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Saadaval 59 keeles" > <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 keelt</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 – araabia" lang="ar" hreflang="ar" data-title="تماكل" data-language-autonym="العربية" data-language-local-name="araabia" 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 – astuuria" lang="ast" hreflang="ast" data-title="Isomorfismu" data-language-autonym="Asturianu" data-language-local-name="astuuria" 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 – aserbaidžaani" lang="az" hreflang="az" data-title="İzomorfluq" data-language-autonym="Azərbaycanca" data-language-local-name="aserbaidžaani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – indoneesia" lang="id" hreflang="id" data-title="Isomorfisme" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneesia" class="interlanguage-link-target"><span>Bahasa Indonesia</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 – baškiiri" lang="ba" hreflang="ba" data-title="Изоморфизм" data-language-autonym="Башҡортса" data-language-local-name="baškiiri" 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 – valgevene" lang="be" hreflang="be" data-title="Ізамарфізм" data-language-autonym="Беларуская" data-language-local-name="valgevene" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – bosnia" lang="bs" hreflang="bs" data-title="Izomorfizam" data-language-autonym="Bosanski" data-language-local-name="bosnia" class="interlanguage-link-target"><span>Bosanski</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 – bulgaaria" lang="bg" hreflang="bg" data-title="Изоморфизъм" data-language-autonym="Български" data-language-local-name="bulgaaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – katalaani" lang="ca" hreflang="ca" data-title="Isomorfisme" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Izomorfismus" title="Izomorfismus – tšehhi" lang="cs" hreflang="cs" data-title="Izomorfismus" data-language-autonym="Čeština" data-language-local-name="tšehhi" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Isomorffedd" title="Isomorffedd – kõmri" lang="cy" hreflang="cy" data-title="Isomorffedd" data-language-autonym="Cymraeg" data-language-local-name="kõmri" 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 – taani" lang="da" hreflang="da" data-title="Isomorfi" data-language-autonym="Dansk" data-language-local-name="taani" 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 – saksa" lang="de" hreflang="de" data-title="Isomorphismus" data-language-autonym="Deutsch" data-language-local-name="saksa" 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 – kreeka" lang="el" hreflang="el" data-title="Ισομορφισμός" data-language-autonym="Ελληνικά" data-language-local-name="kreeka" 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 – inglise" lang="en" hreflang="en" data-title="Isomorphism" data-language-autonym="English" data-language-local-name="inglise" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – hispaania" lang="es" hreflang="es" data-title="Isomorfismo" data-language-autonym="Español" data-language-local-name="hispaania" class="interlanguage-link-target"><span>Español</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-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – baski" lang="eu" hreflang="eu" data-title="Isomorfismo" data-language-autonym="Euskara" data-language-local-name="baski" 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 – pärsia" lang="fa" hreflang="fa" data-title="یکریختی" data-language-autonym="فارسی" data-language-local-name="pärsia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Isomorphisme" title="Isomorphisme – prantsuse" lang="fr" hreflang="fr" data-title="Isomorphisme" data-language-autonym="Français" data-language-local-name="prantsuse" 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 – iiri" lang="ga" hreflang="ga" data-title="Iseamorfacht" data-language-autonym="Gaeilge" data-language-local-name="iiri" 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 – galeegi" lang="gl" hreflang="gl" data-title="Isomorfismo" data-language-autonym="Galego" data-language-local-name="galeegi" class="interlanguage-link-target"><span>Galego</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 – korea" lang="ko" hreflang="ko" data-title="동형 사상" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</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) – armeenia" lang="hy" hreflang="hy" data-title="Իզոմորֆություն (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="armeenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – horvaadi" lang="hr" hreflang="hr" data-title="Izomorfizam" data-language-autonym="Hrvatski" data-language-local-name="horvaadi" class="interlanguage-link-target"><span>Hrvatski</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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – itaalia" lang="it" hreflang="it" data-title="Isomorfismo" data-language-autonym="Italiano" data-language-local-name="itaalia" class="interlanguage-link-target"><span>Italiano</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 – heebrea" lang="he" hreflang="he" data-title="איזומורפיזם" data-language-autonym="עברית" data-language-local-name="heebrea" class="interlanguage-link-target"><span>עברית</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) – kasahhi" lang="kk" hreflang="kk" data-title="Изоморфизм (Математика)" data-language-autonym="Қазақша" data-language-local-name="kasahhi" class="interlanguage-link-target"><span>Қазақша</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 – kirgiisi" lang="ky" hreflang="ky" data-title="Изоморфизм" data-language-autonym="Кыргызча" data-language-local-name="kirgiisi" 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 – ladina" lang="la" hreflang="la" data-title="Isomorphismus" data-language-autonym="Latina" data-language-local-name="ladina" 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 – leedu" lang="lt" hreflang="lt" data-title="Izomorfizmas" data-language-autonym="Lietuvių" data-language-local-name="leedu" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Izomorfia" title="Izomorfia – ungari" lang="hu" hreflang="hu" data-title="Izomorfia" data-language-autonym="Magyar" data-language-local-name="ungari" class="interlanguage-link-target"><span>Magyar</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 – mongoli" lang="mn" hreflang="mn" data-title="Изоморф" data-language-autonym="Монгол" data-language-local-name="mongoli" 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 – hollandi" lang="nl" hreflang="nl" data-title="Isomorfisme" data-language-autonym="Nederlands" data-language-local-name="hollandi" class="interlanguage-link-target"><span>Nederlands</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 – jaapani" lang="ja" hreflang="ja" data-title="同型写像" data-language-autonym="日本語" data-language-local-name="jaapani" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – norra bokmål" lang="nb" hreflang="nb" data-title="Isomorfisme" data-language-autonym="Norsk bokmål" data-language-local-name="norra bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Isomorfi" title="Isomorfi – uusnorra" lang="nn" hreflang="nn" data-title="Isomorfi" data-language-autonym="Norsk nynorsk" data-language-local-name="uusnorra" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Izomorfizm_(matematika)" title="Izomorfizm (matematika) – usbeki" lang="uz" hreflang="uz" data-title="Izomorfizm (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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 – pandžabi" lang="pa" hreflang="pa" data-title="ਆਇਸੋਮੌਰਫਿਜ਼ਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Isomorfism" title="Isomorfism – piemonte" lang="pms" hreflang="pms" data-title="Isomorfism" data-language-autonym="Piemontèis" data-language-local-name="piemonte" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Izomorfizm" title="Izomorfizm – poola" lang="pl" hreflang="pl" data-title="Izomorfizm" data-language-autonym="Polski" data-language-local-name="poola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – portugali" lang="pt" hreflang="pt" data-title="Isomorfismo" data-language-autonym="Português" data-language-local-name="portugali" 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 – rumeenia" lang="ro" hreflang="ro" data-title="Izomorfism" data-language-autonym="Română" data-language-local-name="rumeenia" 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 – vene" lang="ru" hreflang="ru" data-title="Изоморфизм" data-language-autonym="Русский" data-language-local-name="vene" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Isomorphism" title="Isomorphism – lihtsustatud inglise" lang="en-simple" hreflang="en-simple" data-title="Isomorphism" data-language-autonym="Simple English" data-language-local-name="lihtsustatud inglise" 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 – sloveeni" lang="sl" hreflang="sl" data-title="Izomorfizem" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" 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) – serbia" lang="sr" hreflang="sr" data-title="Изоморфизам (математика)" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – serbia-horvaadi" lang="sh" hreflang="sh" data-title="Izomorfizam" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbia-horvaadi" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Isomorfismi" title="Isomorfismi – soome" lang="fi" hreflang="fi" data-title="Isomorfismi" data-language-autonym="Suomi" data-language-local-name="soome" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Isomorfi" title="Isomorfi – rootsi" lang="sv" hreflang="sv" data-title="Isomorfi" data-language-autonym="Svenska" data-language-local-name="rootsi" 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 – tamili" lang="ta" hreflang="ta" data-title="ஐசோமார்பிஸம்" data-language-autonym="தமிழ்" data-language-local-name="tamili" class="interlanguage-link-target"><span>தமிழ்</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 – vietnami" lang="vi" hreflang="vi" data-title="Phép đẳng cấu" data-language-autonym="Tiếng Việt" data-language-local-name="vietnami" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0zomorfizma" title="İzomorfizma – türgi" lang="tr" hreflang="tr" data-title="İzomorfizma" data-language-autonym="Türkçe" data-language-local-name="türgi" 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 – ukraina" lang="uk" hreflang="uk" data-title="Ізоморфізм" data-language-autonym="Українська" data-language-local-name="ukraina" 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 – urdu" lang="ur" hreflang="ur" data-title="مشاکلت" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</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 – uu" lang="wuu" hreflang="wuu" data-title="同构" data-language-autonym="吴语" data-language-local-name="uu" class="interlanguage-link-target"><span>吴语</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 – kantoni" lang="yue" hreflang="yue" data-title="同構" data-language-autonym="粵語" data-language-local-name="kantoni" class="interlanguage-link-target"><span>粵語</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 – hiina" lang="zh" hreflang="zh" data-title="同构" data-language-autonym="中文" data-language-local-name="hiina" class="interlanguage-link-target"><span>中文</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="Muuda keelelinke" class="wbc-editpage">Muuda linke</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="Nimeruumid"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Isomorfism" 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<b>vajab <a href="/wiki/Vikipeedia:Artiklite_toimetamine" title="Vikipeedia:Artiklite toimetamine">toimetamist</a></b>.<span class="hide-when-compact"></span><span class="hide-when-compact"> <br /><small>Palun aita <a class="external text" href="https://et.wikipedia.org/w/index.php?title=Isomorfism&action=edit">artiklit toimetada</a>.</small> <small>(<a href="/wiki/Juhend:Toimetusm%C3%A4rkuste_eemaldamine" title="Juhend:Toimetusmärkuste eemaldamine">Kuidas ja millal see märkus eemaldada?</a>)</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <table> <tbody><tr> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/25px-Disambig_gray.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/38px-Disambig_gray.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/50px-Disambig_gray.svg.png 2x" data-file-width="220" data-file-height="168" /></span></span> </td> <td> <i>See artikkel räägib filosoofia ja matemaatika mõistest; mineraloogia mõiste kohta vaata artiklit <a href="/w/index.php?title=Isomorfism_(mineraloogia)&action=edit&redlink=1" class="new" title="Isomorfism (mineraloogia) (pole veel kirjutatud)">Isomorfism (mineraloogia)</a></i> </td></tr></tbody></table> <hr /> <p><b>Isomorfism</b> (<a href="/wiki/Kreeka_keel" title="Kreeka keel">kreeka</a>: <a href="https://et.wiktionary.org/wiki/%E1%BC%B4%CF%83%CE%BF%CF%82" class="extiw" title="wikt:ἴσος">ἴσος</a> <i>isos</i> – ühesugune, ja <a href="https://et.wiktionary.org/wiki/%CE%BC%CE%BF%CF%81%CF%86%CE%AE" class="extiw" title="wikt:μορφή">μορφή</a> <i>morphe</i> – vorm) moodustavad koos <a href="/wiki/Homomorfism" title="Homomorfism">homomorfismiga</a> üldmõiste (sh ka <a href="/w/index.php?title=Filosoofiline_kategooria&action=edit&redlink=1" class="new" title="Filosoofiline kategooria (pole veel kirjutatud)">filosoofilise kategooria</a>), mis iseloomustab vastavust objektide struktuuride vahel <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. </p><p>Mõned spetsiifilise suunitlusega filosoofilised koolkonnad võivad mitte tunnistada nende mõistete kuulumist kategooriate kilda. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Selgitus">Selgitus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfism&veaction=edit&section=1" title="Muuda alaosa "Selgitus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorfism&action=edit&section=1" title="Muuda alaosa "Selgitus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Teadaolevalt võttis isomorfismi termini kasutusele 1857. aastal A. Cayley oma keemiliste isomeeride alastes uuringutes <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>. Isomorfism tähendab vastavust, kus kaks süsteemi, vaadelduna lahus neid moodustavate elementide loomusest, vastab ühe süsteemi igale elemendile ainult üks teise süsteemi element ning ühe süsteemi igale seosele vastab ainult üks seos teises – ja vastupidi. Seega saab isomorfismist rääkida vaid niisuguste objektide puhul, millel on struktuur, st on määratletud selle elemendid (komponendid, osised) ja nendevahelised seosed (suhted). </p><p>Isomorfism on määratletav kui <i>struktuuri säilitav üksühene vastavus objektide vahel</i>. Isomorfsete objektide hulk moodustab '<i>isomorfismiklassi</i>. Kõige piltlikum näide isomorfismist on graafide isomorfism. </p><p>Kaks graafi on isomorfsed, st omavad ühesugust struktuuri, vaatamata nende erinevale "välimusele". </p> <table class="wikitable" style="margin: 1em auto 1em auto"> <tbody><tr> <th>Graaf G </th> <th>Graaf H </th> <th>Isomorfism<br />G ja H vahel </th></tr> <tr> <td style="padding-left:2em;padding-right:2em;"><span typeof="mw:File"><a href="/wiki/Fail:Graph_isomorphism_a.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Graph_isomorphism_a.svg/100px-Graph_isomorphism_a.svg.png" decoding="async" width="100" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Graph_isomorphism_a.svg/150px-Graph_isomorphism_a.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Graph_isomorphism_a.svg/200px-Graph_isomorphism_a.svg.png 2x" data-file-width="200" data-file-height="400" /></a></span> </td> <td style="padding-left:1em;padding-right:1em;"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fail:Graph_isomorphism_b.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Graph_isomorphism_b.svg/400px-Graph_isomorphism_b.svg.png" decoding="async" width="400" height="400" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Graph_isomorphism_b.svg/600px-Graph_isomorphism_b.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Graph_isomorphism_b.svg/800px-Graph_isomorphism_b.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span> </td> <td align="center" style="background-color:white;">ƒ(<i>a</i>) = 1 <p>ƒ(<i>b</i>) = 6 </p><p>ƒ(<i>c</i>) = 8 </p><p>ƒ(<i>d</i>) = 3 </p><p>ƒ(<i>g</i>) = 5 </p><p>ƒ(<i>h</i>) = 2 </p><p>ƒ(<i>i</i>) = 4 </p><p>ƒ(<i>j</i>) = 7 </p> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Isomorfism_matemaatikas">Isomorfism matemaatikas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfism&veaction=edit&section=2" title="Muuda alaosa "Isomorfism matemaatikas"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorfism&action=edit&section=2" title="Muuda alaosa "Isomorfism matemaatikas" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Matemaatikas defineeritakse isomorfismi kui süsteemi niisugust <a href="/wiki/%C3%9Cks%C3%BChene_kujutus" class="mw-redirect" title="Üksühene kujutus">üksühest kujutust</a> sama tüüpi süsteemiks, mille korral säilib süsteemide <a href="/wiki/Struktuur" title="Struktuur">struktuur</a>. Näiteks, kujund ja selle kujundi matemaatiline avaldis <sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>. Isomorfism on pööratav <a href="/w/index.php?title=Morfism&action=edit&redlink=1" class="new" title="Morfism (pole veel kirjutatud)">morfism</a>, millel on <i>pöördmorfism</i>, kus nende korrutis on <i>ühikmorfism</i>. <a href="/wiki/Topoloogia" title="Topoloogia">Topoloogilist</a> isomorfismi nimetatakse <i>homoömorfismiks</i>. </p><p>Isomorfismiprobleem on aktuaalne algebras, kategooria- ja <a href="/wiki/Graafiteooria" title="Graafiteooria">graafiteoorias</a>. <a href="/wiki/Algebra" title="Algebra">Algebras</a> on isomorfism kujutus objektide vahel, selline mis näitab suhet kahe omaduse või operatsiooni vahel.  Kui kahe struktuuri vahel esineb isomorfism, siis öeldakse, et vastavad objektid on <i>isomorfsed</i>.  Isomorfsed objektid on <i>struktuurselt ekvivalentsed</i>, kuid võivad mõnest muus aspektis ka erineda. Teisisõnu, isomorfism on <i>bijektiivne kujutus</i> <i>f</i> niisugune, et <i>f</i> ja selle pöördfunktsioon <i>f</i><sup> −1</sup> on struktuuri säilitavad kujutused kahe algebralise struktuuri vahel, st need mõlemad on <a href="/wiki/Homomorfism" title="Homomorfism">homomorfsed</a>. Isomorfism on algebras samalaadselt defineeritud ka <a href="/wiki/R%C3%BChm_(matemaatika)" title="Rühm (matemaatika)">rühma</a>, <a href="/wiki/Ring" title="Ring">ringi</a> ja teiste struktuuride kohta. </p><p>Isomorfism <a href="/wiki/Graafiteooria" title="Graafiteooria">graafiteoorias</a> tähendab graafide <i>G</i> ja <i>H</i> struktuuri säilitavat tippude bijektsiooni </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon V(G)\to V(H)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mi>H</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon V(G)\to V(H)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f53cc7970d21ac2a09fb0b40b9cd529eb5950a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:17.397ex; height:2.843ex;" alt="{\displaystyle f\colon V(G)\to V(H)\,\!}"></span></dd></dl> <p>niisugust, et kui graafi <i>G</i> mingid kaks tippu <i>u</i> ja <i>v</i> on <i>seotud</i>, siis ja ainult siis on ƒ(<i>u</i>) ja ƒ(<i>v</i>) <i>seotud</i> garaafis <i>H</i>. </p><p>Selle näide on selgituses esitatud. Oluline on siin nende <a href="/wiki/Substitutsioon" title="Substitutsioon">substitutsioonide</a> väljatoomine: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a&b&c&d&g&h&i&j\\1&6&8&3&5&2&4&7\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> <mtd> <mi>g</mi> </mtd> <mtd> <mi>h</mi> </mtd> <mtd> <mi>i</mi> </mtd> <mtd> <mi>j</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a&b&c&d&g&h&i&j\\1&6&8&3&5&2&4&7\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578e42f786b11a0bb62ed9372c46b84263ebba8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.028ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}a&b&c&d&g&h&i&j\\1&6&8&3&5&2&4&7\end{pmatrix}}}"></span> </p><p>Kahe graafi isomorfsust tähistatakse <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\simeq H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≃</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\simeq H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87587df9ddf620b0e064802f01763df4a21e5c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.989ex; height:2.176ex;" alt="{\displaystyle G\simeq H}"></span>. Juhul kui bijektsioon on graafi kujutus iseendasse, st kui <i>G</i> ja <i>H</i> on üks ja sama graaf, siis seda bijektsiooni nimetatakse graafi <i>G</i> <a href="/wiki/Automorfism" title="Automorfism">automorfismiks</a> <i>AutG</i>. </p><p>Graafide isomorfism on <a href="/wiki/Ekvivalentsus" class="mw-disambig" title="Ekvivalentsus">ekvivalentsussuhe</a> ning isomorfsed graafid kujutavad endast ühesugust struktuuri omavaid <i>graafide isomorfismiklasse</i>. </p><p><i>Isomorfismi tuvastamine</i> kujutab endast vastava <a href="/wiki/Algoritm" title="Algoritm">algoritmi</a> konstrueerimist, mida nimetatakse <a href="/wiki/Isomorfismiprobleem" title="Isomorfismiprobleem">isomorfismiprobleemiks</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Isomorfism_eri_valdkondades">Isomorfism eri valdkondades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfism&veaction=edit&section=3" title="Muuda alaosa "Isomorfism eri valdkondades"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorfism&action=edit&section=3" title="Muuda alaosa "Isomorfism eri valdkondades" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Isomorfismi mõistet kasutatakse ka <a href="/wiki/Geoloogia" title="Geoloogia">geoloogias</a>, <a href="/wiki/Bioloogia" title="Bioloogia">bioloogias</a>, <a href="/wiki/F%C3%BC%C3%BCsika" title="Füüsika">füüsikas</a> jm. Korrektne on seda kasutada vaid seal, kus nende spetsiifiliste objektide struktuur ja bijektsioon on määratletav, see tähendab, kui nende geoloogiliste (bioloogiliste, füüsikaliste jt) süsteemide elemendid (komponendid, osised) ja nendevahelised seosed (suhted) on määratletud. Tegelikult ei peeta sellest nõudest alati kinni. </p> <div class="mw-heading mw-heading2"><h2 id="Viited">Viited</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfism&veaction=edit&section=4" title="Muuda alaosa "Viited"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorfism&action=edit&section=4" title="Muuda alaosa "Viited" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <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">Schmitd, Heirich, 1991. <i>Philosophisches Wörerbuch</i>. Stuttgart. <a href="/wiki/Eri:Raamatuotsimine/5250017940" class="internal mw-magiclink-isbn">ISBN 5250017940</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><i>Новая философская энциклопедия</i>. 2001, Москва. <a href="/wiki/Eri:Raamatuotsimine/5244009613" class="internal mw-magiclink-isbn">ISBN 5244009613</a> (00962-1)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">A. Cayley, 1857. <i>On the theory of the analytical forms called trees</i>. Phil. Mag. (4) 13 (1857), 172–176</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Kaasik, Ülo. 2003. <i>Matemaatikaleksikon</i>. Tartu. <a href="/wiki/Eri:Raamatuotsimine/9985941772" class="internal mw-magiclink-isbn">ISBN 9985941772</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Kirjandus">Kirjandus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorfism&veaction=edit&section=5" title="Muuda alaosa "Kirjandus"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Isomorfism&action=edit&section=5" title="Muuda alaosa "Kirjandus" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>McGraw-Hill dictionary of Mathematics, 1997. N. Y., ISBN 007524335.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Pärit leheküljelt "<a dir="ltr" href="https://et.wikipedia.org/w/index.php?title=Isomorfism&oldid=6640599">https://et.wikipedia.org/w/index.php?title=Isomorfism&oldid=6640599</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Eri:Kategooriad" title="Eri:Kategooriad">Kategooriad</a>: <ul><li><a href="/wiki/Kategooria:Filosoofia" title="Kategooria:Filosoofia">Filosoofia</a></li><li><a href="/wiki/Kategooria:Matemaatika" title="Kategooria:Matemaatika">Matemaatika</a></li><li><a href="/wiki/Kategooria:Graafiteooria" title="Kategooria:Graafiteooria">Graafiteooria</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Peidetud kategooriad: <ul><li><a href="/wiki/Kategooria:ISBN-v%C3%B5lulingiga_lehek%C3%BCljed" title="Kategooria:ISBN-võlulingiga leheküljed">ISBN-võlulingiga leheküljed</a></li><li><a href="/wiki/Kategooria:Keeletoimetamist_ootavad_(juuli_2022)" title="Kategooria:Keeletoimetamist ootavad (juuli 2022)">Keeletoimetamist ootavad (juuli 2022)</a></li><li><a href="/wiki/Kategooria:Keeletoimetamist_ootavad" title="Kategooria:Keeletoimetamist ootavad">Keeletoimetamist ootavad</a></li><li><a href="/wiki/Kategooria:Toimetamist_ootavad" title="Kategooria:Toimetamist ootavad">Toimetamist ootavad</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Selle lehekülje viimane muutmine: 19:27, 27. aprill 2024.</li> <li id="footer-info-copyright">Tekst on kasutatav vastavalt Creative Commonsi litsentsile "<a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.et">Autorile viitamine + jagamine samadel tingimustel</a>"; sellele võivad lisanduda täiendavad tingimused. 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