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id="toc-Théorie_des_modèles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exemples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Exemples</span> </div> </a> <ul id="toc-Exemples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isomorphismes_et_morphismes_bijectifs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isomorphismes_et_morphismes_bijectifs"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Isomorphismes et morphismes bijectifs</span> </div> </a> <ul id="toc-Isomorphismes_et_morphismes_bijectifs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriétés" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propriétés"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Propriétés</span> </div> </a> <ul id="toc-Propriétés-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Objets_isomorphes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Objets_isomorphes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Objets isomorphes</span> </div> </a> <ul id="toc-Objets_isomorphes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes et références</span> </div> </a> <button aria-controls="toc-Notes_et_références-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Notes et références</span> </button> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Basculer la table des matières" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Basculer la table des matières</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Isomorphisme</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 59 langues." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-59" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">59 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%85%D8%A7%D9%83%D9%84" title="تماكل – arabe" lang="ar" hreflang="ar" data-title="تماكل" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Isomorfismu" title="Isomorfismu – asturien" lang="ast" hreflang="ast" data-title="Isomorfismu" data-language-autonym="Asturianu" data-language-local-name="asturien" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0zomorfluq" title="İzomorfluq – azerbaïdjanais" lang="az" hreflang="az" data-title="İzomorfluq" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC" title="Изоморфизм – bachkir" lang="ba" hreflang="ba" data-title="Изоморфизм" data-language-autonym="Башҡортса" data-language-local-name="bachkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%B7%D0%B0%D0%BC%D0%B0%D1%80%D1%84%D1%96%D0%B7%D0%BC" title="Ізамарфізм – biélorusse" lang="be" hreflang="be" data-title="Ізамарфізм" data-language-autonym="Беларуская" data-language-local-name="biélorusse" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D1%8A%D0%BC" title="Изоморфизъм – bulgare" lang="bg" hreflang="bg" data-title="Изоморфизъм" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – bosniaque" lang="bs" hreflang="bs" data-title="Izomorfizam" data-language-autonym="Bosanski" data-language-local-name="bosniaque" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – catalan" lang="ca" hreflang="ca" data-title="Isomorfisme" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Izomorfismus" title="Izomorfismus – tchèque" lang="cs" hreflang="cs" data-title="Izomorfismus" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Isomorffedd" title="Isomorffedd – gallois" lang="cy" hreflang="cy" data-title="Isomorffedd" data-language-autonym="Cymraeg" data-language-local-name="gallois" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Isomorfi" title="Isomorfi – danois" lang="da" hreflang="da" data-title="Isomorfi" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Isomorphismus" title="Isomorphismus – allemand" lang="de" hreflang="de" data-title="Isomorphismus" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%99%CF%83%CE%BF%CE%BC%CE%BF%CF%81%CF%86%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Ισομορφισμός – grec" lang="el" hreflang="el" data-title="Ισομορφισμός" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Isomorphism" title="Isomorphism – anglais" lang="en" hreflang="en" data-title="Isomorphism" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Izomorfio" title="Izomorfio – espéranto" lang="eo" hreflang="eo" data-title="Izomorfio" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – espagnol" lang="es" hreflang="es" data-title="Isomorfismo" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Isomorfism" title="Isomorfism – estonien" lang="et" hreflang="et" data-title="Isomorfism" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – basque" lang="eu" hreflang="eu" data-title="Isomorfismo" data-language-autonym="Euskara" data-language-local-name="basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DB%8C%DA%A9%D8%B1%DB%8C%D8%AE%D8%AA%DB%8C" title="یکریختی – persan" lang="fa" hreflang="fa" data-title="یکریختی" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Isomorfismi" title="Isomorfismi – finnois" lang="fi" hreflang="fi" data-title="Isomorfismi" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Iseamorfacht" title="Iseamorfacht – irlandais" lang="ga" hreflang="ga" data-title="Iseamorfacht" data-language-autonym="Gaeilge" data-language-local-name="irlandais" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – galicien" lang="gl" hreflang="gl" data-title="Isomorfismo" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%96%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="איזומורפיזם – hébreu" lang="he" hreflang="he" data-title="איזומורפיזם" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – croate" lang="hr" hreflang="hr" data-title="Izomorfizam" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Izomorfia" title="Izomorfia – hongrois" lang="hu" hreflang="hu" data-title="Izomorfia" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%A6%D5%B8%D5%B4%D5%B8%D6%80%D6%86%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Իզոմորֆություն (մաթեմատիկա) – arménien" lang="hy" hreflang="hy" data-title="Իզոմորֆություն (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Isomorphismo" title="Isomorphismo – interlingua" lang="ia" hreflang="ia" data-title="Isomorphismo" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – indonésien" lang="id" hreflang="id" data-title="Isomorfisme" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – italien" lang="it" hreflang="it" data-title="Isomorfismo" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%90%8C%E5%9E%8B%E5%86%99%E5%83%8F" title="同型写像 – japonais" lang="ja" hreflang="ja" data-title="同型写像" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC_(%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Изоморфизм (Математика) – kazakh" lang="kk" hreflang="kk" data-title="Изоморфизм (Математика)" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8F%99%ED%98%95_%EC%82%AC%EC%83%81" title="동형 사상 – coréen" lang="ko" hreflang="ko" data-title="동형 사상" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC" title="Изоморфизм – kirghize" lang="ky" hreflang="ky" data-title="Изоморфизм" data-language-autonym="Кыргызча" data-language-local-name="kirghize" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Isomorphismus" title="Isomorphismus – latin" lang="la" hreflang="la" data-title="Isomorphismus" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Izomorfizmas" title="Izomorfizmas – lituanien" lang="lt" hreflang="lt" data-title="Izomorfizmas" data-language-autonym="Lietuvių" data-language-local-name="lituanien" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84" title="Изоморф – mongol" lang="mn" hreflang="mn" data-title="Изоморф" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – néerlandais" lang="nl" hreflang="nl" data-title="Isomorfisme" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Isomorfi" title="Isomorfi – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Isomorfi" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – norvégien bokmål" lang="nb" hreflang="nb" data-title="Isomorfisme" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%86%E0%A8%87%E0%A8%B8%E0%A9%8B%E0%A8%AE%E0%A9%8C%E0%A8%B0%E0%A8%AB%E0%A8%BF%E0%A8%9C%E0%A8%BC%E0%A8%AE" title="ਆਇਸੋਮੌਰਫਿਜ਼ਮ – pendjabi" lang="pa" hreflang="pa" data-title="ਆਇਸੋਮੌਰਫਿਜ਼ਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pendjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Izomorfizm" title="Izomorfizm – polonais" lang="pl" hreflang="pl" data-title="Izomorfizm" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Isomorfism" title="Isomorfism – piémontais" lang="pms" hreflang="pms" data-title="Isomorfism" data-language-autonym="Piemontèis" data-language-local-name="piémontais" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – portugais" lang="pt" hreflang="pt" data-title="Isomorfismo" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Izomorfism" title="Izomorfism – roumain" lang="ro" hreflang="ro" data-title="Izomorfism" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC" title="Изоморфизм – russe" lang="ru" hreflang="ru" data-title="Изоморфизм" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – serbo-croate" lang="sh" hreflang="sh" data-title="Izomorfizam" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Isomorphism" title="Isomorphism – Simple English" lang="en-simple" hreflang="en-simple" data-title="Isomorphism" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Izomorfizem" title="Izomorfizem – slovène" lang="sl" hreflang="sl" data-title="Izomorfizem" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%B0%D0%BC_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Изоморфизам (математика) – serbe" lang="sr" hreflang="sr" data-title="Изоморфизам (математика)" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Isomorfi" title="Isomorfi – suédois" lang="sv" hreflang="sv" data-title="Isomorfi" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%90%E0%AE%9A%E0%AF%8B%E0%AE%AE%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%B8%E0%AE%AE%E0%AF%8D" title="ஐசோமார்பிஸம் – tamoul" lang="ta" hreflang="ta" data-title="ஐசோமார்பிஸம்" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0zomorfizma" title="İzomorfizma – turc" lang="tr" hreflang="tr" data-title="İzomorfizma" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D1%96%D0%B7%D0%BC" title="Ізоморфізм – ukrainien" lang="uk" hreflang="uk" data-title="Ізоморфізм" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B4%D8%A7%DA%A9%D9%84%D8%AA" title="مشاکلت – ourdou" lang="ur" hreflang="ur" data-title="مشاکلت" data-language-autonym="اردو" data-language-local-name="ourdou" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Izomorfizm_(matematika)" title="Izomorfizm (matematika) – ouzbek" lang="uz" hreflang="uz" data-title="Izomorfizm (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_%C4%91%E1%BA%B3ng_c%E1%BA%A5u" title="Phép đẳng cấu – vietnamien" lang="vi" hreflang="vi" data-title="Phép đẳng cấu" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%8C%E6%9E%84" title="同构 – wu" lang="wuu" hreflang="wuu" data-title="同构" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%90%8C%E6%9E%84" title="同构 – chinois" lang="zh" hreflang="zh" data-title="同构" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%90%8C%E6%A7%8B" title="同構 – cantonais" lang="yue" hreflang="yue" data-title="同構" data-language-autonym="粵語" data-language-local-name="cantonais" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q189112#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaces de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Isomorphisme" title="Voir le contenu de la page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussion:Isomorphisme" rel="discussion" title="Discussion au sujet de cette page de contenu [t]" accesskey="t"><span>Discussion</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Modifier la variante de langue" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">français</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet 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href="/w/index.php?title=Isomorphisme&action=history" title="Historique des versions de cette page [h]" accesskey="h"><span>Voir l’historique</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Outils de la page"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Outils" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Outils</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Outils</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">masquer</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Plus d’options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Isomorphisme"><span>Lire</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Isomorphisme&veaction=edit" title="Modifier cette page [v]" accesskey="v"><span>Modifier</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Isomorphisme&action=edit" title="Modifier le wikicode de cette page [e]" accesskey="e"><span>Modifier le code</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Isomorphisme&action=history"><span>Voir l’historique</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Général </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_li%C3%A9es/Isomorphisme" title="Liste des pages liées qui pointent sur celle-ci [j]" accesskey="j"><span>Pages liées</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Suivi_des_liens/Isomorphisme" rel="nofollow" title="Liste des modifications récentes des pages appelées par celle-ci [k]" accesskey="k"><span>Suivi des pages liées</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Aide:Importer_un_fichier" title="Téléverser des fichiers [u]" accesskey="u"><span>Téléverser un fichier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_sp%C3%A9ciales" title="Liste de toutes les pages spéciales [q]" accesskey="q"><span>Pages spéciales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Isomorphisme&oldid=210358157" title="Adresse permanente de cette version de cette page"><span>Lien permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Isomorphisme&action=info" title="Davantage d’informations sur cette page"><span>Informations sur la page</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Citer&page=Isomorphisme&id=210358157&wpFormIdentifier=titleform" title="Informations sur la manière de citer cette page"><span>Citer cette page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:UrlShortener&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FIsomorphisme"><span>Obtenir l'URL raccourcie</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:QrCode&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FIsomorphisme"><span>Télécharger le code QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimer / exporter </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Livre&bookcmd=book_creator&referer=Isomorphisme"><span>Créer un livre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:DownloadAsPdf&page=Isomorphisme&action=show-download-screen"><span>Télécharger comme PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Isomorphisme&printable=yes" title="Version imprimable de cette page [p]" accesskey="p"><span>Version imprimable</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dans d’autres projets </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link 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data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</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">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/Isomorphisme_(chimie)" title="Isomorphisme (chimie)">Isomorphisme (chimie)</a> et <a href="/wiki/Isomorphisme_institutionnel" title="Isomorphisme institutionnel">Isomorphisme institutionnel</a>. </p> </div></div> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, un <b>isomorphisme</b> entre deux <a href="/wiki/Ensemble" title="Ensemble">ensembles</a> <a href="/wiki/Structure_(math%C3%A9matiques)" title="Structure (mathématiques)">structurés</a> est une <a href="/wiki/Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</a> <a href="/wiki/Bijection" title="Bijection">bijective</a> qui préserve la structure, et dont la <a href="/wiki/Bijection_r%C3%A9ciproque" title="Bijection réciproque">réciproque</a> préserve aussi la structure<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>N 1<span class="cite_crochet">]</span></a></sup>. Plus généralement, en <a href="/wiki/Th%C3%A9orie_des_cat%C3%A9gories" title="Théorie des catégories">théorie des catégories</a>, un <a href="/wiki/Isomorphisme_(th%C3%A9orie_des_cat%C3%A9gories)" title="Isomorphisme (théorie des catégories)">isomorphisme</a> entre deux <a href="/wiki/Th%C3%A9orie_des_cat%C3%A9gories" title="Théorie des catégories">objets</a> est un <a href="/wiki/Morphisme" title="Morphisme">morphisme</a> admettant un « morphisme inverse ». </p><p>Par exemple, sur l'intervalle <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 [1;100]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>;</mo> <mn>100</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1;100]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/495dfdda1db41b2bdf24a125debcc02a35de2a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.977ex; height:2.843ex;" alt="{\displaystyle [1;100]}"></span> des valeurs <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 a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"></span>... peuvent être remplacées par leur <a href="/wiki/Logarithme" title="Logarithme">logarithme</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,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span>..., et les relations d'ordre entre elles seront conservées. On peut à tout moment retrouver les valeurs <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 a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> et <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> en prenant les <a href="/wiki/Fonction_exponentielle" title="Fonction exponentielle">exponentielles</a> de <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> et <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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>. Le logarithme et l'exponentielle sont des isomorphismes entre ces intervalles. </p><p>D'autres termes peuvent être utilisés pour désigner un isomorphisme en spécifiant la structure, comme l'<a href="/wiki/Hom%C3%A9omorphisme" title="Homéomorphisme">homéomorphisme</a> entre <a href="/wiki/Espace_topologique" title="Espace topologique">espaces topologiques</a> ou le <a href="/wiki/Diff%C3%A9omorphisme" title="Difféomorphisme">difféomorphisme</a> entre <a href="/wiki/Vari%C3%A9t%C3%A9_(g%C3%A9om%C3%A9trie)" title="Variété (géométrie)">variétés</a>. </p><p>Deux objets sont dits <b>isomorphes</b> s'il existe un isomorphisme de l'un vers l'autre. Dans certains contextes, un isomorphisme d'un objet sur lui-même est appelé un <a href="/wiki/Automorphisme" title="Automorphisme">automorphisme</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définitions"><span id="D.C3.A9finitions"></span>Définitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=1" title="Modifier la section : Définitions" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=1" title="Modifier le code source de la section : Définitions"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Algèbre"><span id="Alg.C3.A8bre"></span>Algèbre</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=2" title="Modifier la section : Algèbre" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=2" title="Modifier le code source de la section : Algèbre"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/Alg%C3%A8bre_g%C3%A9n%C3%A9rale" title="Algèbre générale">algèbre</a>, un isomorphisme est un <a href="/wiki/Morphisme" title="Morphisme">morphisme</a> admettant un inverse qui est lui-même un morphisme. </p><p>C'est donc une <a href="/wiki/Bijection" title="Bijection">bijection</a> pour laquelle les relations « algébriques » entre les éléments de l'ensemble d'arrivée sont les mêmes que celles entre leurs antécédents respectifs (la <a href="/wiki/Structure_(math%C3%A9matiques)" title="Structure (mathématiques)">structure</a> <a href="/wiki/Structure_alg%C3%A9brique" title="Structure algébrique">algébrique</a> est préservée). Ce « méta-concept » mathématique admet une définition formelle en <a href="/wiki/Th%C3%A9orie_des_cat%C3%A9gories" title="Théorie des catégories">théorie des catégories</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Catégorie"><span id="Cat.C3.A9gorie"></span>Catégorie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=3" title="Modifier la section : Catégorie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=3" title="Modifier le code source de la section : Catégorie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans une catégorie donnée, un isomorphisme est un morphisme <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 f:A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20040a52d9391f2fe271f0aaa300bf7887a0c7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\to B}"></span> tel qu'il existe un morphisme <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 g:B\to A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:B\to A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83702ca79adb23095c7e6e470827cc2fc749d979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.174ex; height:2.509ex;" alt="{\displaystyle g:B\to A}"></span> qui soit « inverse » de <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> à la fois à gauche <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 (g\circ f=\mathrm {id} _{A})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\circ f=\mathrm {id} _{A})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5947d0a8aa31d6828887931ecace71490c230c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.901ex; height:2.843ex;" alt="{\displaystyle (g\circ f=\mathrm {id} _{A})}"></span> et à droite <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 (f\circ g=\mathrm {id} _{B}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g=\mathrm {id} _{B}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16795edf3a1e8e57fef8dcaf824de44aa85be0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.563ex; height:2.843ex;" alt="{\displaystyle (f\circ g=\mathrm {id} _{B}).}"></span> </p><p>Il suffit pour cela que <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> possède d'une part un « inverse à gauche » <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> et d'autre part un « inverse à droite » <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>. En effet, on a alors </p> <dl><dd><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 g=g\circ \mathrm {id} _{B}=g\circ (f\circ h)=(g\circ f)\circ h=\mathrm {id} _{A}\circ h=h,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>∘</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mi>g</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>h</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>∘</mo> <mi>h</mi> <mo>=</mo> <mi>h</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=g\circ \mathrm {id} _{B}=g\circ (f\circ h)=(g\circ f)\circ h=\mathrm {id} _{A}\circ h=h,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67da0192699a87e76aa62f9ce36f7a2d96d59030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.126ex; height:2.843ex;" alt="{\displaystyle g=g\circ \mathrm {id} _{B}=g\circ (f\circ h)=(g\circ f)\circ h=\mathrm {id} _{A}\circ h=h,}"></span></dd></dl> <p>ce qui prouve en outre l'unicité de l'inverse. </p><p>En revanche, l'une ou l'autre de ces deux conditions, à elle seule, ne suffit pas. </p> <div class="mw-heading mw-heading3"><h3 id="Théorie_des_modèles"><span id="Th.C3.A9orie_des_mod.C3.A8les"></span>Théorie des modèles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=4" title="Modifier la section : Théorie des modèles" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=4" title="Modifier le code source de la section : Théorie des modèles"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/Th%C3%A9orie_des_mod%C3%A8les" title="Théorie des modèles">théorie des modèles</a>, un homomorphisme concerne deux structures <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 {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> et <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 {\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f939c87a07b7af23e09792e9edb2c7caebb18864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}"></span> dans un même langage <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span>. Un homomorphisme <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> de <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 {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> dans <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 {\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f939c87a07b7af23e09792e9edb2c7caebb18864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}"></span> est une application de <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 |{\mathfrak {A}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\mathfrak {A}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b0dcbd6a83d740ea7217770085bd29bac044ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.962ex; height:2.843ex;" alt="{\displaystyle |{\mathfrak {A}}|}"></span> (l'univers ou domaine de <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 {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span>) dans <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 |{\mathfrak {B}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\mathfrak {B}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3335b4bd0bfcf1f453b3486a8c9288e5f79fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.348ex; height:2.843ex;" alt="{\displaystyle |{\mathfrak {B}}|}"></span> qui satisfait les conditions suivantes : </p> <ul><li>pour tout entier <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, pour tout prédicat <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> de <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> d'arité <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, pour tout <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 a_{1},\dots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852381be25b656d697c7a4a9634d3dc4c182d833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\dots ,a_{n}}"></span> de <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 {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> : <dl><dd>si <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 (a_{1},\dots ,a_{n})\in P^{\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},\dots ,a_{n})\in P^{\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23123d1a824f6e3813cadb83acf1c98ce6b09d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.794ex; height:3.176ex;" alt="{\displaystyle (a_{1},\dots ,a_{n})\in P^{\mathfrak {A}}}"></span>, alors<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 (h(a_{1}),\dots ,h(a_{n}))\in P^{\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h(a_{1}),\dots ,h(a_{n}))\in P^{\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73d0ef0c78ef97cdd090c34105cbd0cc07e4ea3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.363ex; height:3.176ex;" alt="{\displaystyle (h(a_{1}),\dots ,h(a_{n}))\in P^{\mathfrak {B}}}"></span> ;</dd></dl></li> <li>pour tout entier <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, pour toute fonction <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> de <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> d'arité <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, pour tout <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 a_{1},\dots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852381be25b656d697c7a4a9634d3dc4c182d833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\dots ,a_{n}}"></span> de <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 {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> : <dl><dd><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 h(F^{\mathfrak {A}}(a_{1},\dots ,a_{n}))=F^{\mathfrak {B}}(h(a_{1}),\dots ,h(a_{n}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(F^{\mathfrak {A}}(a_{1},\dots ,a_{n}))=F^{\mathfrak {B}}(h(a_{1}),\dots ,h(a_{n}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd45512cca27d95ea1311a08ee94856596f8d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.71ex; height:3.176ex;" alt="{\displaystyle h(F^{\mathfrak {A}}(a_{1},\dots ,a_{n}))=F^{\mathfrak {B}}(h(a_{1}),\dots ,h(a_{n}))}"></span> ;</dd></dl></li> <li>pour toute constante <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> de <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> : <dl><dd><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 h\left(c^{\mathfrak {A}}\right)=c^{\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mrow> <mo>(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\left(c^{\mathfrak {A}}\right)=c^{\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72da81d6cefc820e2066d215c6624e3f3eb256b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.065ex; height:3.343ex;" alt="{\displaystyle h\left(c^{\mathfrak {A}}\right)=c^{\mathfrak {B}}}"></span>.</dd></dl></li></ul> <p>Un homomorphisme bijectif est un isomorphisme. S'il existe un isomorphisme entre deux structures, on dit qu'elles sont isomorphes. Un important théorème assure qu'alors, pour tout entier <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, tout prédicat <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> de <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> d'arité <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> et toute <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span>-formule <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> : </p> <dl><dd><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 \models _{\mathfrak {A}}\phi [a_{1},\dots ,a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>⊨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models _{\mathfrak {A}}\phi [a_{1},\dots ,a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea00f2acc03f0a2abbc37e998eb5be62a42e58d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.662ex; height:2.843ex;" alt="{\displaystyle \models _{\mathfrak {A}}\phi [a_{1},\dots ,a_{n}]}"></span> si et seulement si <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 \models _{\mathfrak {B}}\phi [h(a_{1}),\dots ,h(a_{n})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>⊨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">[</mo> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models _{\mathfrak {B}}\phi [h(a_{1}),\dots ,h(a_{n})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3811412c9747d6b020cb897a5d32f434f64603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.231ex; height:2.843ex;" alt="{\displaystyle \models _{\mathfrak {B}}\phi [h(a_{1}),\dots ,h(a_{n})]}"></span>.</dd></dl> <p>En particulier, les deux structures satisfont les mêmes énoncés. Ainsi, deux structures isomorphes sont <a href="/wiki/%C3%89quivalence_%C3%A9l%C3%A9mentaire" title="Équivalence élémentaire">élémentairement équivalentes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Exemples">Exemples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=5" title="Modifier la section : Exemples" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=5" title="Modifier le code source de la section : Exemples"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Dans la <a href="/wiki/Cat%C3%A9gorie_des_ensembles" title="Catégorie des ensembles">catégorie des ensembles</a>, les isomorphismes sont les bijections.</li> <li>Dans la <a href="/wiki/Cat%C3%A9gorie_des_groupes" title="Catégorie des groupes">catégorie des groupes</a>, les isomorphismes sont les <a href="/wiki/Morphisme_de_groupes" title="Morphisme de groupes">morphismes de groupes</a> bijectifs.</li> <li>Dans la <a href="/wiki/Cat%C3%A9gorie_des_espaces_topologiques" title="Catégorie des espaces topologiques">catégorie des espaces topologiques</a>, un isomorphisme est une <a href="/wiki/Bijection" title="Bijection">bijection</a> continue dont l'<a href="/wiki/Bijection_r%C3%A9ciproque" title="Bijection réciproque">inverse</a> est continue, aussi appelée <a href="/wiki/Hom%C3%A9omorphisme" title="Homéomorphisme">homéomorphisme</a>.</li> <li>De la même façon, un isomorphisme entre <a href="/wiki/Vari%C3%A9t%C3%A9_diff%C3%A9rentielle" title="Variété différentielle">variétés différentielles</a> (par exemple, entre des <a href="/wiki/Ouvert_(topologie)" title="Ouvert (topologie)">ouverts</a> de ℝ<sup><i>n</i></sup>) est un <a href="/wiki/Diff%C3%A9omorphisme" title="Difféomorphisme">difféomorphisme</a>, c'est-à-dire une bijection <a href="/wiki/Diff%C3%A9rentiable" class="mw-redirect" title="Différentiable">différentiable</a> dont l'inverse est différentiable. Plus précisément, si l'on considère une structure <a href="/wiki/Classe_de_r%C3%A9gularit%C3%A9" title="Classe de régularité">C<sup><i>k</i></sup></a> sur une variété, alors on parle de C<sup><i>k</i></sup>-difféomorphisme.</li> <li>Un <a href="/wiki/Isomorphisme_d%27ensembles_ordonn%C3%A9s" class="mw-redirect" title="Isomorphisme d'ensembles ordonnés">isomorphisme d'ensembles ordonnés</a> est une bijection croissante dont la réciproque est croissante.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Isomorphismes_et_morphismes_bijectifs">Isomorphismes et morphismes bijectifs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=6" title="Modifier la section : Isomorphismes et morphismes bijectifs" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=6" title="Modifier le code source de la section : Isomorphismes et morphismes bijectifs"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans une <a href="/wiki/Cat%C3%A9gorie_concr%C3%A8te" title="Catégorie concrète">catégorie concrète</a> (c'est-à-dire, <i>grosso modo</i>, une catégorie dont les objets sont des ensembles et les morphismes, des applications entre ces ensembles), comme la catégorie des espaces topologiques ou les catégories d'objets algébriques comme les groupes, les anneaux et les modules, un isomorphisme doit être bijectif. Dans les catégories algébriques (en particulier, les catégories des <a href="/wiki/Vari%C3%A9t%C3%A9_(alg%C3%A8bre)" title="Variété (algèbre)">variétés au sens de l'algèbre universelle</a>), un isomorphisme est un homomorphisme bijectif. Toutefois, il existe des catégories concrètes dans lesquelles les morphismes bijectifs ne sont pas nécessairement des isomorphismes (comme la <a href="/wiki/Cat%C3%A9gorie_des_espaces_topologiques" title="Catégorie des espaces topologiques">catégorie des espaces topologiques</a>), et dans certaines catégories où tout objet admet un ensemble sous-jacent, les isomorphismes ne sont pas forcément bijectifs (comme la catégorie d'homotopie des <a href="/wiki/CW-complexe" title="CW-complexe">CW-complexes</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=7" title="Modifier la section : Propriétés" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=7" title="Modifier le code source de la section : Propriétés"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un isomorphisme est à la fois un <a href="/wiki/%C3%89pimorphisme" title="Épimorphisme">épimorphisme</a> et un <a href="/wiki/Monomorphisme" title="Monomorphisme">monomorphisme</a>, mais la <a href="/wiki/R%C3%A9ciproque" class="mw-redirect" title="Réciproque">réciproque</a> est fausse en général : il existe des morphismes à la fois épiques et moniques qui ne sont pas des isomorphismes. </p><p><i>Pour plus de détails, voir : <a href="/wiki/Th%C3%A9orie_des_cat%C3%A9gories#Monomorphismes,_épimorphismes_et_isomorphismes" title="Théorie des catégories">Propriétés des morphismes dans les catégories</a>.</i> </p> <div class="mw-heading mw-heading2"><h2 id="Objets_isomorphes">Objets isomorphes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=8" title="Modifier la section : Objets isomorphes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=8" title="Modifier le code source de la section : Objets isomorphes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Deux objets reliés par un isomorphisme sont dits isomorphes. </p><p>Par exemple, le <a href="/wiki/Groupe_de_Klein" title="Groupe de Klein">groupe de Klein</a> est isomorphe à ℤ/2ℤ × ℤ/2ℤ. </p><p>Savoir que deux objets sont isomorphes présente un grand intérêt car cela permet de transposer des résultats et propriétés démontrés de l'un à l'autre. </p><p>Selon certains points de vue, deux objets isomorphes peuvent être considérés comme identiques, ou du moins indiscernables. En effet, bien souvent, les propriétés intéressantes d'un objet seront partagées par tous les objets isomorphes de la catégorie. Ainsi, on parle souvent d'unicité ou d'identité « <a href="/wiki/%C3%80_quelque_chose_pr%C3%A8s" title="À quelque chose près">à un isomorphisme près</a> ». </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=9" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=9" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=10" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=10" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">Si, pour beaucoup de structures en <a href="/wiki/Alg%C3%A8bre" title="Algèbre">algèbre</a>, cette seconde condition est automatiquement remplie, ce n'est pas le cas en <a href="/wiki/Topologie" title="Topologie">topologie</a> par exemple où une bijection peut être <a href="/wiki/Continuit%C3%A9_(math%C3%A9matiques)" title="Continuité (mathématiques)">continue</a> sans que sa réciproque le soit.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Articles_connexes">Articles connexes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphisme&veaction=edit&section=11" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorphisme&action=edit&section=11" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Th%C3%A9or%C3%A8mes_d%27isomorphisme" title="Théorèmes d'isomorphisme">Théorèmes d'isomorphisme</a></li> <li><a href="/wiki/Isomorphisme_de_graphes" title="Isomorphisme de graphes">Isomorphisme de graphes</a></li> <li><a href="/wiki/Quasi-isomorphisme" title="Quasi-isomorphisme">Quasi-isomorphisme</a></li> <li><a href="/wiki/Surjection" title="Surjection">Surjection</a></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail des mathématiques"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/24px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/36px-Racine_carr%C3%A9e_bleue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/48px-Racine_carr%C3%A9e_bleue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail:Mathématiques">Portail des mathématiques</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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