; height:2.843ex;" alt="{\displaystyle (x,y)=x\epsilon +y,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span> is a unit infinitesimal: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span> but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon <y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon <y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a00de8d52005f55d0890be331516e554e5555ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.198ex; height:2.176ex;" alt="{\displaystyle \epsilon <y}"></span> for any positive real number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c973e3cbfee5d7ab9ca2348b578b6ec19a8c019a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y>0}"></span>. <a href="/wiki/Non-Archimedean_ordered_field" title="Non-Archimedean ordered field">Non-Archimedean ordered fields</a> can be defined similarly, and their additive groups are non-Archimedean ordered groups. These are used in <a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">non-standard analysis</a>, and include the <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal numbers</a> and <a href="/wiki/Surreal_number" title="Surreal number">surreal numbers</a>. </p><p>While non-Archimedean ordered groups cannot be <a href="/wiki/Embedding" title="Embedding">embedded</a> in the real numbers, they can be embedded in a power of the real numbers, with lexicographic order, by the <a href="/wiki/Hahn_embedding_theorem" title="Hahn embedding theorem">Hahn embedding theorem</a>; the example above is the 2-dimensional case. </p> <div class="mw-heading mw-heading2"><h2 id="Additional_properties">Additional properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Archimedean_group&action=edit&section=4" title="Edit section: Additional properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every Archimedean group has the property that, for every <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind cut</a> of the group, and every group element ε > 0, there exists another group element <i>x</i> with <i>x</i> on the lower side of the cut and <i>x</i> + 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.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalisations">Generalisations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Archimedean_group&action=edit&section=5" title="Edit section: Generalisations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Archimedean groups can be generalised to <b>Archimedean monoids</b>, <a href="/wiki/Linear_order" class="mw-redirect" title="Linear order">linearly ordered</a> <a href="/wiki/Monoid" title="Monoid">monoids</a> that obey the <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a>. Examples include the natural numbers, the non-negative rational numbers, and the non-negative real numbers, with the usual <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> and order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span>. Through a similar <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> as for Archimedean groups, Archimedean monoids can be shown to be <a href="/wiki/Commutative_monoid" class="mw-redirect" title="Commutative monoid">commutative</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Archimedean_group&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Archimedean_equivalence" class="mw-redirect" title="Archimedean equivalence">Archimedean equivalence</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Archimedean_group&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMarvin2012" class="citation cs2">Marvin, Stephen (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=22x0cpypf3EC&pg=PA17"><i>Dictionary of Scientific Principles</i></a>, John Wiley & Sons, p. 17, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781118582244" title="Special:BookSources/9781118582244"><bdi>9781118582244</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionary+of+Scientific+Principles&rft.pages=17&rft.pub=John+Wiley+%26+Sons&rft.date=2012&rft.isbn=9781118582244&rft.aulast=Marvin&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D22x0cpypf3EC%26pg%3DPA17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Additive notation for groups is usually only used for <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a>, in which the addition operation is <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>. The definition here does not assume commutativity, but it will turn out to follow from the Archimedean property.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlajbegovicMockor1992" class="citation cs2">Alajbegovic, J.; Mockor, J. (1992), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WjI06938xcoC&pg=PA5"><i>Approximation Theorems in Commutative Algebra: Classical and Categorical Methods</i></a>, NATO ASI Series. Series D, Behavioural and Social Sciences, vol. 59, Springer, p. 5, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780792319481" title="Special:BookSources/9780792319481"><bdi>9780792319481</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Approximation+Theorems+in+Commutative+Algebra%3A+Classical+and+Categorical+Methods&rft.series=NATO+ASI+Series.+Series+D%2C+Behavioural+and+Social+Sciences&rft.pages=5&rft.pub=Springer&rft.date=1992&rft.isbn=9780792319481&rft.aulast=Alajbegovic&rft.aufirst=J.&rft.au=Mockor%2C+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWjI06938xcoC%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBelegradek2002" class="citation cs2">Belegradek, Oleg (2002), "Poly-regular ordered abelian groups", <i>Logic and algebra</i>, Contemp. Math., vol. 302, Amer. Math. Soc., Providence, RI, pp. 101–111, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fconm%2F302%2F05049">10.1090/conm/302/05049</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1928386">1928386</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Poly-regular+ordered+abelian+groups&rft.btitle=Logic+and+algebra&rft.series=Contemp.+Math.&rft.pages=101-111&rft.pub=Amer.+Math.+Soc.%2C+Providence%2C+RI&rft.date=2002&rft_id=info%3Adoi%2F10.1090%2Fconm%2F302%2F05049&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1928386%23id-name%3DMR&rft.aulast=Belegradek&rft.aufirst=Oleg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span>.</span> </li> <li id="cite_note-monnd-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-monnd_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-monnd_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuchsSalce2001" class="citation cs2">Fuchs, László; Salce, Luigi (2001), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4NzTiOuirN4C&pg=PA61"><i>Modules over non-Noetherian domains</i></a>, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, p. 61, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1963-0" title="Special:BookSources/978-0-8218-1963-0"><bdi>978-0-8218-1963-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1794715">1794715</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modules+over+non-Noetherian+domains&rft.place=Providence%2C+R.I.&rft.series=Mathematical+Surveys+and+Monographs&rft.pages=61&rft.pub=American+Mathematical+Society&rft.date=2001&rft.isbn=978-0-8218-1963-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1794715%23id-name%3DMR&rft.aulast=Fuchs&rft.aufirst=L%C3%A1szl%C3%B3&rft.au=Salce%2C+Luigi&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4NzTiOuirN4C%26pg%3DPA61&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuchs2011" class="citation book cs1">Fuchs, László (2011) [1963]. <i>Partially ordered algebraic systems</i>. Mineola, New York: Dover Publications. pp. 45–46. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-48387-0" title="Special:BookSources/978-0-486-48387-0"><bdi>978-0-486-48387-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partially+ordered+algebraic+systems&rft.place=Mineola%2C+New+York&rft.pages=45-46&rft.pub=Dover+Publications&rft.date=2011&rft.isbn=978-0-486-48387-0&rft.aulast=Fuchs&rft.aufirst=L%C3%A1szl%C3%B3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKopytovMedvedev1996" class="citation cs2">Kopytov, V. M.; Medvedev, N. Ya. (1996), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xyGHECthEGEC&pg=PA33"><i>Right-Ordered Groups</i></a>, Siberian School of Algebra and Logic, Springer, pp. 33–34, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780306110603" title="Special:BookSources/9780306110603"><bdi>9780306110603</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Right-Ordered+Groups&rft.series=Siberian+School+of+Algebra+and+Logic&rft.pages=33-34&rft.pub=Springer&rft.date=1996&rft.isbn=9780306110603&rft.aulast=Kopytov&rft.aufirst=V.+M.&rft.au=Medvedev%2C+N.+Ya.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxyGHECthEGEC%26pg%3DPA33&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">For a <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> for abelian groups, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1999" class="citation cs2"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kL1HpN1KPZsC&pg=PA60"><i>The Theory of Classical Valuations</i></a>, Monographs in Mathematics, Springer, p. 60, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387985251" title="Special:BookSources/9780387985251"><bdi>9780387985251</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Classical+Valuations&rft.series=Monographs+in+Mathematics&rft.pages=60&rft.pub=Springer&rft.date=1999&rft.isbn=9780387985251&rft.aulast=Ribenboim&rft.aufirst=Paulo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkL1HpN1KPZsC%26pg%3DPA60&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrupka2000" class="citation cs2">Krupka, Demeter (2000), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oyu0DzN6xk8C&pg=PA8"><i>Introduction to Global Variational Geometry</i></a>, North-Holland Mathematical Library, vol. 13, Elsevier, p. 8, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780080954202" title="Special:BookSources/9780080954202"><bdi>9780080954202</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Global+Variational+Geometry&rft.series=North-Holland+Mathematical+Library&rft.pages=8&rft.pub=Elsevier&rft.date=2000&rft.isbn=9780080954202&rft.aulast=Krupka&rft.aufirst=Demeter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Doyu0DzN6xk8C%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVinogradov1967" class="citation cs2 cs1-prop-foreign-lang-source">Vinogradov, A. A. (1967), "Ordered algebraic systems", <i>Algebra, Topology, Geometry, 1965 (Russian)</i> (in Russian), Akad. Nauk SSSR Inst. Naučn. Tehn. Informacii, Moscow, pp. 83–131, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0215761">0215761</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ordered+algebraic+systems&rft.btitle=Algebra%2C+Topology%2C+Geometry%2C+1965+%28Russian%29&rft.pages=83-131&rft.pub=Akad.+Nauk+SSSR+Inst.+Nau%C4%8Dn.+Tehn.+Informacii%2C+Moscow&rft.date=1967&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0215761%23id-name%3DMR&rft.aulast=Vinogradov&rft.aufirst=A.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span>. Translated into English in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFilippov1970" class="citation cs2">Filippov, N. D., ed. (1970), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_m4yIzOuVXsC&pg=PA69"><i>Ten papers on algebra and functional analysis</i></a>, American Mathematical Society Translations, Series 2, vol. 96, American Mathematical Society, Providence, R.I., pp. 69–118, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821896662" title="Special:BookSources/9780821896662"><bdi>9780821896662</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0268000">0268000</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ten+papers+on+algebra+and+functional+analysis&rft.series=American+Mathematical+Society+Translations%2C+Series+2&rft.pages=69-118&rft.pub=American+Mathematical+Society%2C+Providence%2C+R.I.&rft.date=1970&rft.isbn=9780821896662&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0268000%23id-name%3DMR&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_m4yIzOuVXsC%26pg%3DPA69&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArchimedean+group" class="Z3988"></span>.</span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Archimedean_group&oldid=1210555147">https://en.wikipedia.org/w/index.php?title=Archimedean_group&oldid=1210555147</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Ordered_groups" title="Category:Ordered groups">Ordered groups</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_Russian-language_sources_(ru)" title="Category:CS1 Russian-language sources (ru)">CS1 Russian-language sources (ru)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 February 2024, at 05:43<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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