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Endomorphism - Revision history
2024-03-28T15:51:15Z
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en>Omnipaedista: per WP:APPENDIX
2014-12-22T00:48:15Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:48, 22 December 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Let me initial begin by introducing myself</del>. <del style="font-weight: bold; text-decoration: none;">My name </del>is <del style="font-weight: bold; text-decoration: none;">Boyd Butts even though </del>it <del style="font-weight: bold; text-decoration: none;">is not </del>the <del style="font-weight: bold; text-decoration: none;">name on my beginning certificate</del>. <del style="font-weight: bold; text-decoration: none;">The thing she adores most </del>is <del style="font-weight: bold; text-decoration: none;">body building </del>and <del style="font-weight: bold; text-decoration: none;">now she is attempting </del>to <del style="font-weight: bold; text-decoration: none;">earn cash with it</del>. <del style="font-weight: bold; text-decoration: none;">He utilized </del>to be <del style="font-weight: bold; text-decoration: none;">unemployed but now he </del>is <del style="font-weight: bold; text-decoration: none;">a meter reader</del>. <del style="font-weight: bold; text-decoration: none;">Puerto Rico </del>is <del style="font-weight: bold; text-decoration: none;">where he and his wife live</del>.<br><br><del style="font-weight: bold; text-decoration: none;">My web site: </del>[http://<del style="font-weight: bold; text-decoration: none;">social</del>.<del style="font-weight: bold; text-decoration: none;">tradingkakis</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/<del style="font-weight: bold; text-decoration: none;">profile/cemusselma std testing at home</del>]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If you are interested in getting a [http://answers.yahoo.com/search/search_result?p=high-quality&submit-go=Search+Y!+Answers high-quality] product, then you should definitely consider a GHD straightener. There are a number of different sites offering GHD hair straightening products. The GHD straightener has been designed considering the latest in hair care research and development, providing you with smooth and silky hair</ins>. <ins style="font-weight: bold; text-decoration: none;">It does not damage your hair, as no excessive heat </ins>is <ins style="font-weight: bold; text-decoration: none;">involved. On top of that, the GHD straightener comes in a variety of different designs, from trendy to classic and chic, fitting to any woman's personality and lifestyle.<br><br>Because of its growing popularity, </ins>it <ins style="font-weight: bold; text-decoration: none;">can be quite a challenge to make sure that you are availing the original products. To make sure [http://www.aireys.co.nz/cheapghd.html ghd nz] that you are getting your money's worth, here are just some of </ins>the <ins style="font-weight: bold; text-decoration: none;">best places to find GHD hair straighteners</ins>.<ins style="font-weight: bold; text-decoration: none;"><br><br>Health and Beauty Stores<br>One of the best places to find GHD straightener </ins>is <ins style="font-weight: bold; text-decoration: none;">your local health </ins>and <ins style="font-weight: bold; text-decoration: none;">beauty stores. Here, you would be able </ins>to <ins style="font-weight: bold; text-decoration: none;">find a wide variety of different straighteners from GHD which would fit your lifestyle and needs</ins>. <ins style="font-weight: bold; text-decoration: none;">There are a number of store assistants who would be more than happy </ins>to <ins style="font-weight: bold; text-decoration: none;">help you select the right model of hair straightener to suite your needs. On top of that, some beauty stores would </ins>be <ins style="font-weight: bold; text-decoration: none;">willing to demonstrate the product for you to allow you to first try it out before making the purchase.<br><br>The GHD Website<br>Another place where you can find original GHD hair straighteners </ins>is <ins style="font-weight: bold; text-decoration: none;">of course the GHD official website. Apart from allowing you to purchase hair straighteners, you can also choose from their wide range of other products. They also provide you with tips and product registration for warranty and product exchange and return purposes</ins>. <ins style="font-weight: bold; text-decoration: none;">The downside of purchasing products from the official website </ins>is <ins style="font-weight: bold; text-decoration: none;">that the rates are currently quoted in British Pounds, which can be rather difficult if you are not residing in the United Kingdom</ins>.<br><br><ins style="font-weight: bold; text-decoration: none;">Third Party Website<br>The Internet is now the one-stop shop where you can buy almost anything and everything no matter where you are located. If you are living outside of the United Kingdom, you can still purchase GHD hair straighteners through any of the numerous third-party websites offering a wide variety </ins>[http://<ins style="font-weight: bold; text-decoration: none;">www</ins>.<ins style="font-weight: bold; text-decoration: none;">aireys</ins>.<ins style="font-weight: bold; text-decoration: none;">co.nz</ins>/<ins style="font-weight: bold; text-decoration: none;">cheapghd.html ghd nz</ins>] <ins style="font-weight: bold; text-decoration: none;">of different hair styling products. Unfortunately, there are some websites that have been flagged as scam websites. As such, it is important to make sure that you purchase products that are reliable and benefit from secure transactions. There are many websites that offer great GHD hair products. But only some will offer a list of genuine GHD straightener and other accessories to choose from. 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en>Omnipaedista
https://en.formulasearchengine.com/index.php?title=Endomorphism&diff=219358&oldid=prev
152.3.194.201: added dash in nearring
2014-03-04T15:55:12Z
<p>added dash in nearring</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:55, 4 March 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{About|the mathematical concept|the '''endomorphic''' body type|Somatotype}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Let me initial begin by introducing myself</ins>. <ins style="font-weight: bold; text-decoration: none;">My name </ins>is <ins style="font-weight: bold; text-decoration: none;">Boyd Butts even though it </ins>is <ins style="font-weight: bold; text-decoration: none;">not </ins>the <ins style="font-weight: bold; text-decoration: none;">name on my beginning certificate</ins>. The <ins style="font-weight: bold; text-decoration: none;">thing she adores most </ins>is <ins style="font-weight: bold; text-decoration: none;">body building </ins>and <ins style="font-weight: bold; text-decoration: none;">now she </ins>is <ins style="font-weight: bold; text-decoration: none;">attempting to earn cash </ins>with <ins style="font-weight: bold; text-decoration: none;">it</ins>. <ins style="font-weight: bold; text-decoration: none;">He utilized </ins>to be <ins style="font-weight: bold; text-decoration: none;">unemployed but now he </ins>is a <ins style="font-weight: bold; text-decoration: none;">meter reader</ins>. <ins style="font-weight: bold; text-decoration: none;">Puerto Rico </ins>is <ins style="font-weight: bold; text-decoration: none;">where he and his wife live</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">My web site: </ins>[<ins style="font-weight: bold; text-decoration: none;">http:</ins>//<ins style="font-weight: bold; text-decoration: none;">social</ins>.<ins style="font-weight: bold; text-decoration: none;">tradingkakis</ins>.<ins style="font-weight: bold; text-decoration: none;">com</ins>/<ins style="font-weight: bold; text-decoration: none;">profile</ins>/<ins style="font-weight: bold; text-decoration: none;">cemusselma std testing at home</ins>]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In [[mathematics]], an '''endomorphism''' is a [[morphism]] (or [[homomorphism]]) from a [[mathematical object]] to itself</del>. <del style="font-weight: bold; text-decoration: none;"> For example, an endomorphism of a [[vector space]] ''V'' </del>is <del style="font-weight: bold; text-decoration: none;">a [[linear map]] ƒ:&nbsp;''V''&nbsp;→&nbsp;''V'', and an endomorphism of a [[group (mathematics)|group]] ''G'' </del>is <del style="font-weight: bold; text-decoration: none;">a [[group homomorphism]] ƒ:&nbsp;''G''&nbsp;→&nbsp;''G''. In general, we can talk about endomorphisms in any [[category theory|category]]. In </del>the <del style="font-weight: bold; text-decoration: none;">category of [[Set (mathematics)|sets]], endomorphisms are simply functions from a set ''S'' into itself.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In any category, the [[function composition|composition]] of any two endomorphisms of ''X'' is again an endomorphism of ''X''. It follows that the set of all endomorphisms of ''X'' forms a [[monoid]], denoted End(''X'') (or End<sub>''C''</sub>(''X'') to emphasize the category ''C'').</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">An [[inverse element|invertible]] endomorphism of ''X'' is called an [[automorphism]]</del>. <del style="font-weight: bold; text-decoration: none;"> </del>The <del style="font-weight: bold; text-decoration: none;">set of all automorphisms </del>is <del style="font-weight: bold; text-decoration: none;">a [[subset]] of End(''X'') with a [[group (mathematics)|group]] structure, called the [[automorphism group]] of ''X'' </del>and <del style="font-weight: bold; text-decoration: none;">denoted Aut(''X''). In the following diagram, the arrows denote implication:</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{| border="0"</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| align="center" width="42%" | [[automorphism]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| align="center" width="16%" | <math>\Rightarrow</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| align="center" width="42%" | [[isomorphism]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|-</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| align="center" | <math>\Downarrow</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| align="center" | <math>\Downarrow</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|-</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| align="center" | endomorphism</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| align="center" | <math>\Rightarrow</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| align="center" | [[homomorphism|(homo)morphism]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Any two endomorphisms of an [[abelian group]] ''A'' can be added together by the rule (ƒ&nbsp;+&nbsp;''g'')(''a'')&nbsp;=&nbsp;ƒ(''a'')&nbsp;+&nbsp;''g''(''a''). Under this addition, the endomorphisms of an abelian group form a [[ring (mathematics)|ring]] (the [[endomorphism ring]]). For example, the set of endomorphisms of '''Z'''<sup>''n''</sup> </del>is <del style="font-weight: bold; text-decoration: none;">the ring of all ''n''&nbsp;×&nbsp;''n'' matrices with integer entries. The endomorphisms of a vector space or [[module (mathematics)|module]] also form a ring, as do the endomorphisms of any object in a [[preadditive category]]. The endomorphisms of a nonabelian group generate an algebraic structure known as a [[nearring]]. Every ring </del>with <del style="font-weight: bold; text-decoration: none;">one is the endomorphism ring of its [[regular module]], and so is a subring of an endomorphism ring of an abelian group,<ref>Jacobson (2009), p</del>. <del style="font-weight: bold; text-decoration: none;">162, Theorem 3.2.</ref> however there are rings which are not the endomorphism ring of any abelian group.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Operator theory==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In any [[concrete category]], especially for [[vector space]]s, endomorphisms are maps from a set into itself, and may be interpreted as [[unary operator]]s on that set, [[action (group theory)|acting]] on the elements, and allowing </del>to <del style="font-weight: bold; text-decoration: none;">define the notion of [[orbit (group theory)|orbit]]s of elements, etc.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Depending on the additional structure defined for the category at hand ([[topology]], [[metric (mathematics)|metric]], ...), such operators can have properties like [[continuous function (topology)|continuity]], [[boundedness]], and so on. More details should </del>be <del style="font-weight: bold; text-decoration: none;">found in the article about [[operator theory]].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Endofunctions in mathematics==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In [[mathematics]], an '''endofunction''' is a [[function (mathematics)|function]] whose [[codomain]] </del>is <del style="font-weight: bold; text-decoration: none;">equal to its [[domain of </del>a <del style="font-weight: bold; text-decoration: none;">function|domain]]</del>. <del style="font-weight: bold; text-decoration: none;">A [[homomorphism|homomorphic]] endofunction </del>is <del style="font-weight: bold; text-decoration: none;">an endomorphism</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Let ''S'' be an arbitrary set. Among endofunctions on ''S'' one finds [[permutation]]s of ''S'' and constant functions associating to each </del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">x\in S</del><<del style="font-weight: bold; text-decoration: none;">/math</del>> <del style="font-weight: bold; text-decoration: none;">a given <math>c\in S</math>. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Every permutation of ''S'' has the codomain equal to its domain and is </del>[<del style="font-weight: bold; text-decoration: none;">[bijection|bijective]] and invertible. A constant function on ''S'', if ''S'' has more than 1 element, has a codomain that is a proper subset of its domain, is not bijective (and non invertible). The function associating to each natural integer ''n'' the floor of ''n''</del>/<del style="font-weight: bold; text-decoration: none;">2 has its codomain equal to its domain and is not invertible.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Finite endofunctions are equivalent to monogeneous digraphs, i.e. digraphs having all nodes with outdegree equal to 1, and can be easily described. For sets of size ''n'', there are ''n''<sup>''n''<</del>/<del style="font-weight: bold; text-decoration: none;">sup> endofunctions on the set</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Particular bijective endofunctions are the [[Involution (mathematics)|involution]]s, i</del>.<del style="font-weight: bold; text-decoration: none;">e. the functions coinciding with their inverses.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Notes==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><references </del>/<del style="font-weight: bold; text-decoration: none;">></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==See also==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Adjoint endomorphism]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Frobenius endomorphism]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==References==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | series= | publisher=Dover| isbn = 978-0-486-47189-1}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==External links==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* {{springer|title=Endomorphism|id=p</del>/<del style="font-weight: bold; text-decoration: none;">e035600}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*{{planetmath reference|id=7462|title=Endomorphism}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Morphisms]</del>]</div></td><td colspan="2" class="diff-side-added"></td></tr>
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152.3.194.201
https://en.formulasearchengine.com/index.php?title=Endomorphism&diff=326&oldid=prev
en>Tobias Bergemann: /* See also */ Remove morphism as there already is a link to morphism in the very first sentence of this article. Sort alphabetically.
2013-04-26T07:40:22Z
<p><span dir="auto"><span class="autocomment">See also: </span> Remove <a href="/index.php?title=Morphism&action=edit&redlink=1" class="new" title="Morphism (page does not exist)">morphism</a> as there already is a link to <a href="/index.php?title=Morphism&action=edit&redlink=1" class="new" title="Morphism (page does not exist)">morphism</a> in the very first sentence of this article. Sort alphabetically.</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:40, 26 April 2013</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">When my dad started have hand tremors we suspected there was </del>a <del style="font-weight: bold; text-decoration: none;">problem</del>. <del style="font-weight: bold; text-decoration: none;">And when my aunt suggested surely has him tested for Parkinson</del>'<del style="font-weight: bold; text-decoration: none;">s</del>' <del style="font-weight: bold; text-decoration: none;">Disease</del>, <del style="font-weight: bold; text-decoration: none;">our fears were confirmed.<br><br>One </del>of <del style="font-weight: bold; text-decoration: none;">the most effective things you can do to give your home </del>'<del style="font-weight: bold; text-decoration: none;">curb appeal</del>' is <del style="font-weight: bold; text-decoration: none;">to utilize </del>a <del style="font-weight: bold; text-decoration: none;">fancy garage back door</del>. <del style="font-weight: bold; text-decoration: none;">This is a relatively easy technique to add a lot </del>of <del style="font-weight: bold; text-decoration: none;">class to an otherwise dull property. 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The <del style="font-weight: bold; text-decoration: none;">majority </del>of <del style="font-weight: bold; text-decoration: none;">the people I saw looked surprisingly normal, which made it seem like they were </del>all <del style="font-weight: bold; text-decoration: none;">the weirder. On one for whites of me was one person in his 60s wearing dark-rimmed glasses and reading the Wall Street Journal as his wife read Cosmo. A gay couple in their 20s was on the other side. Behind me would be </del>a <del style="font-weight: bold; text-decoration: none;">heavyset man and a darkly tanned woman with leathery your skin. It was nearly a perfect microcosm </del>of <del style="font-weight: bold; text-decoration: none;">your companion I saw every vacation to the store or driving down the road. However, I was still convinced that something was eerily wrong </del>with <del style="font-weight: bold; text-decoration: none;">they. It would take </del>a <del style="font-weight: bold; text-decoration: none;">few more trips before I found that they were, in fact</del>, the <del style="font-weight: bold; text-decoration: none;">really people I saw day by day outside </del>of the <del style="font-weight: bold; text-decoration: none;">beach.</del><<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">Buying </del>a <del style="font-weight: bold; text-decoration: none;">home also </del>a <del style="font-weight: bold; text-decoration: none;">person have insight as </del>a <del style="font-weight: bold; text-decoration: none;">portion </del>of a <del style="font-weight: bold; text-decoration: none;">permanent community</del>. <del style="font-weight: bold; text-decoration: none;">On </del>the <del style="font-weight: bold; text-decoration: none;">other side hand, from a rented apartment or home, one might feel temporary and less involved.</del><<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">Alan Jones was MC and </del>the <del style="font-weight: bold; text-decoration: none;">warm-up included </del>a <del style="font-weight: bold; text-decoration: none;">short film taking </del>a <del style="font-weight: bold; text-decoration: none;">light-weight hearted take </del>a <del style="font-weight: bold; text-decoration: none;">Clinton's last days in office</del>. <del style="font-weight: bold; text-decoration: none;">Scenes included Clinton washing </del>the <del style="font-weight: bold; text-decoration: none;">Presidential car</del>, <del style="font-weight: bold; text-decoration: none;">clipping the hedges </del>and <del style="font-weight: bold; text-decoration: none;">playing switchboard operator each morning Oval Region. 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Saturate the walls with the </del>and <del style="font-weight: bold; text-decoration: none;">white vinegar. This is easy </del>to <del style="font-weight: bold; text-decoration: none;">do with </del>a <del style="font-weight: bold; text-decoration: none;">large sponge</del>. <del style="font-weight: bold; text-decoration: none;">After just several minnutes </del>of <del style="font-weight: bold; text-decoration: none;">soaking, you shouldn</del>'<del style="font-weight: bold; text-decoration: none;">t have any trouble eliminating </del>the <del style="font-weight: bold; text-decoration: none;">wallpaper. Wash off outdated wallpaper glue </del>and <del style="font-weight: bold; text-decoration: none;">let the wall dried out</del>. <del style="font-weight: bold; text-decoration: none;">The final step </del>is <del style="font-weight: bold; text-decoration: none;">to repaint however with </del>a <del style="font-weight: bold; text-decoration: none;">neutral</del>, <del style="font-weight: bold; text-decoration: none;">light color</del>. <del style="font-weight: bold; text-decoration: none;">This will make </del>the <del style="font-weight: bold; text-decoration: none;">room seem larger </del>and <del style="font-weight: bold; text-decoration: none;">won't clash with any furnishings a new owner likely would have</del>.<del style="font-weight: bold; text-decoration: none;"><br><br>At first the disease seemed devastating </del>to <del style="font-weight: bold; text-decoration: none;">my pops</del>. <del style="font-weight: bold; text-decoration: none;">He was used </del>to <del style="font-weight: bold; text-decoration: none;">being independent---not getting help</del>. <del style="font-weight: bold; text-decoration: none;">If anything</del>, <del style="font-weight: bold; text-decoration: none;">he wanted in order to the someone to help your not the opposite way round.</del><<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">Spend any time in winter maintaining your holiday home and you can have less strive and do when somebody to spend an afternoon there </del>the <del style="font-weight: bold; text-decoration: none;">new year</del>. <del style="font-weight: bold; text-decoration: none;">By benefiting from work done now</del>, <del style="font-weight: bold; text-decoration: none;">peaceful breaths </del>. <del style="font-weight: bold; text-decoration: none;">also prevent bigger problems from occurring such like a break in or a burst water pipe</del>.<<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;"><br>If you cherished this write</del>-<del style="font-weight: bold; text-decoration: none;">up and you would like to obtain much more information pertaining to </del>[<del style="font-weight: bold; text-decoration: none;">http</del>:<del style="font-weight: bold; text-decoration: none;">//www.hedgingplants.com/ hedgingplants hedges</del>] <del style="font-weight: bold; text-decoration: none;">kindly take a look at our own web-site.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{About|the mathematical concept|the '''endomorphic''' body type|Somatotype}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In [[mathematics]], an '''endomorphism''' is a [[morphism]] (or [[homomorphism]]) from </ins>a <ins style="font-weight: bold; text-decoration: none;">[[mathematical object]] to itself</ins>. <ins style="font-weight: bold; text-decoration: none;"> For example, an endomorphism of a [[vector space]] ''V'' is a [[linear map]] ƒ:&nbsp;''V''&nbsp;→&nbsp;''V</ins>'', <ins style="font-weight: bold; text-decoration: none;">and an endomorphism </ins>of <ins style="font-weight: bold; text-decoration: none;">a [[group (mathematics)|group]] ''G</ins>'' is a <ins style="font-weight: bold; text-decoration: none;">[[group homomorphism]] ƒ:&nbsp;''G''&nbsp;→&nbsp;''G''</ins>. <ins style="font-weight: bold; text-decoration: none;">In general, we can talk about endomorphisms in any [[category theory|category]]. In the category </ins>of <ins style="font-weight: bold; text-decoration: none;">[[Set (mathematics)|sets]]</ins>, <ins style="font-weight: bold; text-decoration: none;">endomorphisms </ins>are <ins style="font-weight: bold; text-decoration: none;">simply functions from a set ''S'' into itself</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In any category</ins>, <ins style="font-weight: bold; text-decoration: none;">the [[function composition|composition]] of </ins>any <ins style="font-weight: bold; text-decoration: none;">two endomorphisms of ''X'' is again </ins>an <ins style="font-weight: bold; text-decoration: none;">endomorphism of ''X''</ins>. <ins style="font-weight: bold; text-decoration: none;"> It follows that the set of all endomorphisms of ''X'' forms a [[monoid]], denoted End(''X'') (or End</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">''C''</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>><ins style="font-weight: bold; text-decoration: none;">(''X'') to emphasize the category ''C'')</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">An [[inverse element|invertible]] endomorphism of ''X'' is called an [[automorphism]]</ins>. <ins style="font-weight: bold; text-decoration: none;"> </ins>The <ins style="font-weight: bold; text-decoration: none;">set </ins>of all <ins style="font-weight: bold; text-decoration: none;">automorphisms is </ins>a <ins style="font-weight: bold; text-decoration: none;">[[subset]] </ins>of <ins style="font-weight: bold; text-decoration: none;">End(''X'') </ins>with a <ins style="font-weight: bold; text-decoration: none;">[[group (mathematics)|group]] structure</ins>, <ins style="font-weight: bold; text-decoration: none;">called </ins>the <ins style="font-weight: bold; text-decoration: none;">[[automorphism group]] </ins>of <ins style="font-weight: bold; text-decoration: none;">''X'' and denoted Aut(''X''). In the following diagram, </ins>the <ins style="font-weight: bold; text-decoration: none;">arrows denote implication:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{| border="0"</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| align="center" width="42%" | [[automorphism]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| align="center" width="16%" | <math>\Rightarrow</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| align="center" width="42%" | [[isomorphism]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| align="center" | <math>\Downarrow</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| align="center" | </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">\Downarrow</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| align="center" | endomorphism</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| align="center" | </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">\Rightarrow</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| align="center" | [[homomorphism|(homo)morphism]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Any two endomorphisms of an [[abelian group]] ''A'' can be added together by the rule (ƒ&nbsp;+&nbsp;''g'')(''</ins>a<ins style="font-weight: bold; text-decoration: none;">'')&nbsp;=&nbsp;ƒ(''</ins>a<ins style="font-weight: bold; text-decoration: none;">'')&nbsp;+&nbsp;''g''(''</ins>a<ins style="font-weight: bold; text-decoration: none;">''). Under this addition, the endomorphisms </ins>of <ins style="font-weight: bold; text-decoration: none;">an abelian group form </ins>a <ins style="font-weight: bold; text-decoration: none;">[[ring (mathematics)|ring]] (the [[endomorphism ring]])</ins>. <ins style="font-weight: bold; text-decoration: none;"> For example, </ins>the <ins style="font-weight: bold; text-decoration: none;">set of endomorphisms of '''Z'''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">''n''</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>> <ins style="font-weight: bold; text-decoration: none;">is the ring of all ''n''&nbsp;×&nbsp;''n'' matrices with integer entries. The endomorphisms of a vector space or [[module (mathematics)|module]] also form a ring, as do </ins>the <ins style="font-weight: bold; text-decoration: none;">endomorphisms of any object in </ins>a <ins style="font-weight: bold; text-decoration: none;">[[preadditive category]]. The endomorphisms of </ins>a <ins style="font-weight: bold; text-decoration: none;">nonabelian group generate an algebraic structure known as </ins>a <ins style="font-weight: bold; text-decoration: none;">[[nearring]]</ins>. <ins style="font-weight: bold; text-decoration: none;">Every ring with one is </ins>the <ins style="font-weight: bold; text-decoration: none;">endomorphism ring of its [[regular module]]</ins>, and <ins style="font-weight: bold; text-decoration: none;">so is </ins>a <ins style="font-weight: bold; text-decoration: none;">subring of an endomorphism ring of an abelian group,</ins><<ins style="font-weight: bold; text-decoration: none;">ref</ins>><ins style="font-weight: bold; text-decoration: none;">Jacobson (2009), p. 162, Theorem 3.2.</ins><<ins style="font-weight: bold; text-decoration: none;">/ref</ins>> <ins style="font-weight: bold; text-decoration: none;">however there are rings which are not </ins>the <ins style="font-weight: bold; text-decoration: none;">endomorphism ring of any abelian group.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Operator theory==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In any [[concrete category]], especially for [[vector space]]s, endomorphisms are maps from a set into itself</ins>, and <ins style="font-weight: bold; text-decoration: none;">may be interpreted as [[unary operator]]</ins>s <ins style="font-weight: bold; text-decoration: none;">on that set, [[action (group theory)|acting]] on the elements, </ins>and <ins style="font-weight: bold; text-decoration: none;">allowing to define the notion of [[orbit (group theory)|orbit]]s of elements</ins>, <ins style="font-weight: bold; text-decoration: none;">etc</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Depending on the additional structure defined for the category at hand ([[topology]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[metric (mathematics)|metric]]</ins>, ...<ins style="font-weight: bold; text-decoration: none;">), such operators can </ins>have <ins style="font-weight: bold; text-decoration: none;">properties like [[continuous function (topology)|continuity]], [[boundedness]]</ins>, <ins style="font-weight: bold; text-decoration: none;">and so on. More details should be found in </ins>the <ins style="font-weight: bold; text-decoration: none;">article about [[operator theory]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Endofunctions in mathematics==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In [[mathematics]], an '''endofunction''' is a [[function (mathematics)|function]] whose [[codomain]] is equal to its [[domain of a function|domain]]. A [[homomorphism|homomorphic]] endofunction </ins>is <ins style="font-weight: bold; text-decoration: none;">an endomorphism.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Let ''S'' </ins>be <ins style="font-weight: bold; text-decoration: none;">an arbitrary set</ins>. <ins style="font-weight: bold; text-decoration: none;">Among endofunctions on ''S'' one finds [[permutation]]s </ins>of <ins style="font-weight: bold; text-decoration: none;">''S'' </ins>and <ins style="font-weight: bold; text-decoration: none;">constant functions associating </ins>to <ins style="font-weight: bold; text-decoration: none;">each <math>x\in S</math> </ins>a <ins style="font-weight: bold; text-decoration: none;">given <math>c\in S</math></ins>. </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Every permutation </ins>of '<ins style="font-weight: bold; text-decoration: none;">'S'' has </ins>the <ins style="font-weight: bold; text-decoration: none;">codomain equal to its domain and is [[bijection|bijective]] </ins>and <ins style="font-weight: bold; text-decoration: none;">invertible</ins>. <ins style="font-weight: bold; text-decoration: none;">A constant function on ''S'', if ''S'' has more than 1 element, has a codomain that </ins>is a <ins style="font-weight: bold; text-decoration: none;">proper subset of its domain</ins>, <ins style="font-weight: bold; text-decoration: none;">is not bijective (and non invertible)</ins>. <ins style="font-weight: bold; text-decoration: none;">The function associating to each natural integer ''n'' </ins>the <ins style="font-weight: bold; text-decoration: none;">floor of ''n''/2 has its codomain equal to its domain </ins>and <ins style="font-weight: bold; text-decoration: none;">is not invertible</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Finite endofunctions are equivalent </ins>to <ins style="font-weight: bold; text-decoration: none;">monogeneous digraphs, i.e</ins>. <ins style="font-weight: bold; text-decoration: none;">digraphs having all nodes with outdegree equal </ins>to <ins style="font-weight: bold; text-decoration: none;">1, and can be easily described</ins>. <ins style="font-weight: bold; text-decoration: none;">For sets of size ''n''</ins>, <ins style="font-weight: bold; text-decoration: none;">there are ''n''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">''n''</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>> <ins style="font-weight: bold; text-decoration: none;">endofunctions on </ins>the <ins style="font-weight: bold; text-decoration: none;">set</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Particular bijective endofunctions are the [[Involution (mathematics)|involution]]s</ins>, <ins style="font-weight: bold; text-decoration: none;">i</ins>.<ins style="font-weight: bold; text-decoration: none;">e. the functions coinciding with their inverses</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Notes==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins style="font-weight: bold; text-decoration: none;">references /</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==See also==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Adjoint endomorphism]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Frobenius endomorphism]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==References==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | series= | publisher=Dover| isbn = 978-0-486-47189</ins>-<ins style="font-weight: bold; text-decoration: none;">1}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==External links==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* {{springer|title=Endomorphism|id=p/e035600}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{planetmath reference|id=7462|title=Endomorphism}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[</ins>[<ins style="font-weight: bold; text-decoration: none;">Category</ins>:<ins style="font-weight: bold; text-decoration: none;">Morphisms]</ins>]</div></td></tr>
</table>
en>Tobias Bergemann
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en>RibotBOT: Robot: Adding pt:Endomorfismo
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