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id="toc-Common_algebraic_structures-sublist" class="vector-toc-list"> <li id="toc-One_set_with_operations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#One_set_with_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>One set with operations</span> </div> </a> <ul id="toc-One_set_with_operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two_sets_with_operations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two_sets_with_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Two sets with operations</span> </div> </a> <ul id="toc-Two_sets_with_operations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Hybrid_structures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hybrid_structures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Hybrid structures</span> </div> </a> <ul id="toc-Hybrid_structures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Universal_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Universal_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Universal algebra</span> </div> </a> <ul id="toc-Universal_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Category_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Category_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Category theory</span> </div> </a> <ul id="toc-Category_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Different_meanings_of_"structure"" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Different_meanings_of_"structure""> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Different meanings of "structure"</span> </div> </a> <ul id="toc-Different_meanings_of_"structure"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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">9</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">10</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 class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Algebraic structure</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" 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Available in 49 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-49" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">49 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D9%86%D9%8A%D8%A9_%D8%AC%D8%A8%D8%B1%D9%8A%D8%A9" title="بنية جبرية – Arabic" lang="ar" hreflang="ar" data-title="بنية جبرية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Estructura_algebraica" title="Estructura algebraica – Catalan" lang="ca" hreflang="ca" data-title="Estructura algebraica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Algebraick%C3%A1_struktura" title="Algebraická struktura – Czech" lang="cs" hreflang="cs" data-title="Algebraická struktura" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algebraische_Struktur" title="Algebraische Struktur – German" lang="de" hreflang="de" data-title="Algebraische Struktur" 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/Algebraline_struktuur" title="Algebraline struktuur – Estonian" lang="et" hreflang="et" data-title="Algebraline struktuur" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B9%CE%BA%CE%AE_%CE%B4%CE%BF%CE%BC%CE%AE" title="Αλγεβρική δομή – Greek" lang="el" hreflang="el" data-title="Αλγεβρική δομή" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Estructura_algebraica" title="Estructura algebraica – Spanish" lang="es" hreflang="es" data-title="Estructura algebraica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Algebra_strukturo" title="Algebra strukturo – Esperanto" lang="eo" hreflang="eo" data-title="Algebra strukturo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Egitura_aljebraiko" title="Egitura aljebraiko – Basque" lang="eu" hreflang="eu" data-title="Egitura aljebraiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B3%D8%A7%D8%AE%D8%AA%D8%A7%D8%B1_%D8%AC%D8%A8%D8%B1%DB%8C" title="ساختار جبری – Persian" lang="fa" hreflang="fa" data-title="ساختار جبری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Structure_alg%C3%A9brique" title="Structure algébrique – French" lang="fr" hreflang="fr" data-title="Structure algébrique" 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/Estrutura_alx%C3%A9brica" title="Estrutura alxébrica – Galician" lang="gl" hreflang="gl" data-title="Estrutura alxébrica" 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/%EB%8C%80%EC%88%98_%EA%B5%AC%EC%A1%B0" title="대수 구조 – Korean" lang="ko" hreflang="ko" data-title="대수 구조" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%B0%E0%A4%9A%E0%A4%A8%E0%A4%BE" title="बीजगणितीय संरचना – Hindi" lang="hi" hreflang="hi" data-title="बीजगणितीय संरचना" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Algebrala_strukturo" title="Algebrala strukturo – Ido" lang="io" hreflang="io" data-title="Algebrala strukturo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Struktur_aljabar" title="Struktur aljabar – Indonesian" lang="id" hreflang="id" data-title="Struktur aljabar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Structura_algebric" title="Structura algebric – Interlingua" lang="ia" hreflang="ia" data-title="Structura algebric" 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/Struttura_algebrica" title="Struttura algebrica – Italian" lang="it" hreflang="it" data-title="Struttura algebrica" 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%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99" title="מבנה אלגברי – Hebrew" lang="he" hreflang="he" data-title="מבנה אלגברי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Algebresch_Struktur" title="Algebresch Struktur – Luxembourgish" lang="lb" hreflang="lb" data-title="Algebresch Struktur" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Algebrin%C4%97_strukt%C5%ABra" title="Algebrinė struktūra – Lithuanian" lang="lt" hreflang="lt" data-title="Algebrinė struktūra" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%80%E0%B4%9C%E0%B5%80%E0%B4%AF%E0%B4%98%E0%B4%9F%E0%B4%A8" title="ബീജീയഘടന – Malayalam" lang="ml" hreflang="ml" data-title="ബീജീയഘടന" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Struktur_algebra" title="Struktur algebra – Malay" lang="ms" hreflang="ms" data-title="Struktur algebra" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8%D0%B9%D0%BD_%D0%B1%D2%AF%D1%82%D1%8D%D1%86" title="Алгебрийн бүтэц – Mongolian" lang="mn" hreflang="mn" data-title="Алгебрийн бүтэц" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algebra%C3%AFsche_structuur" title="Algebraïsche structuur – Dutch" lang="nl" hreflang="nl" data-title="Algebraïsche structuur" 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/%E4%BB%A3%E6%95%B0%E7%9A%84%E6%A7%8B%E9%80%A0" title="代数的構造 – Japanese" lang="ja" hreflang="ja" data-title="代数的構造" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Algebraisk_struktur" title="Algebraisk struktur – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Algebraisk struktur" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Algebraisk_struktur" title="Algebraisk struktur – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Algebraisk struktur" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Estructura_algebrica" title="Estructura algebrica – Occitan" lang="oc" hreflang="oc" data-title="Estructura algebrica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Strutura_alg%C3%A9brica" title="Strutura algébrica – Piedmontese" lang="pms" hreflang="pms" data-title="Strutura algébrica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_og%C3%B3lna" title="Algebra ogólna – Polish" lang="pl" hreflang="pl" data-title="Algebra ogólna" 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/Estrutura_alg%C3%A9brica" title="Estrutura algébrica – Portuguese" lang="pt" hreflang="pt" data-title="Estrutura algébrica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Structur%C4%83_algebric%C4%83" title="Structură algebrică – Romanian" lang="ro" hreflang="ro" data-title="Structură algebrică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D1%83%D0%BA%D1%82%D1%83%D1%80%D0%B0" title="Алгебраическая структура – Russian" lang="ru" hreflang="ru" data-title="Алгебраическая структура" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Struktura_algjebrike" title="Struktura algjebrike – Albanian" lang="sq" hreflang="sq" data-title="Struktura algjebrike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Algebraic_structure" title="Algebraic structure – Simple English" lang="en-simple" hreflang="en-simple" data-title="Algebraic structure" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Algebrick%C3%A1_%C5%A1trukt%C3%BAra" title="Algebrická štruktúra – Slovak" lang="sk" hreflang="sk" data-title="Algebrická štruktúra" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Algebrska_struktura" title="Algebrska struktura – Slovenian" lang="sl" hreflang="sl" data-title="Algebrska struktura" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%DB%8E%DA%A9%DA%BE%D8%A7%D8%AA%DB%95%DB%8C_%D8%AC%DB%95%D8%A8%D8%B1%DB%8C" title="پێکھاتەی جەبری – Central Kurdish" lang="ckb" hreflang="ckb" data-title="پێکھاتەی جەبری" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D0%B0%D1%80%D1%81%D0%BA%D0%B0_%D1%81%D1%82%D1%80%D1%83%D0%BA%D1%82%D1%83%D1%80%D0%B0" title="Алгебарска структура – Serbian" lang="sr" hreflang="sr" data-title="Алгебарска структура" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Algebarska_struktura" title="Algebarska struktura – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Algebarska struktura" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Algebrallinen_rakenne" title="Algebrallinen rakenne – Finnish" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="display:block;margin-bottom:0.35em;"><a class="mw-selflink selflink">Algebraic structures</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a></li> <li><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> / <a href="/wiki/Monoid" title="Monoid">Monoid</a></li> <li><a href="/wiki/Racks_and_quandles" title="Racks and quandles">Rack and quandle</a></li> <li><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup and loop</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a></li> <li><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li></ul> </div> <i><a href="/wiki/Group_theory" title="Group theory">Group theory</a></i></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a></li> <li><a href="/wiki/Rng_(algebra)" title="Rng (algebra)">Rng</a></li> <li><a href="/wiki/Semiring" title="Semiring">Semiring</a></li> <li><a href="/wiki/Near-ring" title="Near-ring">Near-ring</a></li> <li><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a></li> <li><a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">Domain</a></li> <li><a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a></li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a></li> <li><a href="/wiki/Division_ring" title="Division ring">Division ring</a></li> <li><a href="/wiki/Lie_algebra#Lie_ring" title="Lie algebra">Lie ring</a></li></ul> </div> <i><a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></i></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented lattice</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li></ul> </div> <ul><li><a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a></li> <li><i><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice theory</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></li> <li><a href="/wiki/Group_with_operators" title="Group with operators">Group with operators</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div> <ul><li><i><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Associative_algebra" title="Associative algebra">Associative</a></li> <li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative</a></li> <li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a></li> <li><a href="/wiki/Graded_ring" title="Graded ring">Graded</a></li> <li><a href="/wiki/Bialgebra" title="Bialgebra">Bialgebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebraic_structures" title="Template:Algebraic structures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_structures" title="Template talk:Algebraic structures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_structures" title="Special:EditPage/Template:Algebraic structures"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>algebraic structure</b> or <b>algebraic system</b><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> consists of a nonempty <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>A</i> (called the <b>underlying set</b>, <b>carrier set</b> or <b>domain</b>), a collection of <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> on <i>A</i> (typically <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a> such as addition and multiplication), and a finite set of <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a> (known as <a href="/wiki/Axiom#Non-logical_axioms" title="Axiom"><i>axioms</i></a>) that these operations must satisfy. </p><p>An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a <a href="/wiki/Vector_space" title="Vector space">vector space</a> involves a second structure called a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, and an operation called <i>scalar multiplication</i> between elements of the field (called <i><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a></i>), and elements of the vector space (called <i><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a></i>). </p><p><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a> is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>. <a href="/wiki/Category_theory" title="Category theory">Category theory</a> is another formalization that includes also other <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a> and <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> between structures of the same type (<a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a>). </p><p>In universal algebra, an algebraic structure is called an <i>algebra</i>;<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> this term may be ambiguous, since, in other contexts, <a href="/wiki/An_algebra" class="mw-redirect" title="An algebra">an algebra</a> is an algebraic structure that is a vector space over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> or a <a href="/wiki/Module_(ring_theory)" class="mw-redirect" title="Module (ring theory)">module</a> over a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>. </p><p>The collection of all structures of a given type (same operations and same laws) is called a <a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">variety</a> in universal algebra; this term is also used with a completely different meaning in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, as an abbreviation of <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a>. In category theory, the collection of all structures of a given type and homomorphisms between them form a <a href="/wiki/Concrete_category" title="Concrete category">concrete category</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Addition" title="Addition">Addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> are prototypical examples of <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, <span class="texhtml"><i>a</i> + (<i>b</i> + <i>c</i>) = (<i>a</i> + <i>b</i>) + <i>c</i></span> and <span class="texhtml"><i>a</i>(<i>bc</i>) = (<i>ab</i>)<i>c</i></span> are <a href="/wiki/Associative_law" class="mw-redirect" title="Associative law">associative laws</a>, and <span class="texhtml"><i>a</i> + <i>b</i> = <i>b</i> + <i>a</i></span> and <span class="texhtml"><i>ab</i> = <i>ba</i></span> are <a href="/wiki/Commutative_law" class="mw-redirect" title="Commutative law">commutative laws</a>. Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, the possible moves of an object in three-dimensional space can be combined by performing a first move of the object, and then a second move from its new position. Such moves, formally called <a href="/wiki/Rigid_motion" class="mw-redirect" title="Rigid motion">rigid motions</a>, obey the associative law, but fail to satisfy the commutative law. </p><p>Sets with one or more operations that obey specific laws are called <i>algebraic structures</i>. When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem. </p><p>In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher <a href="/wiki/Arity" title="Arity">arity</a> operations) and operations that take only one <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a> (<a href="/wiki/Unary_operation" title="Unary operation">unary operations</a>) or even zero arguments (<a href="/wiki/Nullary_operation" class="mw-redirect" title="Nullary operation">nullary operations</a>). The examples listed below are by no means a complete list, but include the most common structures taught in undergraduate courses. </p> <div class="mw-heading mw-heading2"><h2 id="Common_axioms">Common axioms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=2" title="Edit section: Common axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Equational_axioms">Equational axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=3" title="Edit section: Equational axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An axiom of an algebraic structure often has the form of an <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a>, that is, an <a href="/wiki/Equation_(mathematics)" class="mw-redirect" title="Equation (mathematics)">equation</a> such that the two sides of the <a href="/wiki/Equals_sign" title="Equals sign">equals sign</a> are <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expressions</a> that involve operations of the algebraic structure and <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>. If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true. Here are some common examples. </p> <dl><dt><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a></dt> <dd>An operation <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> is <i>commutative</i> if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x*y=y*x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>∗</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x*y=y*x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b24dba3d0ce7a781c3b05150ff4296209c3afd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.458ex; height:2.009ex;" alt="{\displaystyle x*y=y*x}" /></span> for every <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> in the algebraic structure.</dd> <dt><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a></dt> <dd>An operation <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> is <i>associative</i> if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x*y)*z=x*(y*z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x*y)*z=x*(y*z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43534cf00082214688f06873ec9ea32caa7e3305" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.643ex; height:2.843ex;" alt="{\displaystyle (x*y)*z=x*(y*z)}" /></span> for every <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml mvar" style="font-style:italic;">z</span> in the algebraic structure.</dd> <dt><a href="/wiki/Left_distributivity" class="mw-redirect" title="Left distributivity">Left distributivity</a></dt> <dd>An operation <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> is <i>left distributive</i> with respect to another operation <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x*(y+z)=(x*y)+(x*z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x*(y+z)=(x*y)+(x*z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/766b0c4db887e79e8cfa9b4ec5b9acc192dbd36e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.268ex; height:2.843ex;" alt="{\displaystyle x*(y+z)=(x*y)+(x*z)}" /></span> for every <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml mvar" style="font-style:italic;">z</span> in the algebraic structure (the second operation is denoted here as <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /></span>, because the second operation is addition in many common examples).</dd> <dt><a href="/wiki/Right_distributivity" class="mw-redirect" title="Right distributivity">Right distributivity</a></dt> <dd>An operation <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> is <i>right distributive</i> with respect to another operation <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y+z)*x=(y*x)+(z*x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∗</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>∗</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y+z)*x=(y*x)+(z*x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0b0bfaa89ce127a4c0c51b8adfba6105b67c7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.268ex; height:2.843ex;" alt="{\displaystyle (y+z)*x=(y*x)+(z*x)}" /></span> for every <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml mvar" style="font-style:italic;">z</span> in the algebraic structure.</dd> <dt><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a></dt> <dd>An operation <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> is <i>distributive</i> with respect to another operation <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /></span> if it is both left distributive and right distributive. If the operation <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> is commutative, left and right distributivity are both equivalent to distributivity.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Existential_axioms">Existential axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=4" title="Edit section: Existential axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some common axioms contain an <a href="/wiki/Existential_clause" title="Existential clause">existential clause</a>. In general, such a clause can be avoided by introducing further operations, and replacing the existential clause by an identity involving the new operation. More precisely, let us consider an axiom of the form <i>"for all <span class="texhtml mvar" style="font-style:italic;">X</span> there is <span class="texhtml mvar" style="font-style:italic;">y</span> such that</i> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X,y)=g(X,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X,y)=g(X,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5005a74e7e0e46e5db804c244d0a23ff467586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.451ex; height:2.843ex;" alt="{\displaystyle f(X,y)=g(X,y)}" /></span>",</span> where <span class="texhtml mvar" style="font-style:italic;">X</span> is a <span class="texhtml mvar" style="font-style:italic;">k</span>-<a href="/wiki/Tuple" title="Tuple">tuple</a> of variables. Choosing a specific value of <span class="texhtml mvar" style="font-style:italic;">y</span> for each value of <span class="texhtml mvar" style="font-style:italic;">X</span> defines a function <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 \varphi :X\mapsto y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">↦</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :X\mapsto y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc40258413803d25f71770965baca4013b11bfa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.854ex; height:2.676ex;" alt="{\displaystyle \varphi :X\mapsto y,}" /></span> which can be viewed as an operation of <a href="/wiki/Arity" title="Arity">arity</a> <span class="texhtml mvar" style="font-style:italic;">k</span>, and the axiom becomes the identity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X,\varphi (X))=g(X,\varphi (X)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X,\varphi (X))=g(X,\varphi (X)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916de53c39daa817b1bc921e4d740e4e372814d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.405ex; height:2.843ex;" alt="{\displaystyle f(X,\varphi (X))=g(X,\varphi (X)).}" /></span> </p><p>The introduction of such auxiliary operation complicates slightly the statement of an axiom, but has some advantages. Given a specific algebraic structure, the proof that an existential axiom is satisfied consists generally of the definition of the auxiliary function, completed with straightforward verifications. Also, when computing in an algebraic structure, one generally uses explicitly the auxiliary operations. For example, in the case of <a href="/wiki/Number" title="Number">numbers</a>, the <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> is provided by the unary minus operation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto -x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo>−</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto -x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/031889143fdbe0b9689c8e4446f2e120397dbdfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.728ex; height:2.176ex;" alt="{\displaystyle x\mapsto -x.}" /></span> </p><p>Also, in <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>, a <a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">variety</a> is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities. What precedes shows that existential axioms of the above form are accepted in the definition of a variety. </p><p>Here are some of the most common existential axioms. </p> <dl><dt><a href="/wiki/Identity_element" title="Identity element">Identity element</a></dt> <dd>A <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> has an identity element if there is an element <span class="texhtml mvar" style="font-style:italic;">e</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x*e=x\quad {\text{and}}\quad e*x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∗</mo> <mi>e</mi> <mo>=</mo> <mi>x</mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em"></mspace> <mi>e</mi> <mo>∗</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x*e=x\quad {\text{and}}\quad e*x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b14f413c7e52f74ed66e483531d3dec1c93f430" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.465ex; height:2.176ex;" alt="{\displaystyle x*e=x\quad {\text{and}}\quad e*x=x}" /></span> for all <span class="texhtml mvar" style="font-style:italic;">x</span> in the structure. Here, the auxiliary operation is the operation of arity zero that has <span class="texhtml mvar" style="font-style:italic;">e</span> as its result.</dd> <dt><a href="/wiki/Inverse_element" title="Inverse element">Inverse element</a></dt> <dd>Given a binary operation <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> that has an identity element <span class="texhtml mvar" style="font-style:italic;">e</span>, an element <span class="texhtml mvar" style="font-style:italic;">x</span> is <i>invertible</i> if it has an inverse element, that is, if there exists an element <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 \operatorname {inv} (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>inv</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {inv} (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f77cac76353750f5941a238db6cc13a5e0fedf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.306ex; height:2.843ex;" alt="{\displaystyle \operatorname {inv} (x)}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {inv} (x)*x=e\quad {\text{and}}\quad x*\operatorname {inv} (x)=e.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>inv</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>x</mi> <mo>=</mo> <mi>e</mi> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em"></mspace> <mi>x</mi> <mo>∗</mo> <mi>inv</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>e</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {inv} (x)*x=e\quad {\text{and}}\quad x*\operatorname {inv} (x)=e.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120997a672d2134b29b3e0dec09ba0cee6721c44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.064ex; height:2.843ex;" alt="{\displaystyle \operatorname {inv} (x)*x=e\quad {\text{and}}\quad x*\operatorname {inv} (x)=e.}" /></span>For example, a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> is an algebraic structure with a binary operation that is associative, has an identity element, and for which all elements are invertible.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Non-equational_axioms">Non-equational axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=5" title="Edit section: Non-equational axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The axioms of an algebraic structure can be any <a href="/wiki/First-order_logic" title="First-order logic">first-order formula</a>, that is a formula involving <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a> (such as <i>"and"</i>, <i>"or"</i> and <i>"not"</i>), and <a href="/wiki/Logical_quantifier" class="mw-redirect" title="Logical quantifier">logical quantifiers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall ,\exists }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo>,</mo> <mi mathvariant="normal">∃</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall ,\exists }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c65054c57a79c67f0583388db198e359b2e24cc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle \forall ,\exists }" /></span>) that apply to elements (not to subsets) of the structure. </p><p>Such a typical axiom is inversion in <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>. This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form a <a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">variety</a> in the sense of <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>.) It can be stated: <i>"Every nonzero element of a field is <a href="/wiki/Invertible_element" class="mw-redirect" title="Invertible element">invertible</a>;"</i> or, equivalently: <i>the structure has a <a href="/wiki/Unary_operation" title="Unary operation">unary operation</a> <span class="texhtml">inv</span> such that</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,\quad x=0\quad {\text{or}}\quad x\cdot \operatorname {inv} (x)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mspace width="1em"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em"></mspace> <mi>x</mi> <mo>⋅</mo> <mi>inv</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,\quad x=0\quad {\text{or}}\quad x\cdot \operatorname {inv} (x)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f2fed65662253ebe9b600d9f2048ca79560fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.511ex; height:2.843ex;" alt="{\displaystyle \forall x,\quad x=0\quad {\text{or}}\quad x\cdot \operatorname {inv} (x)=1.}" /></span></dd></dl> <p>The operation <span class="texhtml">inv</span> can be viewed either as a <a href="/wiki/Partial_operation" class="mw-redirect" title="Partial operation">partial operation</a> that is not defined for <span class="texhtml"><i>x</i> = 0</span>; or as an ordinary function whose value at 0 is arbitrary and must not be used. </p> <div class="mw-heading mw-heading2"><h2 id="Common_algebraic_structures">Common algebraic structures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=6" title="Edit section: Common algebraic structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Outline_of_algebraic_structures#Types_of_algebraic_structures" title="Outline of algebraic structures">Outline of algebraic structures § Types of algebraic structures</a></div> <div class="mw-heading mw-heading3"><h3 id="One_set_with_operations">One set with operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=7" title="Edit section: One set with operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Simple structures</b>: <b>no</b> <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a>: </p> <ul><li><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a>: a degenerate algebraic structure <i>S</i> having no operations.</li></ul> <p><b>Group-like structures</b>: <b>one</b> binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. </p> <ul><li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a>: a <a href="/wiki/Monoid" title="Monoid">monoid</a> with a unary operation (inverse), giving rise to <a href="/wiki/Inverse_element" title="Inverse element">inverse elements</a>.</li> <li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a>: a group whose binary operation is <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>.</li></ul> <p><b>Ring-like structures</b> or <b>Ringoids</b>: <b>two</b> binary operations, often called <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, with multiplication <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributing</a> over addition. </p> <ul><li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a>: a semiring whose additive monoid is an abelian group.</li> <li><a href="/wiki/Division_ring" title="Division ring">Division ring</a>: a <a href="/wiki/Zero_ring" title="Zero ring">nontrivial</a> ring in which <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> by nonzero elements is defined.</li> <li><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a>: a ring in which the multiplication operation is commutative.</li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a>: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element).</li></ul> <p><b>Lattice structures</b>: <b>two</b> or more binary operations, including operations called <a href="/wiki/Meet_and_join" class="mw-redirect" title="Meet and join">meet and join</a>, connected by the <a href="/wiki/Absorption_law" title="Absorption law">absorption law</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete lattice</a>: a lattice in which arbitrary <a href="/wiki/Meet_and_join" class="mw-redirect" title="Meet and join">meet and joins</a> exist.</li> <li><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded lattice</a>: a lattice with a <a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a> and least element.</li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive lattice</a>: a lattice in which each of meet and join <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributes</a> over the other. A <a href="/wiki/Power_set" title="Power set">power set</a> under union and intersection forms a distributive lattice.</li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a>: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Two_sets_with_operations">Two sets with operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=8" title="Edit section: Two sets with operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a>: an abelian group <i>M</i> and a ring <i>R</i> acting as operators on <i>M</i>. The members of <i>R</i> are sometimes called <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>, and the binary operation of <i>scalar multiplication</i> is a function <i>R</i> × <i>M</i> → <i>M</i>, which satisfies several axioms. Counting the ring operations these systems have at least three operations.</li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a>: a module where the ring <i>R</i> is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> or, in some contexts, a <a href="/wiki/Division_ring" title="Division ring">division ring</a>.</li></ul> <ul><li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra over a field</a>: a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and <a href="/wiki/Bilinear_map" title="Bilinear map">linearity</a> with respect to multiplication.</li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a>: a field <i>F</i> and vector space <i>V</i> with a <a href="/wiki/Definite_bilinear_form" class="mw-redirect" title="Definite bilinear form">definite bilinear form</a> <span class="nowrap"><i>V</i> × <i>V</i> → <i>F</i></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Hybrid_structures">Hybrid structures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=9" title="Edit section: Hybrid structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algebraic structures can also coexist with added structure of non-algebraic nature, such as <a href="/wiki/Partially_ordered_set#Formal_definition" title="Partially ordered set">partial order</a> or a <a href="/wiki/Topology" title="Topology">topology</a>. The added structure must be compatible, in some sense, with the algebraic structure. </p> <ul><li><a href="/wiki/Topological_group" title="Topological group">Topological group</a>: a group with a topology compatible with the group operation.</li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a>: a topological group with a compatible smooth <a href="/wiki/Manifold" title="Manifold">manifold</a> structure.</li> <li><a href="/wiki/Ordered_group" class="mw-redirect" title="Ordered group">Ordered groups</a>, <a href="/wiki/Ordered_ring" title="Ordered ring">ordered rings</a> and <a href="/wiki/Ordered_field" title="Ordered field">ordered fields</a>: each type of structure with a compatible <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a>.</li> <li><a href="/wiki/Archimedean_group" title="Archimedean group">Archimedean group</a>: a linearly ordered group for which the <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a> holds.</li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector space</a>: a vector space whose <i>M</i> has a compatible topology.</li> <li><a href="/wiki/Normed_vector_space" title="Normed vector space">Normed vector space</a>: a vector space with a compatible <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>. If such a space is <a href="/wiki/Complete_metric_space" title="Complete metric space">complete</a> (as a metric space) then it is called a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>.</li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure.</li> <li><a href="/wiki/Vertex_operator_algebra" title="Vertex operator algebra">Vertex operator algebra</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a>: a *-algebra of operators on a Hilbert space equipped with the <a href="/wiki/Weak_operator_topology" title="Weak operator topology">weak operator topology</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Universal_algebra">Universal algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=10" title="Edit section: Universal algebra"><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/Universal_algebra" title="Universal algebra">Universal algebra</a></div> <p>Algebraic structures are defined through different configurations of <a href="/wiki/Axiom" title="Axiom">axioms</a>. <a href="/wiki/Universal_algebra" title="Universal algebra">Universal algebra</a> abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by <i>identities</i> and structures that are not. If all axioms defining a class of algebras are identities, then this class is a <a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">variety</a> (not to be confused with <a href="/wiki/Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a> of <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>). </p><p>Identities are equations formulated using only the operations the structure allows, and variables that are tacitly <a href="/wiki/Universal_quantifier" class="mw-redirect" title="Universal quantifier">universally quantified</a> over the relevant <a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">universe</a>. Identities contain no <a href="/wiki/Logical_connective" title="Logical connective">connectives</a>, <a href="/wiki/Quantification_(science)" title="Quantification (science)">existentially quantified variables</a>, or <a href="/wiki/Finitary_relation" title="Finitary relation">relations</a> of any kind other than the allowed operations. The study of varieties is an important part of <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>. An algebraic structure in a variety may be understood as the <a href="/wiki/Quotient_(universal_algebra)" title="Quotient (universal algebra)">quotient algebra</a> of term algebra (also called "absolutely <a href="/wiki/Free_object" title="Free object">free algebra</a>") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given <a href="/wiki/Signature_(logic)" title="Signature (logic)">signatures</a> generate a free algebra, the <a href="/wiki/Term_algebra" title="Term algebra">term algebra</a> <i>T</i>. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure <i>E</i>. The quotient algebra <i>T</i>/<i>E</i> is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator <i>m</i>, taking two arguments, and the inverse operator <i>i</i>, taking one argument, and the identity element <i>e</i>, a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables <i>x</i>, <i>y</i>, <i>z</i>, etc. the term algebra is the collection of all possible <a href="/wiki/Term_(logic)" title="Term (logic)">terms</a> involving <i>m</i>, <i>i</i>, <i>e</i> and the variables; so for example, <i>m</i>(<i>i</i>(<i>x</i>), <i>m</i>(<i>x</i>, <i>m</i>(<i>y</i>,<i>e</i>))) would be an element of the term algebra. One of the axioms defining a group is the identity <i>m</i>(<i>x</i>, <i>i</i>(<i>x</i>)) = <i>e</i>; another is <i>m</i>(<i>x</i>,<i>e</i>) = <i>x</i>. The axioms can be represented as <a rel="nofollow" class="external text" href="http://ncatlab.org/nlab/show/variety+of+algebras#examples_4">trees</a>. These equations induce <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> on the free algebra; the quotient algebra then has the algebraic structure of a group. </p><p>Some structures do not form varieties, because either: </p> <ol><li>It is necessary that 0 ≠ 1, 0 being the additive <a href="/wiki/Identity_element" title="Identity element">identity element</a> and 1 being a multiplicative identity element, but this is a nonidentity;</li> <li>Structures such as fields have some axioms that hold only for nonzero members of <i>S</i>. For an algebraic structure to be a variety, its operations must be defined for <i>all</i> members of <i>S</i>; there can be no partial operations.</li></ol> <p>Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> and <a href="/wiki/Division_ring" title="Division ring">division rings</a>. Structures with nonidentities present challenges varieties do not. For example, the <a href="/wiki/Direct_product" title="Direct product">direct product</a> of two <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> is not a field, because <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 (1,0)\cdot (0,1)=(0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0)\cdot (0,1)=(0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768a1fceb4398bd0ad0622a41ccea52b6e63df06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.282ex; height:2.843ex;" alt="{\displaystyle (1,0)\cdot (0,1)=(0,0)}" /></span>, but fields do not have <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisors</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Category_theory">Category theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=11" title="Edit section: Category theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Category_theory" title="Category theory">Category theory</a> is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of <i>objects</i> with associated <i>morphisms.</i> Every algebraic structure has its own notion of <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a>, namely any <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. For example, the <a href="/wiki/Category_of_groups" title="Category of groups">category of groups</a> has all <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> as objects and all <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphisms</a> as morphisms. This <a href="/wiki/Concrete_category" title="Concrete category">concrete category</a> may be seen as a <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a> with added category-theoretic structure. Likewise, the category of <a href="/wiki/Topological_group" title="Topological group">topological groups</a> (whose morphisms are the continuous group homomorphisms) is a <a href="/wiki/Category_of_topological_spaces" title="Category of topological spaces">category of topological spaces</a> with extra structure. A <a href="/wiki/Forgetful_functor" title="Forgetful functor">forgetful functor</a> between categories of algebraic structures "forgets" a part of a structure. </p><p>There are various concepts in category theory that try to capture the algebraic character of a context, for instance </p> <ul><li><a href="/wiki/Algebraic_category" class="mw-redirect" title="Algebraic category">algebraic category</a></li> <li><a href="/w/index.php?title=Essentially_algebraic_category&action=edit&redlink=1" class="new" title="Essentially algebraic category (page does not exist)">essentially algebraic category</a></li> <li><a href="/w/index.php?title=Presentable_category&action=edit&redlink=1" class="new" title="Presentable category (page does not exist)">presentable category</a></li> <li><a href="/wiki/Locally_presentable_category" class="mw-redirect" title="Locally presentable category">locally presentable category</a></li> <li><a href="/wiki/Monad_(category_theory)" title="Monad (category theory)">monadic</a> functors and categories</li> <li><a href="/wiki/Universal_property" title="Universal property">universal property</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Different_meanings_of_"structure""><span id="Different_meanings_of_.22structure.22"></span>Different meanings of "structure"</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_structure&action=edit&section=12" title="Edit section: Different meanings of "structure""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a slight <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a>, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring <i>structure</i> on the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>", means that we have defined <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <i>operations</i> on the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>. For another example, the group <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} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} ,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/910eaae0a8267ccb04d4846f6a28f02ce6ab8ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.202ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} ,+)}" /></span> can be seen as a set <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" /></span> that is equipped with an <i>algebraic structure,</i> namely the <i>operation</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /></span>. </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=Algebraic_structure&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 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(logic)">Signature (logic)</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Structure (mathematical logic)</a></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=Algebraic_structure&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <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"><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="CITEREFF.-V._Kuhlmann_(originator)" class="citation book cs1">F.-V. Kuhlmann (originator). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/index.php?title=Structure&oldid=14175">"Structure"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics"><i>Encyclopedia of Mathematics</i></a>. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1402006098" title="Special:BookSources/1402006098"><bdi>1402006098</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Structure&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=Springer-Verlag&rft.isbn=1402006098&rft.au=F.-V.+Kuhlmann+%28originator%29&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Findex.php%3Ftitle%3DStructure%26oldid%3D14175&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+structure" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">P.M. Cohn. (1981) <i>Universal Algebra</i>, Springer, p. 41.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Ringoids and <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattices</a> can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributive law</a>; in the case of lattices, they are linked by the <a href="/wiki/Absorption_law" title="Absorption law">absorption law</a>. Ringoids also tend to have numerical <a href="/wiki/Model_theory" title="Model theory">models</a>, while lattices tend to have <a href="/wiki/Set_theory" title="Set theory">set-theoretic</a> models.</span> </li> </ol></div></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=Algebraic_structure&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMac_LaneBirkhoff1999" class="citation cs2"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a>; <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, Garrett</a> (1999), <i>Algebra</i> (2nd ed.), AMS Chelsea, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1646-2" title="Special:BookSources/978-0-8218-1646-2"><bdi>978-0-8218-1646-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=2nd&rft.pub=AMS+Chelsea&rft.date=1999&rft.isbn=978-0-8218-1646-2&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft.au=Birkhoff%2C+Garrett&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+structure" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMichelHerget1993" class="citation cs2">Michel, Anthony N.; Herget, Charles J. (1993), <i>Applied Algebra and Functional Analysis</i>, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-67598-5" title="Special:BookSources/978-0-486-67598-5"><bdi>978-0-486-67598-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Algebra+and+Functional+Analysis&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1993&rft.isbn=978-0-486-67598-5&rft.aulast=Michel&rft.aufirst=Anthony+N.&rft.au=Herget%2C+Charles+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+structure" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBurrisSankappanavar1981" class="citation cs2">Burris, Stanley N.; Sankappanavar, H. P. (1981), <a rel="nofollow" class="external text" href="http://www.thoralf.uwaterloo.ca/htdocs/ualg.html"><i>A Course in Universal Algebra</i></a>, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-90578-3" title="Special:BookSources/978-3-540-90578-3"><bdi>978-3-540-90578-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Universal+Algebra&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1981&rft.isbn=978-3-540-90578-3&rft.aulast=Burris&rft.aufirst=Stanley+N.&rft.au=Sankappanavar%2C+H.+P.&rft_id=http%3A%2F%2Fwww.thoralf.uwaterloo.ca%2Fhtdocs%2Fualg.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+structure" class="Z3988"></span></li></ul> <dl><dt>Category theory</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMac_Lane1998" class="citation cs2"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1998), <i><a href="/wiki/Categories_for_the_Working_Mathematician" title="Categories for the Working Mathematician">Categories for the Working Mathematician</a></i> (2nd ed.), Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98403-2" title="Special:BookSources/978-0-387-98403-2"><bdi>978-0-387-98403-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Categories+for+the+Working+Mathematician&rft.place=Berlin%2C+New+York&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=978-0-387-98403-2&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+structure" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTaylor1999" class="citation cs2">Taylor, Paul (1999), <i>Practical foundations of mathematics</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-63107-5" title="Special:BookSources/978-0-521-63107-5"><bdi>978-0-521-63107-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+foundations+of+mathematics&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-63107-5&rft.aulast=Taylor&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+structure" class="Z3988"></span></li></ul> <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=Algebraic_structure&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://math.chapman.edu/~jipsen/structures/doku.php">Jipsen's algebra structures.</a> Includes many structures not mentioned here.</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/topics/Algebra.html">Mathworld</a> page on abstract algebra.</li> <li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/algebra/">Algebra</a> by <a href="/wiki/Vaughan_Pratt" title="Vaughan Pratt">Vaughan Pratt</a>.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebra" title="Template:Algebra"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebra" title="Template talk:Algebra"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebra" title="Special:EditPage/Template:Algebra"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Algebra304" style="font-size:114%;margin:0 4em"><a href="/wiki/Algebra" title="Algebra">Algebra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_algebra" title="Outline of algebra">Outline</a></li> <li><a href="/wiki/History_of_algebra" title="History of algebra">History</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Areas</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> <ul><li><a href="/wiki/Algebraic_variety" title="Algebraic variety">Algebraic variety</a></li> <li><a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">Scheme</a></li></ul></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary algebra</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological algebra</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li> <li><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_expression" title="Algebraic expression">Algebraic expression</a></li> <li><a href="/wiki/Equation" title="Equation">Equation</a> (<a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a>, <a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a>)</li> <li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a> (<a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">Polynomial function</a>)</li> <li><a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">Inequality</a> (<a href="/wiki/Linear_inequality" title="Linear inequality">Linear inequality</a>)</li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> (<a href="/wiki/Addition" title="Addition">Addition</a>, <a href="/wiki/Multiplication" title="Multiplication">Multiplication</a>)</li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a>)</li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">Variable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic structures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> (<a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">theory</a>)</li> <li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a> (<a href="/wiki/Group_theory" title="Group theory">theory</a>)</li> <li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a> (<a href="/wiki/Commutative_algebra" title="Commutative algebra">theory</a>)</li> <li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a> (<a href="/wiki/Ring_theory" title="Ring theory">theory</a>)</li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a> (<a href="/wiki/Coordinate_vector" title="Coordinate vector">Vector</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Linear and <br />multilinear algebra</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a> (<a href="/wiki/Dot_product" title="Dot product">Dot product</a>)</li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a> (<a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a>)</li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Linear subspace</a> (<a href="/wiki/Affine_space" title="Affine space">Affine space</a>)</li> <li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">Norm</a> (<a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>)</li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a>)</li> <li><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">Rank</a></li> <li><a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">Trace</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Free_object" title="Free object">Free object</a> (<a href="/wiki/Free_group" title="Free group">Free group</a>, ...)</li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a> (<a href="/wiki/Multivector" title="Multivector">Multivector</a>)</li> <li><a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a> (<a href="/wiki/Polynomial" title="Polynomial">Polynomial</a>)</li> <li><a href="/wiki/Quotient_object" class="mw-redirect" title="Quotient object">Quotient object</a> (<a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a>, ...)</li> <li><a href="/wiki/Symmetric_algebra" title="Symmetric algebra">Symmetric algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li></ul> </div></td></tr><tr><th scope="row" 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