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id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menú principal" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menú principal</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header 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accesskey="z"><span>Portada</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Article_aleatori" title="Carrega una pàgina a l’atzar [x]" accesskey="x"><span>Article a l'atzar</span></a></li><li id="n-Articles-de-qualitat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Articles_de_qualitat"><span>Articles de qualitat</span></a></li> </ul> </div> </div> <div id="p-Comunitat" class="vector-menu mw-portlet mw-portlet-Comunitat" > <div class="vector-menu-heading"> Comunitat </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Portal" title="Sobre el projecte, què podeu fer, on trobareu les coses"><span>Portal viquipedista</span></a></li><li id="n-Agenda-d'actes" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Trobades"><span>Agenda d'actes</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Canvis_recents" title="Una llista dels canvis recents al wiki [r]" accesskey="r"><span>Canvis recents</span></a></li><li id="n-La-taverna" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:La_taverna"><span>La taverna</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Contacte"><span>Contacte</span></a></li><li id="n-Xat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Canals_IRC"><span>Xat</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Ajuda" title="El lloc per a saber més coses"><span>Ajuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Viquipèdia" src="/static/images/mobile/copyright/wikipedia-wordmark-ca.svg" style="width: 7.5em; height: 1.4375em;"> <img class="mw-logo-tagline" 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class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Cerca a Viquipèdia" aria-label="Cerca a Viquipèdia" autocapitalize="sentences" title="Cerca a la Viquipèdia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Cerca"> </div> <button class="cdx-button cdx-search-input__end-button">Cerca</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Eines personals"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aparença" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aparença</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ca.wikipedia.org&uselang=ca" class=""><span>Donatius</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crea_compte&returnto=Isomorfisme" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori" class=""><span>Crea un compte</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Isomorfisme" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o" class=""><span>Inicia la sessió</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Més opcions" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Eines personals" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Eines personals</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú d'usuari" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ca.wikipedia.org&uselang=ca"><span>Donatius</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crea_compte&returnto=Isomorfisme" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crea un compte</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Isomorfisme" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Inicia la sessió</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pàgines per a editors no registrats <a href="/wiki/Ajuda:Introducci%C3%B3" aria-label="Vegeu més informació sobre l'edició"><span>més informació</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:Contribucions_pr%C3%B2pies" title="Una llista de les modificacions fetes des d'aquesta adreça IP [y]" accesskey="y"><span>Contribucions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Discussi%C3%B3_personal" title="Discussió sobre les edicions per aquesta adreça ip. [n]" accesskey="n"><span>Discussió per aquest IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contingut" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inici</div> </a> </li> <li id="toc-Definició_formal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definició_formal"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definició formal</span> </div> </a> <ul id="toc-Definició_formal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propietats_en_els_ordres_totals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propietats_en_els_ordres_totals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Propietats en els ordres totals</span> </div> </a> <ul id="toc-Propietats_en_els_ordres_totals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Història_i_concepte" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Història_i_concepte"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Història i concepte</span> </div> </a> <ul id="toc-Història_i_concepte-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isomorfisme_parcial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isomorfisme_parcial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Isomorfisme parcial</span> </div> </a> <ul id="toc-Isomorfisme_parcial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Categories" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Categories"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Categories</span> </div> </a> <ul id="toc-Categories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conjunts_isomorfs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conjunts_isomorfs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Conjunts isomorfs</span> </div> </a> <ul id="toc-Conjunts_isomorfs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-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="Contingut" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Isomorfisme</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="Vés a un article en una altra llengua. Disponible en 59 llengües" > <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 llengües</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 - àrab" lang="ar" hreflang="ar" data-title="تماكل" data-language-autonym="العربية" data-language-local-name="àrab" 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 - asturià" lang="ast" hreflang="ast" data-title="Isomorfismu" data-language-autonym="Asturianu" data-language-local-name="asturià" 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 - azerbaidjanès" lang="az" hreflang="az" data-title="İzomorfluq" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidjanès" 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 - baixkir" lang="ba" hreflang="ba" data-title="Изоморфизм" data-language-autonym="Башҡортса" data-language-local-name="baixkir" 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 - belarús" lang="be" hreflang="be" data-title="Ізамарфізм" data-language-autonym="Беларуская" data-language-local-name="belarús" 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 - búlgar" lang="bg" hreflang="bg" data-title="Изоморфизъм" data-language-autonym="Български" data-language-local-name="búlgar" 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 - bosnià" lang="bs" hreflang="bs" data-title="Izomorfizam" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Izomorfismus" title="Izomorfismus - txec" lang="cs" hreflang="cs" data-title="Izomorfismus" data-language-autonym="Čeština" data-language-local-name="txec" 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 - gal·lès" lang="cy" hreflang="cy" data-title="Isomorffedd" data-language-autonym="Cymraeg" data-language-local-name="gal·lès" 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 - danès" lang="da" hreflang="da" data-title="Isomorfi" data-language-autonym="Dansk" data-language-local-name="danès" 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 - alemany" lang="de" hreflang="de" data-title="Isomorphismus" data-language-autonym="Deutsch" data-language-local-name="alemany" 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 - anglès" lang="en" hreflang="en" data-title="Isomorphism" data-language-autonym="English" data-language-local-name="anglès" 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 - esperanto" lang="eo" hreflang="eo" data-title="Izomorfio" data-language-autonym="Esperanto" data-language-local-name="esperanto" 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 - espanyol" lang="es" hreflang="es" data-title="Isomorfismo" data-language-autonym="Español" data-language-local-name="espanyol" 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 - estonià" lang="et" hreflang="et" data-title="Isomorfism" data-language-autonym="Eesti" data-language-local-name="estonià" 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 - basc" lang="eu" hreflang="eu" data-title="Isomorfismo" data-language-autonym="Euskara" data-language-local-name="basc" 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 - persa" lang="fa" hreflang="fa" data-title="یکریختی" data-language-autonym="فارسی" data-language-local-name="persa" 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 - finès" lang="fi" hreflang="fi" data-title="Isomorfismi" data-language-autonym="Suomi" data-language-local-name="finès" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Isomorphisme" title="Isomorphisme - francès" lang="fr" hreflang="fr" data-title="Isomorphisme" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Iseamorfacht" title="Iseamorfacht - irlandès" lang="ga" hreflang="ga" data-title="Iseamorfacht" data-language-autonym="Gaeilge" data-language-local-name="irlandès" 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 - gallec" lang="gl" hreflang="gl" data-title="Isomorfismo" data-language-autonym="Galego" data-language-local-name="gallec" 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 - hebreu" lang="he" hreflang="he" data-title="איזומורפיזם" data-language-autonym="עברית" data-language-local-name="hebreu" 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 - croat" lang="hr" hreflang="hr" data-title="Izomorfizam" data-language-autonym="Hrvatski" data-language-local-name="croat" 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 - hongarès" lang="hu" hreflang="hu" data-title="Izomorfia" data-language-autonym="Magyar" data-language-local-name="hongarès" 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) - armeni" lang="hy" hreflang="hy" data-title="Իզոմորֆություն (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="armeni" 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 - indonesi" lang="id" hreflang="id" data-title="Isomorfisme" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" 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 - italià" lang="it" hreflang="it" data-title="Isomorfismo" data-language-autonym="Italiano" data-language-local-name="italià" 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 - japonès" lang="ja" hreflang="ja" data-title="同型写像" data-language-autonym="日本語" data-language-local-name="japonès" 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 - coreà" lang="ko" hreflang="ko" data-title="동형 사상" data-language-autonym="한국어" data-language-local-name="coreà" 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 - kirguís" lang="ky" hreflang="ky" data-title="Изоморфизм" data-language-autonym="Кыргызча" data-language-local-name="kirguís" 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 - llatí" lang="la" hreflang="la" data-title="Isomorphismus" data-language-autonym="Latina" data-language-local-name="llatí" 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 - lituà" lang="lt" hreflang="lt" data-title="Izomorfizmas" data-language-autonym="Lietuvių" data-language-local-name="lituà" 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 - neerlandès" lang="nl" hreflang="nl" data-title="Isomorfisme" data-language-autonym="Nederlands" data-language-local-name="neerlandès" 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 - noruec nynorsk" lang="nn" hreflang="nn" data-title="Isomorfi" data-language-autonym="Norsk nynorsk" data-language-local-name="noruec 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 - noruec bokmål" lang="nb" hreflang="nb" data-title="Isomorfisme" data-language-autonym="Norsk bokmål" data-language-local-name="noruec 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 - panjabi" lang="pa" hreflang="pa" data-title="ਆਇਸੋਮੌਰਫਿਜ਼ਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" 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 - polonès" lang="pl" hreflang="pl" data-title="Izomorfizm" data-language-autonym="Polski" data-language-local-name="polonès" 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 - piemontès" lang="pms" hreflang="pms" data-title="Isomorfism" data-language-autonym="Piemontèis" data-language-local-name="piemontès" 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 - portuguès" lang="pt" hreflang="pt" data-title="Isomorfismo" data-language-autonym="Português" data-language-local-name="portuguès" 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 - romanès" lang="ro" hreflang="ro" data-title="Izomorfism" data-language-autonym="Română" data-language-local-name="romanès" 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 - rus" lang="ru" hreflang="ru" data-title="Изоморфизм" data-language-autonym="Русский" data-language-local-name="rus" 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 - serbocroat" lang="sh" hreflang="sh" data-title="Izomorfizam" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" 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 - eslovè" lang="sl" hreflang="sl" data-title="Izomorfizem" data-language-autonym="Slovenščina" data-language-local-name="eslovè" 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) - serbi" lang="sr" hreflang="sr" data-title="Изоморфизам (математика)" data-language-autonym="Српски / srpski" data-language-local-name="serbi" 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 - suec" lang="sv" hreflang="sv" data-title="Isomorfi" data-language-autonym="Svenska" data-language-local-name="suec" 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 - tàmil" lang="ta" hreflang="ta" data-title="ஐசோமார்பிஸம்" data-language-autonym="தமிழ்" data-language-local-name="tàmil" 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 - ucraïnès" lang="uk" hreflang="uk" data-title="Ізоморфізм" data-language-autonym="Українська" data-language-local-name="ucraïnès" 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 - urdú" lang="ur" hreflang="ur" data-title="مشاکلت" data-language-autonym="اردو" data-language-local-name="urdú" 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) - uzbek" lang="uz" hreflang="uz" data-title="Izomorfizm (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbek" 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" 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class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">amaga</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">De la Viquipèdia, l'enciclopèdia lliure</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="ca" dir="ltr"><p>En <a href="/wiki/Matem%C3%A0tiques" title="Matemàtiques">matemàtiques</a>, un <b>isomorfisme</b> és un <a href="/wiki/Morfisme" title="Morfisme">morfisme</a> que admet un <a href="/wiki/Funci%C3%B3_inversa" title="Funció inversa">invers</a>, que és també un <a href="/wiki/Morfisme" title="Morfisme">morfisme</a>.<sup id="cite_ref-Milne_2012_p._20_1-0" class="reference"><a href="#cite_note-Milne_2012_p._20-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Conseqüentment, un <b>isomorfisme</b> és una <a href="/wiki/Bijecci%C3%B3" class="mw-redirect" title="Bijecció">bijecció</a>, ja que les <a href="/wiki/Relacions" class="mw-redirect" title="Relacions">relacions</a> <i>algebraiques</i> entre els <a href="/wiki/Element_(matem%C3%A0tiques)" title="Element (matemàtiques)">elements</a> del <a href="/wiki/Conjunt_d%27arribada" class="mw-redirect" title="Conjunt d'arribada">conjunt d'arribada</a> són les mateixes que els seus antecedents respectius, i l'<a href="/wiki/Estructura_algebraica" title="Estructura algebraica">estructura algebraica</a> es conserva.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definició_formal"><span id="Definici.C3.B3_formal"></span>Definició formal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfisme&action=edit&section=1" title="Modifica la secció: Definició formal"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un isomorfisme es pot definit concisament com un <a href="/wiki/Homomorfisme" class="mw-redirect" title="Homomorfisme">homomorfisme</a> <a href="/wiki/Bijecci%C3%B3" class="mw-redirect" title="Bijecció">bijectiu</a> tal que la seva inversa és també un <a href="/wiki/Homomorfisme" class="mw-redirect" title="Homomorfisme">homomorfisme</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> És a dir:<sup id="cite_ref-Definición_formal_7-0" class="reference"><a href="#cite_note-Definición_formal-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-propiedades_y_definición_8-0" class="reference"><a href="#cite_note-propiedades_y_definición-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <table style="margin-right:4em; margin-right:6em; min-width:50%; max-width:77%;"> <tbody><tr> <td><blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #400000; background-color: #FFFFFF;"> <p>Un <i><b>isomorfisme</b></i> entre dos conjunts <a href="/wiki/Relaci%C3%B3_d%27ordre" title="Relació d'ordre">ordenats</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 (P,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8fe8855b84572c55012b0544255beb8d64b16a" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (P,\leq )}"></span> i <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 (Q,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Q,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e438e7de530cdd6e1f5e576ebd996c32932a768" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.174ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (Q,\leq ')}"></span> és una <a href="/wiki/Funci%C3%B3_bijectiva" title="Funció bijectiva">funció bijectiva</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 {\begin{array}{rrcl}h:P\to Q\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>h</mi> <mo>:</mo> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rrcl}h:P\to Q\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49e0252a9598b27c979ff9c950fe693d89848d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.225ex; height:2.843ex;" alt="{\displaystyle {\begin{array}{rrcl}h:P\to Q\\\end{array}}}"></span> tal que:<br /> Per a tot <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_{1},p_{2}\in P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2}\in P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563a77e08396f8f8509c9b1353000cc92390a244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.157ex; height:2.509ex;" alt="{\displaystyle p_{1},p_{2}\in P}"></span> es té 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 p_{1}\leq p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}\leq p_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f500f2c72ebd224059323a2dad9a7b1742c8b3c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.635ex; height:2.343ex;" aria-hidden="true" alt="{\displaystyle p_{1}\leq p_{2}}"></span> <a href="/wiki/Si_i_nom%C3%A9s_si" title="Si i només si">si i només si</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 h(p_{1})\leq 'h(p_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(p_{1})\leq 'h(p_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd83bcb229613fd7b8d40e16a62e91dd559b6cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:14.527ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle h(p_{1})\leq 'h(p_{2})}"></span>. </p> </blockquote> </td></tr></tbody></table> <p>Si existeix un isomorfisme entre <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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8fe8855b84572c55012b0544255beb8d64b16a" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (P,\leq )}"></span> i <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 (Q,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Q,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e438e7de530cdd6e1f5e576ebd996c32932a768" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.174ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (Q,\leq ')}"></span>, llavors <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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8fe8855b84572c55012b0544255beb8d64b16a" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (P,\leq )}"></span> i <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 (Q,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Q,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e438e7de530cdd6e1f5e576ebd996c32932a768" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.174ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (Q,\leq ')}"></span> s'anomenen <i>isomorfes</i> i la bijecció <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> es coneix com <i>isomorfisme</i> entre <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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8fe8855b84572c55012b0544255beb8d64b16a" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (P,\leq )}"></span> i <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 (Q,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Q,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e438e7de530cdd6e1f5e576ebd996c32932a768" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.174ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (Q,\leq ')}"></span>. A més, <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> i <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> reben el nom de <i><b>similars</b></i> entre sí.<sup id="cite_ref-Definición_formal_7-1" class="reference"><a href="#cite_note-Definición_formal-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>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 P=Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abc7e2c5a78e9e6cb7a2a907279953f9b4a3f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P=Q}"></span>, es diu que l'isomorfisme és un <i><b><a href="/wiki/Automorfisme" title="Automorfisme">automorfisme</a></b></i>.<sup id="cite_ref-Tarrida_2008_p._109_10-0" class="reference"><a href="#cite_note-Tarrida_2008_p._109-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Es pot demostrar que donat un <a href="/w/index.php?title=Conjunt_ben_ordenat&action=edit&redlink=1" class="new" title="Conjunt ben ordenat (encara no existeix)">conjunt ben ordenat</a>, l'únic automorfisme possible és la <a href="/wiki/Funci%C3%B3_identitat" title="Funció identitat">funció identitat</a>.<sup id="cite_ref-propiedades_y_definición_8-1" class="reference"><a href="#cite_note-propiedades_y_definición-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Propietats_en_els_ordres_totals">Propietats en els ordres totals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfisme&action=edit&section=2" title="Modifica la secció: Propietats en els ordres totals"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Els isomorfismes en conjunts <a href="/wiki/Ordre_total" title="Ordre total">linealment ordenats</a> tenen una <a href="/wiki/Relaci%C3%B3_d%27equival%C3%A8ncia" title="Relació d'equivalència">relació d'equivalència</a>, és a dir, cumpleixen la <a href="/wiki/Relaci%C3%B3_reflexiva" title="Relació reflexiva">reflexivitat</a>, la <a href="/wiki/Relaci%C3%B3_sim%C3%A8trica" title="Relació simètrica">simetria</a> i la <a href="/wiki/Relaci%C3%B3_transitiva" title="Relació transitiva">transitivitat</a>, és a dir:<sup id="cite_ref-propiedades_y_definición_8-2" class="reference"><a href="#cite_note-propiedades_y_definición-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <table style="margin-right:4em; margin-right:6em; min-width:50%; max-width:77%;"> <tbody><tr> <td><blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #49768C; background-color: #FFFFFF;"> <p>Siguin <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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dccbed0945476c5f14b21d76e3d936f4acf547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (A,\leq )}"></span>, <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 (B,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b248ec6ed417179fd8f63aa4702015cc33ce509" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.1ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (B,\leq ')}"></span> i <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,\leq '')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>″</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C,\leq '')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bafabe30b74061ae6c22dd19d8f25088807f08cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.555ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (C,\leq '')}"></span> conjunts linealment ordenats, llavors:<br /> </p> <ul><li><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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dccbed0945476c5f14b21d76e3d936f4acf547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (A,\leq )}"></span> és isomorf 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 (A,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dccbed0945476c5f14b21d76e3d936f4acf547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (A,\leq )}"></span>.</li> <li>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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dccbed0945476c5f14b21d76e3d936f4acf547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (A,\leq )}"></span> és isomorf 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 (B,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b248ec6ed417179fd8f63aa4702015cc33ce509" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.1ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (B,\leq ')}"></span>, llavors <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 (B,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b248ec6ed417179fd8f63aa4702015cc33ce509" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.1ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (B,\leq ')}"></span> és isomorf 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 (A,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dccbed0945476c5f14b21d76e3d936f4acf547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (A,\leq )}"></span>.</li> <li>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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dccbed0945476c5f14b21d76e3d936f4acf547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (A,\leq )}"></span> és isomorfo 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 (B,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b248ec6ed417179fd8f63aa4702015cc33ce509" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.1ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (B,\leq ')}"></span> i alhora, <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 (B,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b248ec6ed417179fd8f63aa4702015cc33ce509" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.1ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (B,\leq ')}"></span> és isomorf 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 (C,\leq '')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>″</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C,\leq '')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bafabe30b74061ae6c22dd19d8f25088807f08cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.555ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (C,\leq '')}"></span> llavors <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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dccbed0945476c5f14b21d76e3d936f4acf547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (A,\leq )}"></span> és isomorf 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 (C,\leq '')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>″</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C,\leq '')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bafabe30b74061ae6c22dd19d8f25088807f08cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.555ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (C,\leq '')}"></span>.</li></ul> </blockquote> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Història_i_concepte"><span id="Hist.C3.B2ria_i_concepte"></span>Història i concepte</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfisme&action=edit&section=3" title="Modifica la secció: Història i concepte"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En el segle XX, es va precisar en matemàtiques la noció intuïtiva d'<a href="/wiki/Estructura_matem%C3%A0tica" title="Estructura matemàtica">estructura</a>, seguint la concepció d'<a href="/wiki/Arist%C3%B2til" title="Aristòtil">Aristòtil</a> de la matèria i la forma, segons la qual cada estructura és un <a href="/wiki/Conjunt" title="Conjunt">conjunt</a> X dotat de certes operacions (com la suma o el producte) o de certes relacions (com una <a href="/wiki/Relaci%C3%B3_d%27ordre" title="Relació d'ordre">ordenació</a>) o certs subconjunts (com en el cas de la <a href="/wiki/Topologia" title="Topologia">topologia</a>), etc. En aquest cas, el conjunt X és la matèria i les operacions, relacions, etc., en ell definides, són la forma. </p><p>El descobrimient de <a href="/wiki/Plat%C3%B3" title="Plató">Plató</a>, que la forma és el que importa, es recull en matemàtiques amb el concepte d'isomorfisme. Una <a href="/wiki/Funci%C3%B3" title="Funció">aplicació</a> f:X→Y entre dos conjunts dotats del mateix tipus d'estructura és un isomorfisme quan cada element d'Y prové d'un únic element de X i f transforma les operacions, relacions, etc., que hi ha en X en les que hi ha en Y. Quan entre dues estructures hi ha un isomorfisme, ambdues són indistingibles, tenen les mateixes propietats, i qualsevol enunciat és simultàniament cert o fals en les dues. Per això en matemàtiques les estructures s'han de classificar <i><a href="/wiki/Llevat_de" title="Llevat de">llevat dels</a> isomorfismes</i>. </p><p>En el segle XX el biòleg i filòsof de la ciència austríac, <a href="/w/index.php?title=Ludwig_von_Bertalanffy&action=edit&redlink=1" class="new" title="Ludwig von Bertalanffy (encara no existeix)">Ludwig von Bertalanffy</a>, va recuperar aquest concepte com a element en la formulació de la seva <a href="/wiki/Teoria_de_sistemes" title="Teoria de sistemes">teoria general de sistemes</a>.<sup id="cite_ref-Voennaya_mysl_2006_p._2-PA159_11-0" class="reference"><a href="#cite_note-Voennaya_mysl_2006_p._2-PA159-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Per a aquest autor, existien un seguit de coincidències en l'evolució dels processos que es duen a terme en diferents camps del coneixement (la biologia, la demografia, la física, les ciències socials, etc.) que va denominar isomorfismes.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Resultava important per al plantejament de la nova teoria, atès que «l'isomorfisme trobat entre diferents terrenys es basa en l'existència de principis generals de sistemes, d'una teoria general dels sistemes més o menys ben desenvolupada».<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Isomorfisme_parcial">Isomorfisme parcial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfisme&action=edit&section=4" title="Modifica la secció: Isomorfisme parcial"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Està definit com:<sup id="cite_ref-propiedades_y_definición_8-3" class="reference"><a href="#cite_note-propiedades_y_definición-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <table style="margin-right:4em; margin-right:6em; min-width:50%; max-width:77%;"> <tbody><tr> <td><blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #400000; background-color: #FFFFFF;"> <p>Un <i><b>isomorfisme parcial</b></i> entre dos conjunts ordenats <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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8fe8855b84572c55012b0544255beb8d64b16a" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle (P,\leq )}"></span> i <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 (Q,\leq ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>,</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Q,\leq ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e438e7de530cdd6e1f5e576ebd996c32932a768" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:7.174ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle (Q,\leq ')}"></span> és una <a href="/wiki/Funci%C3%B3_bijectiva" title="Funció bijectiva">funció bijectiva</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 {\begin{array}{rrcl}h:X\to Q\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>h</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rrcl}h:X\to Q\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f75d3519b69046d95a1644ae76bcb67e756519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.46ex; height:2.843ex;" alt="{\displaystyle {\begin{array}{rrcl}h:X\to Q\\\end{array}}}"></span> amb <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\subseteq P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊆</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\subseteq P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b77cc0aa5954980cb224eb41fa480843af1a32c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.824ex; height:2.343ex;" alt="{\displaystyle X\subseteq P}"></span> tal que per a tot <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_{1},p_{2}\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2}\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba3ab90c5c5ecf0812dea49366dde433e24d902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.391ex; height:2.509ex;" alt="{\displaystyle p_{1},p_{2}\in X}"></span> es té 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 p_{1}\leq p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}\leq p_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f500f2c72ebd224059323a2dad9a7b1742c8b3c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.635ex; height:2.343ex;" aria-hidden="true" alt="{\displaystyle p_{1}\leq p_{2}}"></span> si i només 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 h(p_{1})\leq 'h(p_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo>≤</mo> <mo>′</mo> </msup> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(p_{1})\leq 'h(p_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd83bcb229613fd7b8d40e16a62e91dd559b6cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:14.527ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle h(p_{1})\leq 'h(p_{2})}"></span>. </p> </blockquote> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Categories">Categories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfisme&action=edit&section=5" title="Modifica la secció: Categories"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De forma més general, en la <a href="/wiki/Teoria_de_les_categories" class="mw-redirect" title="Teoria de les categories">teoria de les categories</a>, un <b>isomorfisme</b> és un <a href="/wiki/Morfisme" title="Morfisme">morfisme</a> que té un <b>invers a la dreta</b> i un <b>invers a l'esquerra</b>. </p> <ul><li><i>Invers a la dreta</i>: 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 f:A\rightarrow 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\rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2fb4d5e9d282ee5442719053c46ad1ad96f2ca" 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\rightarrow B}"></span>, llavors, existeix <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\rightarrow 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\rightarrow A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7baa77b641a7a411ab38856d75cf9990e41cb0b" 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\rightarrow A}"></span> tal 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\circ g=I_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g=I_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80f18d7cce1a916dd119a4d8930bbb0924d4f734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.19ex; height:2.509ex;" alt="{\displaystyle f\circ g=I_{B}}"></span></li> <li><i>Invers a l'esquerra</i>: 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 f:A\rightarrow 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\rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2fb4d5e9d282ee5442719053c46ad1ad96f2ca" 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\rightarrow B}"></span>, llavors, existeix <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\rightarrow 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\rightarrow A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7baa77b641a7a411ab38856d75cf9990e41cb0b" 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\rightarrow A}"></span> tal 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 g\circ f=I_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f=I_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f6adc514b704ff5e702ee606f19b3b2f5650ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.176ex; height:2.509ex;" alt="{\displaystyle g\circ f=I_{A}}"></span></li></ul> <p>Cal observar que, generalment, l'existència de l'<b>invers a la dreta</b> no comporta l'existència de l'<b>invers a l'esquerra</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Conjunts_isomorfs">Conjunts isomorfs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfisme&action=edit&section=6" title="Modifica la secció: Conjunts isomorfs"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dos <a href="/wiki/Conjunt" title="Conjunt">conjunts</a> enllaçats per un <b>isomorfisme</b> s'anomenen <b>isomorfs</b>. </p><p>Des de molts punts de vista, dos conjunts <b>isomorfs</b> poden ser considerats <i>idèntics</i>. En efecte, normalment les <a href="/wiki/Propietat_(ontologia)" title="Propietat (ontologia)">propietats</a> interessants d'un conjunt seran compartides per tots els seus conjunts <b>isomorfs</b> de la categoria. </p> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfisme&action=edit&section=7" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Milne_2012_p._20-1"><span class="mw-cite-backlink"><a href="#cite_ref-Milne_2012_p._20_1-0">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFMilne2012"><span style="font-variant: small-caps;">Milne</span>, J.S.. <a rel="nofollow" class="external text" href="https://books.google.cat/books?id=Nj9ZLgOEzE4C&pg=PA20"><i>Algebraic Geometry</i></a> (en alemany).  Allied Publishers, 2012, p. 20. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-81-7764-454-8" title="Especial:Fonts bibliogràfiques/978-81-7764-454-8">ISBN 978-81-7764-454-8</a></span> [Consulta: 28 juliol 2023].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.aulast=Milne&rft.aufirst=J.S.&rft.date=2012&rft.pub=Allied+Publishers&rft.pages=20&rft.isbn=978-81-7764-454-8&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%3Fid%3DNj9ZLgOEzE4C%26pg%3DPA20"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Isomorphism.html">Isomorphism</a>. MathWorld</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/index.php?title=Isomorphism">Isomorphism - Encyclopedia of Mathematics</a>». [Consulta: 21 gener 2022].</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://matematica.laguia2000.com/general/isomorfismo">Isomorfismo | La Guía de Matemática</a>». [Consulta: 21 gener 2022].</span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://www.britannica.com/science/isomorphism-mathematics">Isomorphism | Group Theory, Algebraic Structures, Equivalence Relations | Britannica</a>» (en anglès). [Consulta: 18 juliol 2023].</span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Isomorphism.html">Mathworld</a></span> </li> <li id="cite_note-Definición_formal-7"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Definición_formal_7-0">7,0</a></sup> <sup><a href="#cite_ref-Definición_formal_7-1">7,1</a></sup></span> <span class="reference-text"><span class="citation" style="font-style:normal" id="CITEREFCasanovas1998"><span style="font-variant: small-caps;">Casanovas</span>, E. «<a rel="nofollow" class="external text" href="http://www.ub.edu/modeltheory/documentos/T.C.pdf">Teoría axiomática de conjuntos</a>». <i>Universidad de Barcelona</i>, 1998, pàg. 5, 6, 7 [Consulta: 23 abril 2013].</span></span> </li> <li id="cite_note-propiedades_y_definición-8"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-propiedades_y_definición_8-0">8,0</a></sup> <sup><a href="#cite_ref-propiedades_y_definición_8-1">8,1</a></sup> <sup><a href="#cite_ref-propiedades_y_definición_8-2">8,2</a></sup> <sup><a href="#cite_ref-propiedades_y_definición_8-3">8,3</a></sup></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFHrbecek1999"><span style="font-variant: small-caps;">Hrbecek</span>, Karel. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontose00hrba"><i>Introduction to Set Theory</i></a> (en anglès).  Marcel Dekker, Inc, 1999, p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontose00hrba/page/n52">36</a>, 58.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Set+Theory&rft.aulast=Hrbecek&rft.aufirst=Karel&rft.date=1999&rft.pub=Marcel+Dekker%2C+Inc&rft.pages=%5Bhttps%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontose00hrba%2Fpage%2Fn52+36%5D%2C+58&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontose00hrba"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFHernández_Hernández1998"><span style="font-variant: small-caps;">Hernández Hernández</span>, Fernando. <i>Teoría de conjuntos: Una introducción</i>.  Sociedad Matemática Mexicana, 1998, p. 84,85.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teor%C3%ADa+de+conjuntos%3A+Una+introducci%C3%B3n&rft.aulast=Hern%C3%A1ndez+Hern%C3%A1ndez&rft.aufirst=Fernando&rft.date=1998&rft.pub=Sociedad+Matem%C3%A1tica+Mexicana&rft.pages=84%2C85"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Tarrida_2008_p._109-10"><span class="mw-cite-backlink"><a href="#cite_ref-Tarrida_2008_p._109_10-0">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFTarrida2008"><span style="font-variant: small-caps;">Tarrida</span>, A.R.. <a rel="nofollow" class="external text" href="https://books.google.cat/books?id=j38yHGdm9eoC&pg=PA109"><i>Afinitats, moviments i quàdriques</i></a>.  Universitat Autònoma de Barcelona, Servei de Publicacions, 2008, p. 109. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-84-490-2554-9" title="Especial:Fonts bibliogràfiques/978-84-490-2554-9">ISBN 978-84-490-2554-9</a></span> [Consulta: 28 juliol 2023].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Afinitats%2C+moviments+i+qu%C3%A0driques&rft.aulast=Tarrida&rft.aufirst=A.R.&rft.date=2008&rft.pub=Universitat+Aut%C3%B2noma+de+Barcelona%2C+Servei+de+Publicacions&rft.pages=109&rft.isbn=978-84-490-2554-9&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%3Fid%3Dj38yHGdm9eoC%26pg%3DPA109"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Voennaya_mysl_2006_p._2-PA159-11"><span class="mw-cite-backlink"><a href="#cite_ref-Voennaya_mysl_2006_p._2-PA159_11-0">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://books.google.cat/books?id=eBgKdEzf3P8C&pg=RA2-PA159"><i>Military Thought</i></a>.  Voennaya mysl, 2006, p. 2-PA159 [Consulta: 28 juliol 2023].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Military+Thought&rft.date=2006&rft.pub=Voennaya+mysl&rft.pages=2-PA159&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%3Fid%3DeBgKdEzf3P8C%26pg%3DRA2-PA159"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFVon_Bertalanfffy2009"><span style="font-variant: small-caps;">Von Bertalanfffy</span>, Ludwing. <i>Teoría General de los Sistemas</i>.  Fondo de Cultura Económica, 2009, p. 82-88. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-968-16-0627-5" title="Especial:Fonts bibliogràfiques/978-968-16-0627-5">ISBN 978-968-16-0627-5</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teor%C3%ADa+General+de+los+Sistemas&rft.aulast=Von+Bertalanfffy&rft.aufirst=Ludwing&rft.date=2009&rft.pub=Fondo+de+Cultura+Econ%C3%B3mica&rft.pages=82-88&rft.isbn=978-968-16-0627-5"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFVon_Bertalanffy2009"><span style="font-variant: small-caps;">Von Bertalanffy</span>, Ludwing. <i>Teoría General de los Sistemas</i>.  Fondo de Cultura Económica, 2009, p. 86.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teor%C3%ADa+General+de+los+Sistemas&rft.aulast=Von+Bertalanffy&rft.aufirst=Ludwing&rft.date=2009&rft.pub=Fondo+de+Cultura+Econ%C3%B3mica&rft.pages=86"><span style="display: none;"> </span></span></span> </li> </ol></div></div> <p><span style="display: none;" class="interProject"><a href="https://ca.wiktionary.org/wiki/isomorfisme" class="extiw" title="wikt:isomorfisme">Viccionari</a></span> </p> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Control_d%27autoritats" title="Control d'autoritats">Registres d'autoritat</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/LCCN" class="mw-redirect" title="LCCN">LCCN</a> <span class="uid"> (<a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85068654">1</a>)</span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Isomorfisme&oldid=33972276">https://ca.wikipedia.org/w/index.php?title=Isomorfisme&oldid=33972276</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categoria</a>: <ul><li><a href="/wiki/Categoria:%C3%80lgebra_abstracta" title="Categoria:Àlgebra abstracta">Àlgebra abstracta</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria oculta: <ul><li><a href="/wiki/Categoria:Control_d%27autoritats" title="Categoria:Control d'autoritats">Control d'autoritats</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 20 set 2024 a les 18:46.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. 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