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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrýt</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">(úvod)</div> </a> </li> <li id="toc-Definice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definice</span> </div> </a> <button aria-controls="toc-Definice-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Definice</span> </button> <ul id="toc-Definice-sublist" class="vector-toc-list"> <li id="toc-Definice_z_teorie_množin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definice_z_teorie_množin"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Definice z teorie množin</span> </div> </a> <ul id="toc-Definice_z_teorie_množin-sublist" class="vector-toc-list"> <li id="toc-Význam_definice" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Význam_definice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Význam definice</span> </div> </a> <ul id="toc-Význam_definice-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definice_pro_uspořádané_množiny" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definice_pro_uspořádané_množiny"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definice pro uspořádané množiny</span> </div> </a> <ul id="toc-Definice_pro_uspořádané_množiny-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraická_definice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraická_definice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Algebraická definice</span> </div> </a> <ul id="toc-Algebraická_definice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definice_pro_grafy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definice_pro_grafy"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Definice pro grafy</span> </div> </a> <ul id="toc-Definice_pro_grafy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vztah_k_homomorfismům" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vztah_k_homomorfismům"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vztah k homomorfismům</span> </div> </a> <ul id="toc-Vztah_k_homomorfismům-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Příklady" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Příklady"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Příklady</span> </div> </a> <ul id="toc-Příklady-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-V_teorii_kategorií" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#V_teorii_kategorií"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>V teorii kategorií</span> </div> </a> <ul id="toc-V_teorii_kategorií-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Odkazy</span> </div> </a> <button aria-controls="toc-Odkazy-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Odkazy</span> </button> <ul id="toc-Odkazy-sublist" class="vector-toc-list"> <li id="toc-Poznámky" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poznámky"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Poznámky</span> </div> </a> <ul id="toc-Poznámky-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-sublist" class="vector-toc-list"> </ul> </li> </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="Obsah" 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="Přepnout obsah" > <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">Přepnout obsah</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">Izomorfismus</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="Přejděte k článku v jiném jazyce. Je dostupný v 59 jazycích" > <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 jazyků</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 – arabština" lang="ar" hreflang="ar" data-title="تماكل" data-language-autonym="العربية" data-language-local-name="arabština" 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 – asturština" lang="ast" hreflang="ast" data-title="Isomorfismu" data-language-autonym="Asturianu" data-language-local-name="asturština" 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 – ázerbájdžánština" lang="az" hreflang="az" data-title="İzomorfluq" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" 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 – baškirština" lang="ba" hreflang="ba" data-title="Изоморфизм" data-language-autonym="Башҡортса" data-language-local-name="baškirština" 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 – běloruština" lang="be" hreflang="be" data-title="Ізамарфізм" data-language-autonym="Беларуская" data-language-local-name="běloruština" 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 – bulharština" lang="bg" hreflang="bg" data-title="Изоморфизъм" data-language-autonym="Български" data-language-local-name="bulharština" 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 – bosenština" lang="bs" hreflang="bs" data-title="Izomorfizam" data-language-autonym="Bosanski" data-language-local-name="bosenština" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – katalánština" lang="ca" hreflang="ca" data-title="Isomorfisme" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Isomorffedd" title="Isomorffedd – velština" lang="cy" hreflang="cy" data-title="Isomorffedd" data-language-autonym="Cymraeg" data-language-local-name="velština" 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 – dánština" lang="da" hreflang="da" data-title="Isomorfi" data-language-autonym="Dansk" data-language-local-name="dánština" 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 – němčina" lang="de" hreflang="de" data-title="Isomorphismus" data-language-autonym="Deutsch" data-language-local-name="němčina" 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 – řečtina" lang="el" hreflang="el" data-title="Ισομορφισμός" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" 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 – angličtina" lang="en" hreflang="en" data-title="Isomorphism" data-language-autonym="English" data-language-local-name="angličtina" 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 – španělština" lang="es" hreflang="es" data-title="Isomorfismo" data-language-autonym="Español" data-language-local-name="španělština" 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 – estonština" lang="et" hreflang="et" data-title="Isomorfism" data-language-autonym="Eesti" data-language-local-name="estonština" 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 – baskičtina" lang="eu" hreflang="eu" data-title="Isomorfismo" data-language-autonym="Euskara" data-language-local-name="baskičtina" 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 – perština" lang="fa" hreflang="fa" data-title="یکریختی" data-language-autonym="فارسی" data-language-local-name="perština" 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 – finština" lang="fi" hreflang="fi" data-title="Isomorfismi" data-language-autonym="Suomi" data-language-local-name="finština" 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 – francouzština" lang="fr" hreflang="fr" data-title="Isomorphisme" data-language-autonym="Français" data-language-local-name="francouzština" 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 – irština" lang="ga" hreflang="ga" data-title="Iseamorfacht" data-language-autonym="Gaeilge" data-language-local-name="irština" 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 – galicijština" lang="gl" hreflang="gl" data-title="Isomorfismo" data-language-autonym="Galego" data-language-local-name="galicijština" 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 – hebrejština" lang="he" hreflang="he" data-title="איזומורפיזם" data-language-autonym="עברית" data-language-local-name="hebrejština" 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 – chorvatština" lang="hr" hreflang="hr" data-title="Izomorfizam" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" 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 – maďarština" lang="hu" hreflang="hu" data-title="Izomorfia" data-language-autonym="Magyar" data-language-local-name="maďarština" 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) – arménština" lang="hy" hreflang="hy" data-title="Իզոմորֆություն (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="arménština" 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 – indonéština" lang="id" hreflang="id" data-title="Isomorfisme" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" 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 – italština" lang="it" hreflang="it" data-title="Isomorfismo" data-language-autonym="Italiano" data-language-local-name="italština" 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 – japonština" lang="ja" hreflang="ja" data-title="同型写像" data-language-autonym="日本語" data-language-local-name="japonština" 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) – kazaština" lang="kk" hreflang="kk" data-title="Изоморфизм (Математика)" data-language-autonym="Қазақша" data-language-local-name="kazaština" 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 – korejština" lang="ko" hreflang="ko" data-title="동형 사상" data-language-autonym="한국어" data-language-local-name="korejština" 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 – kyrgyzština" lang="ky" hreflang="ky" data-title="Изоморфизм" data-language-autonym="Кыргызча" data-language-local-name="kyrgyzština" 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 – latina" lang="la" hreflang="la" data-title="Isomorphismus" data-language-autonym="Latina" data-language-local-name="latina" 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 – litevština" lang="lt" hreflang="lt" data-title="Izomorfizmas" data-language-autonym="Lietuvių" data-language-local-name="litevština" 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 – mongolština" lang="mn" hreflang="mn" data-title="Изоморф" data-language-autonym="Монгол" data-language-local-name="mongolština" 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 – nizozemština" lang="nl" hreflang="nl" data-title="Isomorfisme" data-language-autonym="Nederlands" data-language-local-name="nizozemština" 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 – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Isomorfi" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (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 – norština (bokmål)" lang="nb" hreflang="nb" data-title="Isomorfisme" data-language-autonym="Norsk bokmål" data-language-local-name="norština (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 – paňdžábština" lang="pa" hreflang="pa" data-title="ਆਇਸੋਮੌਰਫਿਜ਼ਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="paňdžábština" 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 – polština" lang="pl" hreflang="pl" data-title="Izomorfizm" data-language-autonym="Polski" data-language-local-name="polština" 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 – piemonština" lang="pms" hreflang="pms" data-title="Isomorfism" data-language-autonym="Piemontèis" data-language-local-name="piemonština" 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 – portugalština" lang="pt" hreflang="pt" data-title="Isomorfismo" data-language-autonym="Português" data-language-local-name="portugalština" 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 – rumunština" lang="ro" hreflang="ro" data-title="Izomorfism" data-language-autonym="Română" data-language-local-name="rumunština" 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 – ruština" lang="ru" hreflang="ru" data-title="Изоморфизм" data-language-autonym="Русский" data-language-local-name="ruština" 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 – srbochorvatština" lang="sh" hreflang="sh" data-title="Izomorfizam" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" 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 – slovinština" lang="sl" hreflang="sl" data-title="Izomorfizem" data-language-autonym="Slovenščina" data-language-local-name="slovinština" 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) – srbština" lang="sr" hreflang="sr" data-title="Изоморфизам (математика)" data-language-autonym="Српски / srpski" data-language-local-name="srbština" 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 – švédština" lang="sv" hreflang="sv" data-title="Isomorfi" data-language-autonym="Svenska" data-language-local-name="švédština" 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 – tamilština" lang="ta" hreflang="ta" data-title="ஐசோமார்பிஸம்" data-language-autonym="தமிழ்" data-language-local-name="tamilština" 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 – turečtina" lang="tr" hreflang="tr" data-title="İzomorfizma" data-language-autonym="Türkçe" data-language-local-name="turečtina" 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 – ukrajinština" lang="uk" hreflang="uk" data-title="Ізоморфізм" data-language-autonym="Українська" data-language-local-name="ukrajinština" 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 – urdština" lang="ur" hreflang="ur" data-title="مشاکلت" data-language-autonym="اردو" data-language-local-name="urdština" 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) – uzbečtina" lang="uz" hreflang="uz" data-title="Izomorfizm (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_%C4%91%E1%BA%B3ng_c%E1%BA%A5u" title="Phép đẳng cấu – vietnamština" lang="vi" hreflang="vi" data-title="Phép đẳng cấu" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%8C%E6%9E%84" title="同构 – čínština (dialekty Wu)" lang="wuu" hreflang="wuu" data-title="同构" data-language-autonym="吴语" data-language-local-name="čínština (dialekty Wu)" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%90%8C%E6%9E%84" title="同构 – čínština" lang="zh" hreflang="zh" data-title="同构" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%90%8C%E6%A7%8B" title="同構 – kantonština" lang="yue" hreflang="yue" data-title="同構" data-language-autonym="粵語" data-language-local-name="kantonština" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q189112#sitelinks-wikipedia" title="Editovat mezijazykové odkazy" class="wbc-editpage">Upravit odkazy</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Jmenné prostory"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> 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vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Nástroje</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">skrýt</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Další možnosti" > <div class="vector-menu-heading"> Akce </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Izomorfismus"><span>Číst</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Izomorfismus&veaction=edit" title="Editovat tuto stránku [v]" accesskey="v"><span>Editovat</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Izomorfismus&action=edit" title="Editovat zdrojový kód této stránky [e]" accesskey="e"><span>Editovat zdroj</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Izomorfismus&action=history"><span>Zobrazit historii</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Obecné </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Co_odkazuje_na/Izomorfismus" title="Seznam všech wikistránek, které sem odkazují [j]" accesskey="j"><span>Odkazuje sem</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Souvisej%C3%ADc%C3%AD_zm%C4%9Bny/Izomorfismus" rel="nofollow" title="Nedávné změny stránek, na které je odkazováno [k]" accesskey="k"><span>Související změny</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=cs" title="Nahrát obrázky či jiná multimédia [u]" accesskey="u"><span>Načíst soubor</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Speci%C3%A1ln%C3%AD_str%C3%A1nky" title="Seznam všech speciálních stránek [q]" accesskey="q"><span>Speciální stránky</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Izomorfismus&oldid=24348451" title="Trvalý odkaz na současnou verzi této 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lang="cs" dir="ltr"><div class="uvodni-upozorneni hatnote noprint">Další významy jsou uvedeny na stránce <a href="/wiki/Izomorfismus_(rozcestn%C3%ADk)" class="mw-disambig" title="Izomorfismus (rozcestník)">Izomorfismus (rozcestník)</a>.</div> <p><b>Izomorfismus</b> je <a href="/wiki/Zobrazen%C3%AD_(matematika)" title="Zobrazení (matematika)">zobrazení</a> mezi dvěma <a href="/wiki/Matematick%C3%A1_struktura" title="Matematická struktura">matematickými strukturami</a>, které je <a href="/wiki/Bijekce" title="Bijekce">vzájemně jednoznačné (bijektivní)</a> a zachovává všechny vlastnosti touto strukturou definované. Jinými slovy, každému <a href="/wiki/Prvek_mno%C5%BEiny" title="Prvek množiny">prvku</a> první struktury odpovídá právě jeden prvek struktury druhé a toto přiřazení zachovává vztahy k ostatním prvkům. </p><p>O izomorfismech je možno mluvit mezi <a href="/wiki/Mno%C5%BEina" title="Množina">množinami</a>, <a href="/wiki/Algebraick%C3%A1_struktura" title="Algebraická struktura">algebraickými</a> i <a href="/wiki/Rela%C4%8Dn%C3%AD_struktura" class="mw-redirect" title="Relační struktura">relačními strukturami</a>, <a href="/wiki/Graf_(teorie_graf%C5%AF)" title="Graf (teorie grafů)">grafy</a>, <a href="/wiki/Model_(logika)" title="Model (logika)">modely</a>, <a href="/wiki/Metrick%C3%BD_prostor" title="Metrický prostor">metrickými</a> i <a href="/wiki/Topologick%C3%BD_prostor" title="Topologický prostor">topologickými prostory</a> a mnoha dalšími strukturami. </p><p>Například zobrazení <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(x)=2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a8ebea86ba5d3a71121e0a4156f5ec07b25220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.008ex; height:2.843ex;" alt="{\displaystyle f(x)=2x}"></span> z množiny <a href="/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo">reálných čísel</a> do reálných čísel zachovává sčítání (a je tedy <a href="/wiki/Grupa" title="Grupa">grupovým</a> izomorfismem), ale ne násobení (proto není <a href="/wiki/T%C4%9Bleso_(algebra)" title="Těleso (algebra)">tělesovým</a> izomorfismem) ani vzdálenost (proto není izomorfismem <a href="/wiki/Metrick%C3%BD_prostor" title="Metrický prostor">metrických prostorů</a>, ovšem je <a href="/wiki/Homeomorfismus" title="Homeomorfismus">homeomorfismem</a> neboli <a href="/wiki/Topologie" title="Topologie">topologickým izomorfismem</a>). </p><p>Pokud takové zobrazení existuje (tedy struktury jsou <b>izomorfní</b>), mají obě množiny zcela totožné vlastnosti, takže rozdíl mezi nimi je pouze formální a nepodstatný (z hlediska příslušné teorie). Například funkce <a href="/wiki/Arkus_tangens" title="Arkus tangens">arkus tangens</a> je topologickým, ale ne metrickým izomorfismem mezi intervalem <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 (-\pi ,\pi )\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ,\pi )\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d014bca6722584f56ee3538cbcf0d4bd8469fa43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:7.702ex; height:2.843ex;" alt="{\displaystyle (-\pi ,\pi )\,\!}"></span> a reálnými čísly, takže tyto dvě struktury (množiny vybavené <a href="/wiki/Metrick%C3%BD_prostor" title="Metrický prostor">metrikou</a>) mají zcela shodné všechny topologické vlastnosti, ale ne všechny metrické. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definice">Definice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=1" title="Editace sekce: Definice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=1" title="Editovat zdrojový kód sekce Definice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zde uvedeme definice pro jednotlivé obory matematiky a vztahy mezi nimi. </p> <div class="mw-heading mw-heading3"><h3 id="Definice_z_teorie_množin"><span id="Definice_z_teorie_mno.C5.BEin"></span>Definice z teorie množin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=2" title="Editace sekce: Definice z teorie množin" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=2" title="Editovat zdrojový kód sekce Definice z teorie množin"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Předpokládejme, že na množině <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\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c6c570e3a2c130fc8d968160962b5e0fe4b0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.367ex; height:2.176ex;" alt="{\displaystyle X\,\!}"></span> jsou definovány <a href="/wiki/Relace_(matematika)" title="Relace (matematika)">relace</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 R_{1},R_{2},\ldots ,R_{n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1},R_{2},\ldots ,R_{n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db80ba4c1adb532e1dab5bc02f80d0fdc14741a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:15.218ex; height:2.509ex;" alt="{\displaystyle R_{1},R_{2},\ldots ,R_{n}\,\!}"></span> a na množině <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e65066132416db15d76f148ea4436daf49c863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; margin-right: -0.387ex; width:2.16ex; height:2.009ex;" alt="{\displaystyle Y\,\!}"></span> jsou definovány <a href="/wiki/Relace_(matematika)" title="Relace (matematika)">relace</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 S_{1},S_{2},\ldots ,S_{n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1},S_{2},\ldots ,S_{n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0c8a6e1e32415eaa57a8165db97c2f4f593059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:14.201ex; height:2.509ex;" alt="{\displaystyle S_{1},S_{2},\ldots ,S_{n}\,\!}"></span>. Řekneme, že zobrazení <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec86a832c57e76bd90bfa2600197fd4bb435ff02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.128ex; height:2.176ex;" alt="{\displaystyle F\,\!}"></span> je <b>izomorfismus</b> mezi <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\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c6c570e3a2c130fc8d968160962b5e0fe4b0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.367ex; height:2.176ex;" alt="{\displaystyle X\,\!}"></span> 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 Y\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e65066132416db15d76f148ea4436daf49c863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; margin-right: -0.387ex; width:2.16ex; height:2.009ex;" alt="{\displaystyle Y\,\!}"></span> vzhledem k relacím <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 R_{1},R_{2},\ldots ,R_{n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1},R_{2},\ldots ,R_{n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db80ba4c1adb532e1dab5bc02f80d0fdc14741a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:15.218ex; height:2.509ex;" alt="{\displaystyle R_{1},R_{2},\ldots ,R_{n}\,\!}"></span> 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 S_{1},S_{2},\ldots ,S_{n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1},S_{2},\ldots ,S_{n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0c8a6e1e32415eaa57a8165db97c2f4f593059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:14.201ex; height:2.509ex;" alt="{\displaystyle S_{1},S_{2},\ldots ,S_{n}\,\!}"></span>, pokud platí: </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 F\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec86a832c57e76bd90bfa2600197fd4bb435ff02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.128ex; height:2.176ex;" alt="{\displaystyle F\,\!}"></span> je <a href="/wiki/Bijekce" title="Bijekce">vzájemně jednoznačné zobrazení</a> mezi <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\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c6c570e3a2c130fc8d968160962b5e0fe4b0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.367ex; height:2.176ex;" alt="{\displaystyle X\,\!}"></span> 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 Y\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e65066132416db15d76f148ea4436daf49c863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; margin-right: -0.387ex; width:2.16ex; height:2.009ex;" alt="{\displaystyle Y\,\!}"></span></li> <li>pokud jsou <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 R_{i},S_{i}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i},S_{i}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a56b27f773d5c7e6822e38d8868bacc831a3864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:6.209ex; height:2.509ex;" alt="{\displaystyle R_{i},S_{i}\,\!}"></span> j-ární relace, potom <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 \forall x_{1},x_{2},\ldots ,x_{j}\in X:[x_{1},x_{2},\ldots ,x_{j}]\in R_{i}\Leftrightarrow [F(x_{1}),F(x_{2}),\ldots ,F(x_{j})]\in S_{i}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>∈</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">⇔</mo> <mo stretchy="false">[</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x_{1},x_{2},\ldots ,x_{j}\in X:[x_{1},x_{2},\ldots ,x_{j}]\in R_{i}\Leftrightarrow [F(x_{1}),F(x_{2}),\ldots ,F(x_{j})]\in S_{i}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793753db40a2dca354d797420c5f946d66a2adc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:75.417ex; height:3.009ex;" alt="{\displaystyle \forall x_{1},x_{2},\ldots ,x_{j}\in X:[x_{1},x_{2},\ldots ,x_{j}]\in R_{i}\Leftrightarrow [F(x_{1}),F(x_{2}),\ldots ,F(x_{j})]\in S_{i}\,\!}"></span>.</li></ul> <p>Řekneme, že struktury <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,R_{1},R_{2},\ldots ,R_{n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,R_{1},R_{2},\ldots ,R_{n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb05465a01bdbbcc4d4839df826f84f5418b6ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:18.232ex; height:2.509ex;" alt="{\displaystyle X,R_{1},R_{2},\ldots ,R_{n}\,\!}"></span> 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 Y,S_{1},S_{2},\ldots ,S_{n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,S_{1},S_{2},\ldots ,S_{n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7582cf05066a231bdbf3ffcf8a8905ccf65ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:17.008ex; height:2.509ex;" alt="{\displaystyle Y,S_{1},S_{2},\ldots ,S_{n}\,\!}"></span> jsou <b>izomorfní</b>, pokud mezi nimi existuje nějaký izomorfismus ve smyslu výše uvedené definice. </p> <div class="mw-heading mw-heading4"><h4 id="Význam_definice"><span id="V.C3.BDznam_definice"></span>Význam definice</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=3" title="Editace sekce: Význam definice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=3" title="Editovat zdrojový kód sekce Význam definice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ačkoli definice může působit složitě a nepřehledně, zachycuje přesně intuici řečenou v úvodním přiblížení: </p> <ul><li>V rámci izomorfismu se nesmí žádné prvky ztrácet ani objevovat, obě množiny musí mít stejný počet prvků (v případě <a href="/wiki/Nekone%C4%8Dn%C3%A1_mno%C5%BEina" title="Nekonečná množina">nekonečných množin</a> stejnou <a href="/wiki/Mohutnost" title="Mohutnost">mohutnost</a>).</li> <li>Izomorfismus musí zachovávat všechny vztahy, tj. relace - pokud jsou v původní množině nějaké prvky v nějakém vztahu, musí být v nové množině také v odpovídajícím vztahu a naopak.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Definice_pro_uspořádané_množiny"><span id="Definice_pro_uspo.C5.99.C3.A1dan.C3.A9_mno.C5.BEiny"></span>Definice pro uspořádané množiny</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=4" title="Editace sekce: Definice pro uspořádané množiny" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=4" title="Editovat zdrojový kód sekce Definice pro uspořádané množiny"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uvažujme o množinách <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7643a9df8d529853debbea317ca757ae63bfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:5.174ex; height:2.509ex;" alt="{\displaystyle X,Y\,\!}"></span>, které mají <a href="/wiki/Uspo%C5%99%C3%A1d%C3%A1n%C3%AD" class="mw-redirect mw-disambig" title="Uspořádání">uspořádání</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 R,S\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>S</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,S\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6dba624b45af3d371cd24537a078e826b4513c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:4.684ex; height:2.509ex;" alt="{\displaystyle R,S\,\!}"></span>. Izomorfismus v tomto případě znamená, že pokud je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in X,a\leq _{R}b\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mi>a</mi> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mi>b</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in X,a\leq _{R}b\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0d02fe1432d35c48711671abc334881128b6c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:16.308ex; height:2.509ex;" alt="{\displaystyle a,b\in X,a\leq _{R}b\,\!}"></span>, pak musí být <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)\leq _{S}F(b)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(a)\leq _{S}F(b)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1bf2f30b1f4100107cca5342329062316c094a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:14.105ex; height:2.843ex;" alt="{\displaystyle F(a)\leq _{S}F(b)\,\!}"></span>. </p><p>Dá se snadno ukázat, že v izomorfismu se musí <a href="/wiki/Nejmen%C5%A1%C3%AD_prvek" class="mw-redirect" title="Nejmenší prvek">nejmenší prvek</a> zobrazit opět na nejmenší prvek, <a href="/wiki/Infimum" title="Infimum">infimum</a> na infimum, <a href="/wiki/Maxim%C3%A1ln%C3%AD_a_minim%C3%A1ln%C3%AD_prvek" class="mw-redirect" title="Maximální a minimální prvek">minimální prvek</a> na minimální prvek… </p> <div class="mw-heading mw-heading3"><h3 id="Algebraická_definice"><span id="Algebraick.C3.A1_definice"></span>Algebraická definice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=5" title="Editace sekce: Algebraická definice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=5" title="Editovat zdrojový kód sekce Algebraická definice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>V <a href="/wiki/Algebra" title="Algebra">algebře</a> izomorfismem mezi dvěma <a href="/wiki/Algebra_(struktura)" title="Algebra (struktura)">algebrami</a> rozumíme <a href="/wiki/Bijekce" title="Bijekce">bijektivní</a> <a href="/wiki/Homomorfismus" title="Homomorfismus">homomorfismus</a>, tedy zobrazení <a href="/w/index.php?title=Slu%C4%8Ditelnost&action=edit&redlink=1" class="new" title="Slučitelnost (stránka neexistuje)">slučitelné</a> se všemi <a href="/wiki/Operace_(matematika)" title="Operace (matematika)">operacemi</a> na algebře, které je zároveň bijekcí (každému prvku z jedné množiny přiřadí právě jeden prvek z druhé). </p><p>Opět se jedná o zvláštní případ výše uvedené definice – uvědomme si, že operace není nic jiného, než konkrétní typ relace. </p><p>Dá se snadno ukázat, že v izomorfismu se musí <a href="/wiki/Neutr%C3%A1ln%C3%AD_prvek" title="Neutrální prvek">neutrální prvek</a> operace zobrazit na neutrální prvek jí odpovídající operace v druhé množině, obdobně například <a href="/wiki/Inverzn%C3%AD_prvek" title="Inverzní prvek">inverzní prvek</a> opět na inverzní prvek. </p> <div class="mw-heading mw-heading3"><h3 id="Definice_pro_grafy">Definice pro grafy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=6" title="Editace sekce: Definice pro grafy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=6" title="Editovat zdrojový kód sekce Definice pro grafy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="uvodni-upozorneni hatnote"> Podrobnější informace naleznete v článku <a href="/wiki/Izomorfismus_(graf)" title="Izomorfismus (graf)">Izomorfismus (graf)</a>.</div> <p>V <a href="/wiki/Teorie_graf%C5%AF" title="Teorie grafů">teorii grafů</a> řekneme, že dva grafy jsou izomorfní, pokud <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 \exists \ F\colon V(G)\to V(G'):\{x,y\}\in E(G)\Leftrightarrow \{f(x),f(y)\}\in E(G')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mtext> </mtext> <mi>F</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>:</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo>∈</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>∈</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \ F\colon V(G)\to V(G'):\{x,y\}\in E(G)\Leftrightarrow \{f(x),f(y)\}\in E(G')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c0c1469ad67c3cc8d8e86a51f2714a56e381c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.398ex; height:3.009ex;" alt="{\displaystyle \exists \ F\colon V(G)\to V(G'):\{x,y\}\in E(G)\Leftrightarrow \{f(x),f(y)\}\in E(G')}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Vztah_k_homomorfismům"><span id="Vztah_k_homomorfism.C5.AFm"></span>Vztah k homomorfismům</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=7" title="Editace sekce: Vztah k homomorfismům" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=7" title="Editovat zdrojový kód sekce Vztah k homomorfismům"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>U algebraických struktur jsou izomorfismem právě <a href="/wiki/Bijekce" title="Bijekce">bijektivní</a> <a href="/wiki/Homomorfismus" title="Homomorfismus">homomorfismy</a>. To však neplatí <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>pozn 1<span class="cite-bracket">]</span></a></sup> pro některé jiné struktury, např. relační struktury nebo topologické prostory (v nichž roli homomorfismu plní <a href="/wiki/Spojit%C3%A9_zobrazen%C3%AD" title="Spojité zobrazení">spojitá zobrazení</a>). Obecně však platí, že zobrazení mezi dvěma strukturami je izomorfismem, právě když je bijektivním homomorfismem, jehož <a href="/wiki/Inverzn%C3%AD_zobrazen%C3%AD" title="Inverzní zobrazení">inverzní zobrazení</a> je také homomorfismem. </p> <div class="mw-heading mw-heading2"><h2 id="Příklady"><span id="P.C5.99.C3.ADklady"></span>Příklady</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=8" title="Editace sekce: Příklady" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=8" title="Editovat zdrojový kód sekce Příklady"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Grupa" title="Grupa">Grupa</a> celých čísel s obvyklým sčítáním je izomorfní s množinou všech sudých čísel s obvyklým sčítáním pomocí zobrazení <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(x)=2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a8ebea86ba5d3a71121e0a4156f5ec07b25220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.008ex; height:2.843ex;" alt="{\displaystyle f(x)=2x}"></span>. Celá čísla s operací násobení tvoří <a href="/wiki/Monoid" title="Monoid">monoid</a>, tento monoid však <i>není</i> izomorfní s množinou sudých čísel s obvyklým násobením – například <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(1)\cdot f(1)=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)\cdot f(1)=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b45bf2a2d921dc8e8baf057d32e6180a202ba3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.441ex; height:2.843ex;" alt="{\displaystyle f(1)\cdot f(1)=4}"></span>, ale <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(1\cdot 1)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1\cdot 1)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e00062fd53fc8bc9518aab3855d49fe557fa3dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.353ex; height:2.843ex;" alt="{\displaystyle f(1\cdot 1)=2}"></span>. Pokud bychom ale na sudých číslech zavedli novou operaci <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 \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span> tak, že <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\circ y={\frac {x\cdot y}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∘</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\circ y={\frac {x\cdot y}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fccc12ac2793b1a347d2397758f5fff6603e07cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.779ex; height:4.843ex;" alt="{\displaystyle x\circ y={\frac {x\cdot y}{2}}}"></span>, pak zobrazení <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> již je izomorfismem. Například <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 2\circ 2=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>∘</mo> <mn>2</mn> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\circ 2=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe7749ce878c520bf36676b25bf52aeee0d86e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.781ex; height:2.176ex;" alt="{\displaystyle 2\circ 2=2}"></span> , takže platí <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(1)\cdot f(1)=f(1\cdot 1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)\cdot f(1)=f(1\cdot 1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0347dea489228cab9bade9eed8b68510de0028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.37ex; height:2.843ex;" alt="{\displaystyle f(1)\cdot f(1)=f(1\cdot 1)}"></span> a obecně <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(x)\circ f(y)=f(x\cdot y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\circ f(y)=f(x\cdot y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed53adcd9b2f7b4c8f46558485e1a5752e334db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.206ex; height:2.843ex;" alt="{\displaystyle f(x)\circ f(y)=f(x\cdot y)}"></span>, což je definice izomorfismu.</li></ul> <ul><li>Množina všech <a href="/wiki/P%C5%99irozen%C3%A9_%C4%8D%C3%ADslo" title="Přirozené číslo">přirozených čísel</a> a množina všech <a href="/wiki/Sud%C3%A1_a_lich%C3%A1_%C4%8D%C3%ADsla" title="Sudá a lichá čísla">sudých čísel</a> jsou izomorfní vzhledem k uspořádání podle velikosti podle funkce <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(x)=2.x\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2.</mn> <mi>x</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=2.x\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b33bda2ae8fcfa071554f1808b061a3b7d4c883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:11.504ex; height:2.843ex;" alt="{\displaystyle F(x)=2.x\,\!}"></span>.</li></ul> <ul><li>Pro množiny všech přirozených čísel a všech <a href="/wiki/Cel%C3%A9_%C4%8D%C3%ADslo" title="Celé číslo">celých čísel</a> neexistuje izomorfismus - celá čísla mají prvky menší než 1 (0, -1, -2, ...), zatímco přirozená čísla ne.</li></ul> <ul><li>Algebry zbytkových tříd po dělení sedmi a zbytkových tříd po dělení devíti nejsou izomorfní – to vyplývá z faktu, že nemají stejný počet prvků, takže mezi nimi neexistuje žádné vzájemně jednoznačné zobrazení.</li></ul> <ul><li>Dvouprvková <a href="/wiki/Booleova_algebra" title="Booleova algebra">Booleova algebra</a> s běžnými logickými operacemi <a href="/wiki/Konjunkce_(matematika)" class="mw-redirect" title="Konjunkce (matematika)">konjunkce</a>, <a href="/wiki/Disjunkce" title="Disjunkce">disjunkce</a> a <a href="/wiki/Negace" title="Negace">negace</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 \land ,\vee ,\neg \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land ,\vee ,\neg \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c414732c664b706e02fac2ce2bccd514f2aba07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:7.106ex; height:2.343ex;" alt="{\displaystyle \land ,\vee ,\neg \,\!}"></span> je izomorfní s dvouprvkovou množinou <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=\{\emptyset ,\{\emptyset \}\}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∅</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∅</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\{\emptyset ,\{\emptyset \}\}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b90ea377eaea3c81b05c3d6fac635091727ed8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:13.474ex; height:2.843ex;" alt="{\displaystyle X=\{\emptyset ,\{\emptyset \}\}\,\!}"></span> s množinovými operacemi <a href="/wiki/Sjednocen%C3%AD" title="Sjednocení">sjednocení</a>, <a href="/wiki/Pr%C5%AFnik" title="Průnik">průnik</a> a <a href="/wiki/Dopln%C4%9Bk_mno%C5%BEiny" title="Doplněk množiny">doplňku</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 \cap ,\cup ,neg(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∩</mo> <mo>,</mo> <mo>∪</mo> <mo>,</mo> <mi>n</mi> <mi>e</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cap ,\cup ,neg(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b5799d6ca8381ed963c8db003acc17c86b7e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.802ex; height:2.843ex;" alt="{\displaystyle \cap ,\cup ,neg(a)}"></span>, kde <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 neg(a)=X-a\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>e</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> <mo>−</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle neg(a)=X-a\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1ccbe25153e749474aa84f6c5da786465b716c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:16.169ex; height:2.843ex;" alt="{\displaystyle neg(a)=X-a\,\!}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="V_teorii_kategorií"><span id="V_teorii_kategori.C3.AD"></span>V teorii kategorií</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=9" title="Editace sekce: V teorii kategorií" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=9" title="Editovat zdrojový kód sekce V teorii kategorií"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pojem izomorfismus lze definovat i prostředky <a href="/wiki/Teorie_kategori%C3%AD" title="Teorie kategorií">teorii kategorií</a>, tj. pomocí objektů a morfismů. Objekty <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> jsou izomorfní, pokud mezi nimi existuje izomorfismus, tj. morfismus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20040a52d9391f2fe271f0aaa300bf7887a0c7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\to B}"></span>, k němuž existuje morfismus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:B\to A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:B\to A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83702ca79adb23095c7e6e470827cc2fc749d979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.174ex; height:2.509ex;" alt="{\displaystyle g:B\to A}"></span> takový, že platí <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=1_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f=1_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558458e919f6d3617e6cac40bb2fd05467e6f52f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle g\circ f=1_{A}}"></span> 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 f\circ g=1_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g=1_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c4b7c0ed4ef56cfda70e22f9edabde106c118c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.33ex; height:2.509ex;" alt="{\displaystyle f\circ g=1_{B}}"></span>. </p><p>V <a href="/wiki/Konkr%C3%A9tn%C3%AD_kategorie" title="Konkrétní kategorie">obvyklých kategoriích</a> tato definice splývá s výše uvedenou, protože morfismem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> je <a href="/wiki/Inverzn%C3%AD_zobrazen%C3%AD" title="Inverzní zobrazení">inverzní zobrazení</a> k <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> a jejich složení je <a href="/wiki/Identita_(matematika)" title="Identita (matematika)">identickým zobrazením</a> na <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, resp. <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Odkazy">Odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=10" title="Editace sekce: Odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=10" title="Editovat zdrojový kód sekce Odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Poznámky"><span id="Pozn.C3.A1mky"></span>Poznámky</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=11" title="Editace sekce: Poznámky" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=11" title="Editovat zdrojový kód sekce Poznámky"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Protipříkladem je např. <a href="/wiki/Bijekce" title="Bijekce">bijekce</a> mezi <a href="/wiki/Neorientovan%C3%BD_graf" title="Neorientovaný graf">neorientovanými grafy</a> se dvěma vrcholy, z nichž první (vzor) nemá žádné hrany, ale druhý má jednu. To je bijektivní homomorfismus, ale ne izomorfismus.</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=12" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=12" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Relace_(matematika)" title="Relace (matematika)">Relace (matematika)</a></li> <li><a href="/wiki/Bin%C3%A1rn%C3%AD_relace" title="Binární relace">Binární relace</a></li> <li><a href="/w/index.php?title=Izomorfn%C3%AD_vno%C5%99en%C3%AD&action=edit&redlink=1" class="new" title="Izomorfní vnoření (stránka neexistuje)">Izomorfní vnoření</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Izomorfismus&veaction=edit&section=13" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Izomorfismus&action=edit&section=13" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="sisterproject sisterproject-wiktionary"><span class="sisterproject_image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/16px-Wiktionary-logo-cs.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/24px-Wiktionary-logo-cs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/32px-Wiktionary-logo-cs.svg.png 2x" data-file-width="411" data-file-height="411" /></span></span></span> <span class="sisterproject_text"><span class="sisterproject_text_prefix">Slovníkové heslo </span><span class="sisterproject_text_target"><a href="https://cs.wiktionary.org/wiki/izomorfismus" class="extiw" title="wikt:izomorfismus">izomorfismus</a></span><span class="sisterproject_text_suffix"> ve Wikislovníku</span></span></span></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Isomorphism.html">Izomorfismus</a> v encyklopedii <a href="/wiki/MathWorld" 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