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class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span 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href="https://ca.wikipedia.org/wiki/Endomorfisme" title="Endomorfisme – Catalan" lang="ca" hreflang="ca" data-title="Endomorfisme" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Endomorfi" title="Endomorfi – Danish" lang="da" hreflang="da" data-title="Endomorfi" data-language-autonym="Dansk" data-language-local-name="Danish" 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/Endomorphismus" title="Endomorphismus – German" lang="de" hreflang="de" data-title="Endomorphismus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Endomorfism" title="Endomorfism – Estonian" lang="et" hreflang="et" data-title="Endomorfism" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Endomorfismo" title="Endomorfismo – Spanish" lang="es" hreflang="es" data-title="Endomorfismo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Endomorfismo" title="Endomorfismo – Basque" lang="eu" hreflang="eu" data-title="Endomorfismo" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Endomorphisme" title="Endomorphisme – French" lang="fr" hreflang="fr" data-title="Endomorphisme" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Endomorfismo" title="Endomorfismo – Galician" lang="gl" hreflang="gl" data-title="Endomorfismo" data-language-autonym="Galego" data-language-local-name="Galician" 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/%EC%9E%90%EA%B8%B0_%EC%82%AC%EC%83%81" title="자기 사상 – Korean" lang="ko" hreflang="ko" data-title="자기 사상" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Endomorphismo" title="Endomorphismo – Interlingua" lang="ia" hreflang="ia" data-title="Endomorphismo" 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/Endomorfismo" title="Endomorfismo – Italian" lang="it" hreflang="it" data-title="Endomorfismo" data-language-autonym="Italiano" data-language-local-name="Italian" 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%A0%D7%93%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="אנדומורפיזם – Hebrew" lang="he" hreflang="he" data-title="אנדומורפיזם" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Endomorfisme" title="Endomorfisme – Dutch" lang="nl" hreflang="nl" data-title="Endomorfisme" data-language-autonym="Nederlands" data-language-local-name="Dutch" 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/%E8%87%AA%E5%B7%B1%E6%BA%96%E5%90%8C%E5%9E%8B" title="自己準同型 – Japanese" lang="ja" hreflang="ja" data-title="自己準同型" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%90%E0%A8%82%E0%A8%A1%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="ਐਂਡੋਮੌਰਫਿਜ਼ਮ – Punjabi" lang="pa" hreflang="pa" data-title="ਐਂਡੋਮੌਰਫਿਜ਼ਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" 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/Endomorfizm" title="Endomorfizm – Polish" lang="pl" hreflang="pl" data-title="Endomorfizm" data-language-autonym="Polski" data-language-local-name="Polish" 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/Endomorfismo" title="Endomorfismo – Portuguese" lang="pt" hreflang="pt" data-title="Endomorfismo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BD%D0%B4%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC" title="Эндоморфизм – Russian" lang="ru" hreflang="ru" data-title="Эндоморфизм" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Self-self morphism</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Endomorphic" redirects here. For the Sheldon body type, see <a href="/wiki/Somatotype_and_constitutional_psychology" title="Somatotype and constitutional psychology">Somatotype and constitutional psychology</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-One_source plainlinks metadata ambox ambox-content ambox-one_source" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>relies largely or entirely on a <a href="/wiki/Wikipedia:Articles_with_a_single_source" title="Wikipedia:Articles with a single source">single source</a></b>.<span class="hide-when-compact"> Relevant discussion may be found on the <a href="/wiki/Talk:Endomorphism##" title="Talk:Endomorphism">talk page</a>. Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Endomorphism&action=edit">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">introducing citations to additional sources</a>.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Endomorphism%22">"Endomorphism"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Endomorphism%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Endomorphism%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Endomorphism%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Endomorphism%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Endomorphism%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">March 2024</span>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Orthogonal_projection.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orthogonal_projection.svg/252px-Orthogonal_projection.svg.png" decoding="async" width="252" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orthogonal_projection.svg/378px-Orthogonal_projection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orthogonal_projection.svg/504px-Orthogonal_projection.svg.png 2x" data-file-width="252" data-file-height="252" /></a><figcaption><a href="/wiki/Orthogonal_projection" class="mw-redirect" title="Orthogonal projection">Orthogonal projection</a> onto a line, <span class="texhtml"><i>m</i></span>, is a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> on the plane. This is an example of an endomorphism that is not an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a>.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>endomorphism</b> is a <a href="/wiki/Morphism" title="Morphism">morphism</a> from a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> to itself. An endomorphism that is also an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> is an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a>. For example, an endomorphism of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <span class="texhtml"><i>V</i></span> is a <a href="/wiki/Linear_map" title="Linear map">linear map</a> <span class="texhtml"><i>f</i>: <i>V</i> → <i>V</i></span>, and an endomorphism of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <span class="texhtml"><i>G</i></span> is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> <span class="texhtml"><i>f</i>: <i>G</i> → <i>G</i></span>. In general, we can talk about endomorphisms in any <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. In the <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a>, endomorphisms are <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> from a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>S</i> to itself. </p><p>In any category, the <a href="/wiki/Function_composition" title="Function composition">composition</a> of any two endomorphisms of <span class="texhtml"><i>X</i></span> is again an endomorphism of <span class="texhtml"><i>X</i></span>. It follows that the set of all endomorphisms of <span class="texhtml"><i>X</i></span> forms a <a href="/wiki/Monoid" title="Monoid">monoid</a>, the <a href="/wiki/Full_transformation_monoid" class="mw-redirect" title="Full transformation monoid">full transformation monoid</a>, and denoted <span class="texhtml">End(<i>X</i>)</span> (or <span class="texhtml">End<sub><i>C</i></sub>(<i>X</i>)</span> to emphasize the category <span class="texhtml"><i>C</i></span>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Automorphisms">Automorphisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Endomorphism&action=edit&section=1" title="Edit section: Automorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Automorphism" title="Automorphism">Automorphism</a></div> <p>An <a href="/wiki/Inverse_function" title="Inverse function">invertible</a> endomorphism of <span class="texhtml"><i>X</i></span> is called an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a>. The set of all automorphisms is a <a href="/wiki/Subset" title="Subset">subset</a> of <span class="texhtml">End(<i>X</i>)</span> with a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> structure, called the <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of <span class="texhtml"><i>X</i></span> and denoted <span class="texhtml">Aut(<i>X</i>)</span>. In the following diagram, the arrows denote implication: </p> <table style="border:none;"> <tbody><tr> <td align="center" width="42%"><a href="/wiki/Automorphism" title="Automorphism">Automorphism</a> </td> <td align="center" width="16%">⇒ </td> <td align="center" width="42%"><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a> </td></tr> <tr> <td align="center">⇓ </td> <td> </td> <td align="center">⇓ </td></tr> <tr> <td align="center">Endomorphism </td> <td align="center">⇒ </td> <td align="center"><a href="/wiki/Homomorphism" title="Homomorphism">(Homo)morphism</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Endomorphism_rings">Endomorphism rings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Endomorphism&action=edit&section=2" title="Edit section: Endomorphism rings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">Endomorphism ring</a></div> <p>Any two endomorphisms of an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>, <span class="texhtml"><i>A</i></span>, can be added together by the rule <span class="texhtml">(<i>f</i> + <i>g</i>)(<i>a</i>) = <i>f</i>(<i>a</i>) + <i>g</i>(<i>a</i>)</span>. Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> (the <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a>). For example, the set of endomorphisms of <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 \mathbb {Z} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5de7ced4588982b574fe19894aec6a3ca4c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{n}}"></span> is the ring of all <span class="texhtml"><i>n</i> × <i>n</i></span> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> with <a href="/wiki/Integer" title="Integer">integer</a> entries. The endomorphisms of a vector space or <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> also form a ring, as do the endomorphisms of any object in a <a href="/wiki/Preadditive_category" title="Preadditive category">preadditive category</a>. The endomorphisms of a nonabelian group generate an algebraic structure known as a <a href="/wiki/Near-ring" title="Near-ring">near-ring</a>. Every ring with one is the endomorphism ring of its <a href="/wiki/Regular_module" class="mw-redirect" title="Regular module">regular module</a>, and so is a subring of an endomorphism ring of an abelian group;<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> however there are rings that are not the endomorphism ring of any abelian group. </p> <div class="mw-heading mw-heading2"><h2 id="Operator_theory">Operator theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Endomorphism&action=edit&section=3" title="Edit section: Operator theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In any <a href="/wiki/Concrete_category" title="Concrete category">concrete category</a>, especially for <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, endomorphisms are maps from a set into itself, and may be interpreted as <a href="/wiki/Unary_operator" class="mw-redirect" title="Unary operator">unary operators</a> on that set, <a href="/wiki/Action_(group_theory)" class="mw-redirect" title="Action (group theory)">acting</a> on the elements, and allowing the notion of element <a href="/wiki/Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">orbits</a> to be defined, etc. </p><p>Depending on the additional structure defined for the category at hand (<a href="/wiki/Topology" title="Topology">topology</a>, <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>, ...), such operators can have properties like <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuity</a>, <a href="/wiki/Bounded_function" title="Bounded function">boundedness</a>, and so on. More details should be found in the article about <a href="/wiki/Operator_theory" title="Operator theory">operator theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Endofunctions">Endofunctions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Endomorphism&action=edit&section=4" title="Edit section: Endofunctions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b>endofunction</b> is a function whose <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> is equal to its <a href="/wiki/Codomain" title="Codomain">codomain</a>. A <a href="/wiki/Homomorphism" title="Homomorphism">homomorphic</a> endofunction is an endomorphism. </p><p>Let <span class="texhtml"><i>S</i></span> be an arbitrary set. Among endofunctions on <span class="texhtml"><i>S</i></span> one finds <a href="/wiki/Permutation" title="Permutation">permutations</a> of <span class="texhtml"><i>S</i></span> and constant functions associating to every <span class="texhtml"><i>x</i></span> in <span class="texhtml"><i>S</i></span> the same element <span class="texhtml"><i>c</i></span> in <span class="texhtml"><i>S</i></span>. Every permutation of <span class="texhtml"><i>S</i></span> has the codomain equal to its domain and is <a href="/wiki/Bijection" title="Bijection">bijective</a> and invertible. If <span class="texhtml"><i>S</i></span> has more than one element, a constant function on <span class="texhtml"><i>S</i></span> has an <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> that is a proper subset of its codomain, and thus is not bijective (and hence not invertible). The function associating to each <a href="/wiki/Natural_number" title="Natural number">natural number</a> <span class="texhtml"><i>n</i></span> the floor of <span class="texhtml"><i>n</i>/2</span> has its image equal to its codomain and is not invertible. </p><p>Finite endofunctions are equivalent to <a href="/wiki/Directed_pseudoforest" class="mw-redirect" title="Directed pseudoforest">directed pseudoforests</a>. For sets of size <span class="texhtml"><i>n</i></span> there are <span class="texhtml"><i>n</i><sup><i>n</i></sup></span> endofunctions on the set. </p><p>Particular examples of bijective endofunctions are the <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involutions</a>; i.e., the functions coinciding with their inverses. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Endomorphism&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Adjoint_endomorphism" class="mw-redirect" title="Adjoint endomorphism">Adjoint endomorphism</a></li> <li><a href="/wiki/Epimorphism" title="Epimorphism">Epimorphism</a> (surjective homomorphism)</li> <li><a href="/wiki/Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></li> <li><a href="/wiki/Monomorphism" title="Monomorphism">Monomorphism</a> (injective homomorphism)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Endomorphism&action=edit&section=6" title="Edit section: Notes"><span>edit</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"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Jacobson (2009), p. 162, Theorem 3.2.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Endomorphism&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJacobson2009" class="citation cs2"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (2009), <i>Basic algebra</i>, vol. 1 (2nd ed.), Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47189-1" title="Special:BookSources/978-0-486-47189-1"><bdi>978-0-486-47189-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra&rft.edition=2nd&rft.pub=Dover&rft.date=2009&rft.isbn=978-0-486-47189-1&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEndomorphism" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Endomorphism&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span 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