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[r]" accesskey="r"><span>Perubahan terbaru</span></a></li><li id="n-Artikel-pilihan" class="mw-list-item"><a href="/wiki/Wikipedia:Artikel_pilihan/Topik"><span>Artikel pilihan</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Peristiwa_terkini" title="Temukan informasi tentang peristiwa terkini"><span>Peristiwa terkini</span></a></li><li id="n-newpage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_baru"><span>Halaman baru</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_sembarang" title="Tampilkan sembarang halaman [x]" accesskey="x"><span>Halaman sembarang</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_istimewa"><span>Halaman istimewa</span></a></li> </ul> </div> </div> <div id="p-Komunitas" class="vector-menu mw-portlet mw-portlet-Komunitas" > <div class="vector-menu-heading"> Komunitas </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-Warung-Kopi" class="mw-list-item"><a href="/wiki/Wikipedia:Warung_Kopi"><span>Warung Kopi</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Portal:Komunitas" title="Tentang proyek, apa yang dapat Anda lakukan, di mana untuk mencari sesuatu"><span>Portal komunitas</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Bantuan:Isi" title="Tempat mencari bantuan."><span>Bantuan</span></a></li> </ul> </div> </div> <div id="p-Wikipedia" class="vector-menu mw-portlet mw-portlet-Wikipedia" > <div class="vector-menu-heading"> Wikipedia </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Perihal"><span>Tentang Wikipedia</span></a></li><li id="n-Pancapilar" class="mw-list-item"><a href="/wiki/Wikipedia:Pancapilar"><span>Pancapilar</span></a></li><li id="n-Kebijakan" class="mw-list-item"><a href="/wiki/Wikipedia:Kebijakan_dan_pedoman"><span>Kebijakan</span></a></li><li id="n-Hubungi-kami" class="mw-list-item"><a href="/wiki/Wikipedia:Hubungi_kami"><span>Hubungi kami</span></a></li><li id="n-Bak-pasir" class="mw-list-item"><a href="/wiki/Wikipedia:Bak_pasir"><span>Bak pasir</span></a></li> </ul> </div> </div> <div id="p-Bagikan" class="vector-menu mw-portlet mw-portlet-Bagikan emptyPortlet" > <div class="vector-menu-heading"> Bagikan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Halaman_Utama" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="Ensiklopedia Bebas" src="/static/images/mobile/copyright/wikipedia-tagline-id.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Istimewa:Pencarian" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cari di Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Pencarian</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Telusuri Wikipedia" aria-label="Telusuri Wikipedia" autocapitalize="sentences" title="Cari di Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Istimewa:Pencarian"> </div> <button class="cdx-button cdx-search-input__end-button">Cari</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Perkakas pribadi"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Tampilan" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Tampilan</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=id.wikipedia.org&uselang=id" class=""><span>Menyumbang</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Istimewa:Buat_akun&returnto=Aljabar+universal" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan" class=""><span>Buat akun baru</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Istimewa:Masuk_log&returnto=Aljabar+universal" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o" class=""><span>Masuk log</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Opsi lainnya" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas pribadi" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Perkakas pribadi</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu pengguna" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=id.wikipedia.org&uselang=id"><span>Menyumbang</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Buat_akun&returnto=Aljabar+universal" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Buat akun baru</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Masuk_log&returnto=Aljabar+universal" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Daftar isi" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Awal</div> </a> </li> <li id="toc-Ide_dasar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ide_dasar"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Ide dasar</span> </div> </a> <button aria-controls="toc-Ide_dasar-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Ide dasar</span> </button> <ul id="toc-Ide_dasar-sublist" class="vector-toc-list"> <li id="toc-Persamaan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Persamaan"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Persamaan</span> </div> </a> <ul id="toc-Persamaan-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Varietas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Varietas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Varietas</span> </div> </a> <button aria-controls="toc-Varietas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Varietas</span> </button> <ul id="toc-Varietas-sublist" class="vector-toc-list"> <li id="toc-Contoh" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contoh"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Contoh</span> </div> </a> <ul id="toc-Contoh-sublist" class="vector-toc-list"> <li id="toc-Grup" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Grup"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Grup</span> </div> </a> <ul id="toc-Grup-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contoh_lainnya" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Contoh_lainnya"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Contoh lainnya</span> </div> </a> <ul id="toc-Contoh_lainnya-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Konstruksi_dasar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Konstruksi_dasar"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Konstruksi dasar</span> </div> </a> <ul id="toc-Konstruksi_dasar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beberapa_teorema_dasar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Beberapa_teorema_dasar"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Beberapa teorema dasar</span> </div> </a> <ul id="toc-Beberapa_teorema_dasar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Motivasi_dan_Aplikasi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Motivasi_dan_Aplikasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Motivasi dan Aplikasi</span> </div> </a> <button aria-controls="toc-Motivasi_dan_Aplikasi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Motivasi dan Aplikasi</span> </button> <ul id="toc-Motivasi_dan_Aplikasi-sublist" class="vector-toc-list"> <li id="toc-Masalah_kepuasan_kendala" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Masalah_kepuasan_kendala"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Masalah kepuasan kendala</span> </div> </a> <ul id="toc-Masalah_kepuasan_kendala-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sejarah" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sejarah"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sejarah</span> </div> </a> <ul id="toc-Sejarah-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Catatan_kaki" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Catatan_kaki"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Catatan kaki</span> </div> </a> <ul id="toc-Catatan_kaki-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Referensi</span> </div> </a> <ul id="toc-Referensi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pranala_luar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pranala_luar"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Pranala luar</span> </div> </a> <ul id="toc-Pranala_luar-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="Daftar isi" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Daftar Isi" > <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="Gulingkan daftar isi" > <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">Gulingkan daftar isi</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">Aljabar universal</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Pergi ke artikel dalam bahasa lain. Terdapat 33 bahasa" > <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-33" 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">33 bahasa</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%AC%D8%A8%D8%B1_%D8%B4%D8%A7%D9%85%D9%84" title="جبر شامل – Arab" lang="ar" hreflang="ar" data-title="جبر شامل" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Универсална алгебра – Bulgaria" lang="bg" hreflang="bg" data-title="Универсална алгебра" data-language-autonym="Български" data-language-local-name="Bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_universal" title="Àlgebra universal – Katalan" lang="ca" hreflang="ca" data-title="Àlgebra universal" data-language-autonym="Català" data-language-local-name="Katalan" 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/Univerz%C3%A1ln%C3%AD_algebra" title="Univerzální algebra – Ceko" lang="cs" hreflang="cs" data-title="Univerzální algebra" data-language-autonym="Čeština" data-language-local-name="Ceko" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B0%D0%BB%D0%BB%C4%83_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Универсаллă алгебра – Chuvash" lang="cv" hreflang="cv" data-title="Универсаллă алгебра" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Universelle_Algebra" title="Universelle Algebra – Jerman" lang="de" hreflang="de" data-title="Universelle Algebra" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Universal_algebra" title="Universal algebra – Inggris" lang="en" hreflang="en" data-title="Universal algebra" data-language-autonym="English" data-language-local-name="Inggris" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_universal" title="Álgebra universal – Spanyol" lang="es" hreflang="es" data-title="Álgebra universal" data-language-autonym="Español" data-language-local-name="Spanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Universaalalgebra" title="Universaalalgebra – Estonia" lang="et" hreflang="et" data-title="Universaalalgebra" data-language-autonym="Eesti" data-language-local-name="Estonia" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%AC%D9%87%D8%A7%D9%86%DB%8C" title="جبر جهانی – Persia" lang="fa" hreflang="fa" data-title="جبر جهانی" data-language-autonym="فارسی" data-language-local-name="Persia" 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/Alg%C3%A8bre_universelle" title="Algèbre universelle – Prancis" lang="fr" hreflang="fr" data-title="Algèbre universelle" data-language-autonym="Français" data-language-local-name="Prancis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%90%D7%95%D7%A0%D7%99%D7%91%D7%A8%D7%A1%D7%9C%D7%99%D7%AA" title="אלגברה אוניברסלית – Ibrani" lang="he" hreflang="he" data-title="אלגברה אוניברסלית" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Alj%C3%A8b_iniv%C3%A8s%C3%A8l" title="Aljèb inivèsèl – Kreol Haiti" lang="ht" hreflang="ht" data-title="Aljèb inivèsèl" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Kreol Haiti" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Algebra_universal" title="Algebra universal – Interlingua" lang="ia" hreflang="ia" data-title="Algebra universal" 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/Algebra_universale" title="Algebra universale – Italia" lang="it" hreflang="it" data-title="Algebra universale" data-language-autonym="Italiano" data-language-local-name="Italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%99%AE%E9%81%8D%E4%BB%A3%E6%95%B0%E5%AD%A6" title="普遍代数学 – Jepang" lang="ja" hreflang="ja" data-title="普遍代数学" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A3%E1%83%9C%E1%83%98%E1%83%95%E1%83%94%E1%83%A0%E1%83%A1%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%90%E1%83%9A%E1%83%92%E1%83%94%E1%83%91%E1%83%A0%E1%83%90" title="უნივერსალური ალგებრა – Georgia" lang="ka" hreflang="ka" data-title="უნივერსალური ალგებრა" data-language-autonym="ქართული" data-language-local-name="Georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B4%ED%8E%B8%EB%8C%80%EC%88%98%ED%95%99" title="보편대수학 – Korea" lang="ko" hreflang="ko" data-title="보편대수학" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Algebra_universal" title="Algebra universal – Melayu" lang="ms" hreflang="ms" data-title="Algebra universal" data-language-autonym="Bahasa Melayu" data-language-local-name="Melayu" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Universele_algebra" title="Universele algebra – Belanda" lang="nl" hreflang="nl" data-title="Universele algebra" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_uniwersalna" title="Algebra uniwersalna – Polski" lang="pl" hreflang="pl" data-title="Algebra uniwersalna" data-language-autonym="Polski" data-language-local-name="Polski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B9%D8%A7%D9%84%D9%85%DB%8C_%D8%A7%D9%84%D8%AC%D8%A8%D8%B1%D8%A7" title="عالمی الجبرا – Western Punjabi" lang="pnb" hreflang="pnb" data-title="عالمی الجبرا" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_universal" title="Álgebra universal – Portugis" lang="pt" hreflang="pt" data-title="Álgebra universal" data-language-autonym="Português" data-language-local-name="Portugis" 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/Algebr%C4%83_universal%C4%83" title="Algebră universală – Rumania" lang="ro" hreflang="ro" data-title="Algebră universală" data-language-autonym="Română" data-language-local-name="Rumania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Универсальная алгебра – Rusia" lang="ru" hreflang="ru" data-title="Универсальная алгебра" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Universell_algebra" title="Universell algebra – Swedia" lang="sv" hreflang="sv" data-title="Universell algebra" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Evrensel_cebir" title="Evrensel cebir – Turki" lang="tr" hreflang="tr" data-title="Evrensel cebir" data-language-autonym="Türkçe" data-language-local-name="Turki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A3%D0%BD%D1%96%D0%B2%D0%B5%D1%80%D1%81%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Універсальна алгебра – Ukraina" lang="uk" hreflang="uk" data-title="Універсальна алгебра" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B9%D8%A7%D9%84%D9%85%DB%8C_%D8%A7%D9%84%D8%AC%D8%A8%D8%B1%D8%A7" title="عالمی الجبرا – Urdu" lang="ur" hreflang="ur" data-title="عالمی الجبرا" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1i_s%E1%BB%91_ph%E1%BB%95_d%E1%BB%A5ng" title="Đại số phổ dụng – Vietnam" lang="vi" hreflang="vi" data-title="Đại số phổ dụng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Alhebra_unibersal" title="Alhebra unibersal – Warai" lang="war" hreflang="war" data-title="Alhebra unibersal" data-language-autonym="Winaray" data-language-local-name="Warai" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B3%9B%E4%BB%A3%E6%95%B0" title="泛代数 – Tionghoa" lang="zh" hreflang="zh" data-title="泛代数" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B3%9B%E4%BB%A3%E6%95%B8" title="泛代數 – Kanton" lang="yue" hreflang="yue" data-title="泛代數" data-language-autonym="粵語" data-language-local-name="Kanton" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1636734#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</a></span></div> </div> </div> </div> </header> 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class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Tampilan</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sembunyikan</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="id" dir="ltr"><p><b>Aljabar universal</b> (kadang disebut <b>aljabar umum</b>) adalah bidang <a href="/wiki/Matematika" title="Matematika">matematika</a> yang mempelajari <a href="/wiki/Struktur_aljabar" title="Struktur aljabar">struktur aljabar</a> itu sendiri, bukan contoh ("model") dari struktur aljabar. Misalnya, daripada mengambil <a href="/wiki/Grup_(matematika)" title="Grup (matematika)">grup</a> tertentu sebagai objek studi, dalam aljabar universal seseorang mengambil <a href="/wiki/Kelas_grup" title="Kelas grup">kelas grup</a> sebagai objek studi. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Ide_dasar">Ide dasar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=1" title="Sunting bagian: Ide dasar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=1" title="Sunting kode sumber bagian: Ide dasar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalam aljabar universal, sebuah <b>aljabar</b> (atau <b>aljabar <a href="/wiki/Struktur_(logika_matematika)" title="Struktur (logika matematika)">struktur</a></b>) adalah <a href="/wiki/Himpunan_(matematika)" title="Himpunan (matematika)">himpunan</a> <i>A</i> bersama dengan kumpulan operasi pada <i>A</i>. <b><i>n</i>-<a href="/wiki/Ariti" title="Ariti">ary</a> <a href="/wiki/Operasi_(matematika)" title="Operasi (matematika)">operasi</a></b> pada <i>A</i> adalah <a href="/wiki/Fungsi_(matematika)" title="Fungsi (matematika)">fungsi</a> yang membutuhkan <i> n </i> elemen <i> A </i> dan mengembalikan satu elemen <i> A </i>. Jadi, operasi 0-ary (atau <i> operasi nullari </i>) dapat direpresentasikan secara sederhana sebagai elemen <i> A </i>, atau <i> <a href="/wiki/Konstanta_(matematika)" title="Konstanta (matematika)">konstanta</a> </i>, sering dilambangkan dengan surat seperti <i> a </i>. Operasi 1-ari (atau <i> <a href="/wiki/Operasi_uner" title="Operasi uner">operasi uner</a> </i>) hanyalah fungsi dari <i> A </i> hingga <i> A </i>, sering dilambangkan dengan simbol yang ditempatkan di depan argumennya, seperti ~<i>x</i>. Operasi 2-ary (atau <i> <a href="/wiki/Operasi_biner" title="Operasi biner">operasi biner</a> </i>) sering dilambangkan dengan simbol yang ditempatkan di antara argumennya, seperti <i> x </i> ∗ <i> y </i>. Operasi <i> <a href="/wiki/Ariti" title="Ariti">ariti</a> </i> yang lebih tinggi atau tidak ditentukan biasanya dilambangkan dengan simbol fungsi, dengan argumen ditempatkan dalam tanda kurung dan dipisahkan dengan koma, seperti <i>f</i>(<i>x</i>,<i>y</i>,<i>z</i>) atau <i>f</i>(<i>x</i><sub>1</sub>,...,<i>x</i><sub><i>n</i></sub>). Beberapa peneliti mengizinkan operasi <a href="/wiki/Infiniter" class="mw-redirect" title="Infiniter">infiniter</a>, seperti <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 \textstyle \bigwedge _{\alpha \in J}x_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \bigwedge _{\alpha \in J}x_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810c74faa30a3f92b1b4110818856ff32da9175d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.358ex; height:3.009ex;" alt="{\displaystyle \textstyle \bigwedge _{\alpha \in J}x_{\alpha }}" /></span> di mana <i> J </i> adalah <a href="/wiki/Himpunan_indeks" title="Himpunan indeks">himpunan indeks</a> tak terhingga, sehingga mengarah ke teori aljabar <a href="/wiki/Kisi_kompleks" class="mw-redirect" title="Kisi kompleks">kisi kompleks</a>. Maka, salah satu cara untuk membicarakan aljabar adalah dengan menyebutnya sebagai <a href="/w/index.php?title=Garis_Besar_struktur_aljabar&action=edit&redlink=1" class="new" title="Garis Besar struktur aljabar (halaman belum tersedia)">Aljabar Jenis Tertentu</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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" /></span>, dimana <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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" /></span> adalah urutan bilangan asli yang mewakili aritas operasi aljabar. </p> <div class="mw-heading mw-heading3"><h3 id="Persamaan">Persamaan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=2" title="Sunting bagian: Persamaan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=2" title="Sunting kode sumber bagian: Persamaan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Setelah operasi ditentukan, sifat aljabar selanjutnya ditentukan oleh <a href="/wiki/Aksioma" title="Aksioma">aksioma</a>, yang dalam aljabar universal sering mengambil bentuk <a href="/wiki/Identitas_(matematika)#Logika_dan_aljabar_universal" title="Identitas (matematika)">identitas</a>, atau <b>hukum persamaan.</b> Contohnya adalah aksioma <a href="/wiki/Asosiatif" class="mw-redirect" title="Asosiatif">asosiatif</a> untuk operasi biner, yang diberikan oleh persamaan <i> x </i> ∗ (<i> y </i> ∗ <i> z </i>) = (<i> x </i> ∗ <i> y </i>) ∗ <i> z </i>. </p> <div class="mw-heading mw-heading2"><h2 id="Varietas">Varietas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=3" title="Sunting bagian: Varietas" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=3" title="Sunting kode sumber bagian: Varietas"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r18844875">.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}</style><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/wiki/Varietas_(aljabar_universal)" title="Varietas (aljabar universal)">Varietas (aljabar universal)</a></div> <p>Kumpulan struktur aljabar yang ditentukan oleh identitas disebut sebagai <a href="/wiki/Varietas_(aljabar_universal)" title="Varietas (aljabar universal)">variasi</a> atau <b>kelas persamaan</b>. Beberapa penulis menganggap varietas sebagai fokus utama aljabar universal.<sup class="noprint Inline-Template"><span title="Kalimat yang diikuti tag ini membutuhkan rujukan. since Desember 2020" style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Kutip_sumber_tulisan" title="Wikipedia:Kutip sumber tulisan">butuh rujukan</a></i>]</span></sup> </p><p>Membatasi studi seseorang pada varietas mengesampingkan: </p> <ul><li><a href="/wiki/Kuantifikasi_(logika)" class="mw-redirect" title="Kuantifikasi (logika)">Kuantifikasi</a>, termasuk <a href="/wiki/Kuantifikasi_universal" class="mw-redirect" title="Kuantifikasi universal">kuantifikasi universal</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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \forall }" /></span>) kecuali sebelum persamaan, dan <a href="/wiki/Kuantifikasi_eksistensial" title="Kuantifikasi eksistensial">kuantifikasi eksistensial</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 \exists }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ed842b6b90b2fdd825320cf8e5265fa937b583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }" /></span>), serta <a href="/wiki/Logika_predikat" class="mw-redirect" title="Logika predikat">logika predikat</a></li> <li><a href="/wiki/Hubungan_finiter" class="mw-redirect" title="Hubungan finiter">hubungan</a> selain persamaan, khususnya <a href="/wiki/Pertidaksamaan_(matematika)" class="mw-redirect" title="Pertidaksamaan (matematika)">pertidaksamaan</a>, keduanya <span class="nowrap"><i>a</i> ≠ <i>b</i></span> dan <a href="/wiki/Teori_order" title="Teori order">relasi order</a></li></ul> <p>Studi tentang kelas persamaan dapat dilihat sebagai cabang khusus dari <a href="/wiki/Teori_model" title="Teori model">teori model</a>, biasanya berurusan dengan struktur yang hanya memiliki operasi (yaitu <a href="/wiki/Tanda_tangan_(logika)" class="mw-redirect" title="Tanda tangan (logika)">jenis</a> dapat memiliki simbol untuk fungsi tetapi tidak untuk <a href="/wiki/Relasi_finiter" title="Relasi finiter">hubungan</a> selain persamaan), dan di mana bahasa yang digunakan untuk berbicara tentang struktur ini menggunakan persamaan saja. </p><p>Tidak semua <a href="/wiki/Struktur_aljabar" title="Struktur aljabar">struktur aljabar</a> dalam arti yang lebih luas termasuk dalam lingkup ini. Misalnya, <a href="/wiki/Grup_terurut" title="Grup terurut">grup terurut</a> melibatkan relasi pengurutan, jadi tidak akan termasuk dalam cakupan ini. </p><p>Kelas dari <a href="/wiki/Bidang_(matematika)" class="mw-redirect" title="Bidang (matematika)">bidang</a> bukanlah kelas persamaan karena tidak ada tipe (atau "tanda tangan") di mana semua hukum bidang dapat dituliskan sebagai persamaan (inversi elemen didefinisikan untuk semua elemen <i> bukan nol </i> dalam bidang, jadi inversi tidak dapat ditambahkan ke jenis). </p><p>Satu keuntungan dari pembatasan ini adalah bahwa struktur yang dipelajari dalam aljabar universal dapat didefinisikan dalam <a href="/wiki/Teori_kategori" title="Teori kategori">kategori</a> yang memiliki <i>hingga <a href="/wiki/Produk_(teori_kategori)" title="Produk (teori kategori)">produk</a></i>. Misalnya, <a href="/wiki/Grup_topologi" title="Grup topologi">grup topologi</a> hanyalah grup dalam kategori <a href="/wiki/Ruang_topologi" class="mw-redirect" title="Ruang topologi">ruang topologi</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Contoh">Contoh</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=4" title="Sunting bagian: Contoh" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=4" title="Sunting kode sumber bagian: Contoh"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sebagian besar sistem aljabar matematika biasa adalah contoh varietas, tetapi tidak selalu dengan cara yang jelas, karena definisi biasa sering kali melibatkan penghitungan atau ketidaksamaan. </p> <div class="mw-heading mw-heading4"><h4 id="Grup">Grup</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=5" title="Sunting bagian: Grup" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=5" title="Sunting kode sumber bagian: Grup"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sebagai contoh, perhatikan definisi dari sebuah <a href="/wiki/Grup_(matematika)" title="Grup (matematika)">grup</a>. Biasanya sebuah grup didefinisikan dalam operasi biner tunggal ∗, tunduk pada aksioma: </p> <ul><li><a href="/wiki/Asosiatif" class="mw-redirect" title="Asosiatif">Asosiatif</a> (seperti pada <a href="#Persamaan">bagian sebelumnya</a>): <i>x</i> ∗ (<i>y</i> ∗ <i>z</i>)  =  (<i>x</i> ∗ <i>y</i>) ∗ <i>z</i>;   secara resmi: ∀<i>x</i>,<i>y</i>,<i>z</i>. <i>x</i>∗(<i>y</i>∗<i>z</i>)=(<i>x</i>∗<i>y</i>)∗<i>z</i>.</li> <li><a href="/wiki/Elemen_identitas" class="mw-redirect" title="Elemen identitas">Elemen identitas</a>: Terdapat elemen <i> e </i> sedemikian rupa sehingga untuk setiap elemen <i> x </i>, satu memiliki <i>e</i> ∗ <i>x</i>  =  <i>x</i>  =  <i>x</i> ∗ <i>e</i>;   secara resmi: ∃<i>e</i> ∀<i>x</i>. <i>e</i>∗<i>x</i>=<i>x</i>=<i>x</i>∗<i>e</i>.</li> <li><a href="/wiki/Elemen_invers" title="Elemen invers">Elemen invers</a>: Elemen identitas mudah dilihat sebagai unik, dan biasanya dilambangkan dengan <i> e </i>. Kemudian untuk setiap <i> x </i>, terdapat elemen <i> i </i> sedemikian rupa <i>x</i> ∗ <i>i</i>  =  <i>e</i>  =  <i>i</i> ∗ <i>x</i>;   secara resmi: ∀<i>x</i> ∃<i>i</i>. <i>x</i>∗<i>i</i>=<i>e</i>=<i>i</i>∗<i>x</i>.</li></ul> <p>(Beberapa penulis juga menggunakan aksioma "<a href="/w/index.php?title=Penutupan_(matematika)&action=edit&redlink=1" class="new" title="Penutupan (matematika) (halaman belum tersedia)">penutupan</a>" bahwa <i> x </i> ∗ <i> y </i> milik <i> A </i> setiap kali <i> x </i> dan <i> y </i> lakukan, tapi di sini ini sudah tersirat dengan merumuskan biner ∗. </p><p>Definisi grup ini tidak langsung sesuai dengan sudut pandang aljabar universal, karena aksioma elemen identitas dan inversi tidak dinyatakan murni dalam istilah hukum persamaan yang memegang secara universal elemen "untuk semua ...", tetapi juga melibatkan bilangan eksistensial. Aksioma grup dapat diutarakan sebagai persamaan yang dikuantifikasi secara universal dengan menentukan, selain operasi biner ∗, operasi nullary <i> e </i> dan operasi unary ~, dengan ~<i>x</i> biasanya ditulis sebagai <i>x</i><sup>−1</sup>. Aksioma menjadi: </p> <ul><li>Asosiatif: <span class="nowrap"><i>x</i> ∗ (<i>y</i> ∗ <i>z</i>)  = </span> <span class="nowrap">(<i>x</i> ∗ <i>y</i>) ∗ <i>z</i></span>.</li> <li>Elemen identitas: <span class="nowrap"><i>e</i> ∗ <i>x</i>  = </span> <span class="nowrap"><i>x</i>  = </span> <span class="nowrap"><i>x</i> ∗ <i>e</i></span>; secara resmi: ∀<i>x</i>. <i>e</i>∗<i>x</i>=<i>x</i>=<i>x</i>∗<i>e</i>.</li> <li>Elemen invers: <span class="nowrap"><i>x</i> ∗ (~<i>x</i>)  = </span> <span class="nowrap"><i>e</i>  = </span> <span class="nowrap">(~<i>x</i>) ∗ <i>x</i></span>   secara resmi: ∀<i>x</i>. <i>x</i>∗~<i>x</i>=<i>e</i>=~<i>x</i>∗<i>x</i>.</li></ul> <p>Untuk meringkas, definisi biasa memiliki: </p> <ul><li>operasi biner tunggal (<a href="/wiki/Tanda_tangan_(logika)" class="mw-redirect" title="Tanda tangan (logika)">tanda tangan</a> (2))</li> <li>1 hukum persamaan (asosiatif)</li> <li>2 hukum terukur (identitas dan invers)</li></ul> <p>sedangkan definisi aljabar universal memiliki: </p> <ul><li>3 operasi: satu biner, satu unary, dan satu nullary (<a href="/wiki/Tanda_tangan_(logika)" class="mw-redirect" title="Tanda tangan (logika)">tanda tangan</a> (2,1,0))</li> <li>3 hukum persamaan (asosiativitas, identitas, dan kebalikan)</li> <li>tidak ada hukum terukur (kecuali bilangan universal terluar, yang diperbolehkan dalam varietas)</li></ul> <p>Poin utamanya adalah bahwa operasi tambahan tidak menambahkan informasi, tetapi mengikuti secara unik dari definisi grup yang biasa. Meskipun definisi biasa tidak secara unik menentukan elemen identitas <i> e </i>, latihan mudah menunjukkannya unik, seperti halnya masing-masing <a href="/wiki/Elemen_invers" title="Elemen invers">elemen invers</a>. </p><p>Sudut pandang aljabar universal disesuaikan dengan baik dengan teori kategori. Misalnya, ketika mendefinisikan <a href="/wiki/Objek_grup" title="Objek grup">objek grup</a> dalam teori kategori, di mana objek yang dimaksud mungkin bukan himpunan, seseorang harus menggunakan hukum persamaan (yang masuk akal dalam kategori umum), daripada hukum terkuantifikasi (yang mengacu pada elemen individu). Selanjutnya, kebalikan dan identitas ditentukan sebagai morfisme dalam kategori. Misalnya, dalam sebuah <a href="/wiki/Grup_topologi" title="Grup topologi">grup topologi</a>, invers tidak hanya harus ada berdasarkan elemen, tetapi harus memberikan pemetaan berkelanjutan (morfisme). Beberapa penulis juga mensyaratkan peta identitas menjadi <a href="/wiki/Inklusi_tertutup" class="mw-redirect" title="Inklusi tertutup">inklusi tertutup</a> (<a href="/wiki/Kofibrasi" title="Kofibrasi">kofibrasi</a>). </p> <div class="mw-heading mw-heading4"><h4 id="Contoh_lainnya">Contoh lainnya</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=6" title="Sunting bagian: Contoh lainnya" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=6" title="Sunting kode sumber bagian: Contoh lainnya"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kebanyakan struktur aljabar adalah contoh aljabar universal. </p> <ul><li><a href="/wiki/Gelanggang_(matematika)" title="Gelanggang (matematika)">Gelanggang</a>, <a href="/wiki/Semigrup" title="Semigrup">semigrup</a>, <a href="/wiki/Kuasigrup" title="Kuasigrup">kuasigrup</a>, <a href="/wiki/Grupoid" title="Grupoid">grupoid</a>, <a href="/wiki/Magma_(matematika)" class="mw-redirect" title="Magma (matematika)">magma</a>, <a href="/wiki/Loop_(aljabar)" class="mw-redirect" title="Loop (aljabar)">loop</a>, dan lain-lain.</li> <li><a href="/wiki/Ruang_vektor" title="Ruang vektor">Ruang vektor</a> di atas bidang tetap dan <a href="/wiki/Modul_(matematika)" title="Modul (matematika)">modul</a> di atas gelanggang tetap adalah aljabar universal. Ini memiliki penjumlahan biner dan keluarga operator perkalian skalar unary, satu untuk setiap elemen bidang atau gelanggang.</li></ul> <p>Contoh aljabar relasional termasuk <a href="/wiki/Semikisi" class="mw-redirect" title="Semikisi">semikisi</a>, <a href="/wiki/Kisi_(urutan)" class="mw-redirect" title="Kisi (urutan)">kisi</a>, dan <a href="/wiki/Aljabar_Boolean" class="mw-redirect" title="Aljabar Boolean">Aljabar Boolean</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Konstruksi_dasar">Konstruksi dasar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=7" title="Sunting bagian: Konstruksi dasar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=7" title="Sunting kode sumber bagian: Konstruksi dasar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kami berasumsi bahwa tipe, <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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" /></span>, telah diperbaiki. Kemudian ada tiga konstruksi dasar dalam aljabar universal: citra homomorfik, subaljabar, dan hasil kali. </p><p><a href="/wiki/Homomorfisme" title="Homomorfisme">Homomorfisme</a> antara dua aljabar <i> A </i> dan <i> B </i> adalah <a href="/wiki/Fungsi_(matematika)" title="Fungsi (matematika)">fungsi</a> <i> h </i>: <i>A</i> → <i>B</i> dari himpunan A ke himpunan B sedemikian rupa, untuk setiap operasi <i>f</i><sub><i>A</i></sub> dari A dan yang sesuai <i>f</i><sub><i>B</i></sub> dari B (ariti, ditulis, <i> n </i>), <i>h</i>(<i>f</i><sub><i>A</i></sub>(<i>x</i><sub>1</sub>,...,<i>x</i><sub><i>n</i></sub>)) = <i>f</i><sub><i>B</i></sub>(<i>h</i>(<i>x</i><sub>1</sub>),...,<i>h</i>(<i>x</i><sub><i>n</i></sub>)). (Kadang-kadang subskrip pada <i> f </i> dihapus ketika jelas dari konteks dari mana aljabar fungsi tersebut berasal.) Misalnya, jika <i> e </i> adalah konstanta (operasi nullary), maka <i>h</i>(<i>e</i><sub><i>A</i></sub>) = <i>e</i><sub><i>B</i></sub>. If ~ adalah operasi unari, maka <i>h</i>(~<i>x</i>) = ~<i>h</i>(<i>x</i>). Jika ∗ adalah operasi biner, maka <i>h</i>(<i>x</i> ∗ <i>y</i>) = <i>h</i>(<i>x</i>) ∗ <i>h</i>(<i>y</i>). Dan seterusnya. Beberapa hal yang dapat dilakukan dengan homomorfisme, serta definisi jenis khusus homomorfisme tertentu, terdaftar di bawah entri <a href="/wiki/Homomorfisme" title="Homomorfisme">Homomorfisme</a>. Secara khusus, kita dapat mengambil citra homomorfik dari sebuah aljabar, <i>h</i>(<i>A</i>). </p><p>Subaljabar dari <i> A </i> adalah himpunan bagian dari <i> A </i> yang ditutup di bawah semua operasi <i> A </i>. Produk dari beberapa himpunan struktur aljabar adalah <a href="/wiki/Produk_Kartesius" class="mw-redirect" title="Produk Kartesius">produk Kartesius</a> dari himpunan dengan operasi yang ditentukan secara koordinat. </p> <div class="mw-heading mw-heading2"><h2 id="Beberapa_teorema_dasar">Beberapa teorema dasar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=8" title="Sunting bagian: Beberapa teorema dasar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=8" title="Sunting kode sumber bagian: Beberapa teorema dasar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Teorema_isomorfisme" title="Teorema isomorfisme">Teorema isomorfisme</a>, yang mencakup teorema isomorfisme <a href="/wiki/Grup_(matematika)" title="Grup (matematika)">grup</a>, <a href="/wiki/Gelanggang_(matematika)" title="Gelanggang (matematika)">gelanggang</a>, <a href="/wiki/Modul_(matematika)" title="Modul (matematika)">modul</a>, dll.</li> <li><a href="/wiki/Varietas_(aljabar_universal)#Teorema_Birkhoff" title="Varietas (aljabar universal)">Teorema HSP Birkhoff</a>, yang menyatakan bahwa kelas aljabar adalah <a href="/w/index.php?title=Ragam_(aljabar_universal)&action=edit&redlink=1" class="new" title="Ragam (aljabar universal) (halaman belum tersedia)">ragam</a> jika dan hanya jika ditutup di bawah gambar homomorfik, subaljabar, dan produk langsung sembarang.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Motivasi_dan_Aplikasi">Motivasi dan Aplikasi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=9" title="Sunting bagian: Motivasi dan Aplikasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=9" title="Sunting kode sumber bagian: Motivasi dan Aplikasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r26333525">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Tanpa_referensi plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/Berkas:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Bab atau bagian ini <b>tidak memiliki <a href="/wiki/Wikipedia:Kutip_sumber_tulisan" title="Wikipedia:Kutip sumber tulisan">referensi</a> atau <a href="/wiki/Wikipedia:Sumber_tepercaya" title="Wikipedia:Sumber tepercaya">sumber tepercaya</a></b> sehingga isinya tidak bisa <a href="/wiki/Wikipedia:Pemastian" title="Wikipedia:Pemastian">dipastikan</a>.<span class="hide-when-compact"> Tolong bantu <span class="plainlinks"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Aljabar_universal&action=edit">perbaiki artikel ini</a></span> dengan menambahkan referensi yang layak. <a href="/wiki/Wikipedia:Warung_Kopi/Kembangkan#Bab_atau_bagian" title="Wikipedia:Warung Kopi/Kembangkan">Bab atau bagian</a> ini akan dihapus bila tidak tersedia referensi ke <a href="/wiki/Wikipedia:Sumber_tepercaya" title="Wikipedia:Sumber tepercaya">sumber tepercaya</a> dalam bentuk <a href="/wiki/Bantuan:Catatan_kaki" title="Bantuan:Catatan kaki">catatan kaki</a> atau <a href="/wiki/Wikipedia:Pranala_luar" title="Wikipedia:Pranala luar">pranala luar</a>.</span></div></td></tr></tbody></table> <p>Selain pendekatan pemersatu, aljabar universal juga memberikan teorema yang dalam serta contoh dan contoh tandingan yang penting. Ini memberikan kerangka kerja yang berguna bagi mereka yang berniat untuk memulai studi kelas baru aljabar. </p><p> Ini dapat memungkinkan penggunaan metode yang ditemukan untuk beberapa kelas aljabar tertentu ke kelas aljabar lain, dengan menyusun kembali metode dalam istilah aljabar universal (jika mungkin), dan kemudian menafsirkannya seperti yang diterapkan pada kelas lain. Ini juga memberikan klarifikasi konseptual; sebagai J.D.H. Smith mengatakannya,: <style data-mw-deduplicate="TemplateStyles:r21477316">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 40px}.mw-parser-output .templatequote .templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0}</style></p><blockquote class="templatequote"><p><i>"Apa yang terlihat berantakan dan rumit dalam kerangka kerja tertentu mungkin berubah menjadi sederhana dan jelas dalam kerangka umum yang tepat."</i></p><div class="templatequotecite">— <cite>J.D.H. Smith</cite></div></blockquote> <p>Secara khusus, aljabar universal dapat diterapkan untuk mempelajari <a href="/wiki/Monoid" title="Monoid">monoid</a>, <a href="/wiki/Gelanggang_(aljabar)" class="mw-redirect" title="Gelanggang (aljabar)">gelanggang</a>, dan <a href="/wiki/Kisi_(urutan)" class="mw-redirect" title="Kisi (urutan)">kisi</a>. Sebelum aljabar universal muncul, banyak teorema (terutama <a href="/wiki/Teorema_isomorfisme" title="Teorema isomorfisme">teorema isomorfisme</a>) telah dibuktikan secara terpisah di semua kelas ini, tetapi dengan aljabar universal, mereka dapat dibuktikan sekali dan untuk selamanya untuk setiap jenis sistem aljabar. </p><p>Makalah 1956 oleh Higgins yang dirujuk di bawah ini telah ditindaklanjuti dengan baik untuk kerangka kerjanya untuk berbagai sistem aljabar tertentu, sementara makalahnya pada tahun 1963 terkenal karena pembahasannya tentang aljabar dengan operasi yang hanya didefinisikan sebagian, contoh khas untuk ini adalah kategori dan grupoid. Hal ini mengarah pada subjek <a href="/w/index.php?title=Aljabar_berdimensi_lebih_tinggi&action=edit&redlink=1" class="new" title="Aljabar berdimensi lebih tinggi (halaman belum tersedia)">aljabar berdimensi lebih tinggi</a> yang dapat didefinisikan sebagai studi teori aljabar dengan operasi parsial yang domainnya ditentukan dalam kondisi geometris. Contoh penting dari ini adalah berbagai bentuk kategori dan groupoids berdimensi lebih tinggi. </p> <div class="mw-heading mw-heading3"><h3 id="Masalah_kepuasan_kendala">Masalah kepuasan kendala</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=10" title="Sunting bagian: Masalah kepuasan kendala" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=10" title="Sunting kode sumber bagian: Masalah kepuasan kendala"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875" /><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Masalah_kepuasan_kendala&action=edit&redlink=1" class="new" title="Masalah kepuasan kendala (halaman belum tersedia)">Masalah kepuasan kendala</a></div> <p>Aljabar universal menyediakan bahasa alami untuk <a href="/w/index.php?title=Masalah_kepuasan_kendala&action=edit&redlink=1" class="new" title="Masalah kepuasan kendala (halaman belum tersedia)"> masalah kepuasan kendala (CSP)</a>. CSP merujuk ke kelas penting dari masalah komputasi yang diberi aljabar relasional <span class="texhtml mvar" style="font-style:italic;"> A</span> dan eksistensial <a href="/w/index.php?title=Kalimat_(logika_matematika)&action=edit&redlink=1" class="new" title="Kalimat (logika matematika) (halaman belum tersedia)">kalimat</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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span> di atas aljabar ini, pertanyaannya adalah mencari tahu apakah <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span> bisa dipenuhi di <span class="texhtml mvar" style="font-style:italic;"> A</span>. Aljabar <span class="texhtml mvar" style="font-style:italic;"> A</span> sering kali diperbaiki, sehingga <span class="texhtml" style="white-space: nowrap;">CSP<sub><i>A</i></sub></span> mengacu pada masalah yang instansinya hanya kalimat eksistensial <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span>. </p><p>Terbukti bahwa setiap masalah komputasi dapat dirumuskan sebagai <span class="texhtml" style="white-space: nowrap;">CSP<sub><i>A</i></sub></span> untuk beberapa aljabar <span class="texhtml mvar" style="font-style:italic;"> A</span>. </p><p>Sebagai contoh, masalah <a href="/w/index.php?title=Mewarnai_grafik&action=edit&redlink=1" class="new" title="Mewarnai grafik (halaman belum tersedia)"><i> n </i>-warna</a> dapat dinyatakan sebagai CSP dari aljabar <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 {\big (}\{0,1,\dots ,n-1\},\neq {\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big (}\{0,1,\dots ,n-1\},\neq {\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dafbdd5ead9420791050f88503b209742a0c056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.877ex; height:3.176ex;" alt="{\displaystyle {\big (}\{0,1,\dots ,n-1\},\neq {\big )}}" /></span>, i.e. sebuah aljabar dengan elemen <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> dan satu relasi, pertidaksamaan. </p><p>Dugaan dikotomi (dibuktikan pada April 2017) menyatakan bahwa jika <span class="texhtml mvar" style="font-style:italic;"> A</span> adalah aljabar berhingga, maka <span class="texhtml" style="white-space: nowrap;">CSP<sub><i>A</i></sub></span> bisa berupa <a href="/w/index.php?title=P_(kompleksitas)&action=edit&redlink=1" class="new" title="P (kompleksitas) (halaman belum tersedia)">P</a> atau <a href="/w/index.php?title=Kompleks-NP&action=edit&redlink=1" class="new" title="Kompleks-NP (halaman belum tersedia)">kompleks-NP</a>.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Sejarah">Sejarah</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=11" title="Sunting bagian: Sejarah" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=11" title="Sunting kode sumber bagian: Sejarah"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalam buku <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> <i> A Treatise on Universal Algebra, </i> yang diterbitkan pada tahun 1898, istilah <i> aljabar universal </i> pada dasarnya memiliki arti yang sama dengan yang dimilikinya saat ini. Whitehead memuji <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> dan <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> sebagai pencetus materi pelajaran, dan <a href="/w/index.php?title=James_Joseph_Sylvester&action=edit&redlink=1" class="new" title="James Joseph Sylvester (halaman belum tersedia)">James Joseph Sylvester</a> dengan menciptakan istilah itu sendiri.<sup id="cite_ref-Gratzer.1968_2-0" class="reference"><a href="#cite_note-Gratzer.1968-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference plainlinksneverexpand" style="white-space:nowrap"><sup>:v</sup></span> </p><p>Pada saat struktur seperti <a href="/wiki/Lie_aljabar" class="mw-redirect" title="Lie aljabar">Lie aljabar</a> s dan <a href="/w/index.php?title=Quaternion_hiperbolik&action=edit&redlink=1" class="new" title="Quaternion hiperbolik (halaman belum tersedia)">quaternion hiperbolik</a> s menarik perhatian pada kebutuhan untuk memperluas struktur aljabar di luar kelas perkalian asosiatif. Dalam sebuah ulasan <a href="/w/index.php?title=Alexander_Macfarlane&action=edit&redlink=1" class="new" title="Alexander Macfarlane (halaman belum tersedia)">Alexander Macfarlane</a> menulis: "Ide utama dari karya ini bukanlah penyatuan beberapa metode, atau generalisasi aljabar biasa untuk memasukkannya, melainkan studi komparatif dari beberapa struktur mereka."<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> Pada saat itu, aljabar logika <a href="/wiki/George_Boole" title="George Boole">George Boole</a> membuat tandingan yang kuat dengan aljabar bilangan biasa, sehingga istilah "universal" berfungsi untuk menenangkan perasaan tegang. </p><p>Karya awal Whitehead berusaha untuk menyatukan <a href="/wiki/Kuaternion" title="Kuaternion">kuaternion</a> (karena Hamilton), <a href="/w/index.php?title=Aljabar_Eksterior&action=edit&redlink=1" class="new" title="Aljabar Eksterior (halaman belum tersedia)">Ausdehnungslehre</a>, dan aljabar logika Boole. Whitehead menulis dalam bukunya: </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r21477316" /><blockquote class="templatequote"><p><i>"Aljabar semacam itu memiliki nilai intrinsik untuk studi terperinci yang terpisah; juga mereka layak untuk studi banding, demi cahaya dengan demikian dilemparkan pada teori umum penalaran simbolik, dan simbolisme aljabar pada khususnya. Studi banding perlu mengandaikan beberapa studi terpisah sebelumnya, perbandingan tidak mungkin tanpa pengetahuan."</i><sup id="cite_ref-Gratzer.1968_2-1" class="reference"><a href="#cite_note-Gratzer.1968-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></p><div class="templatequotecite">— <cite>Whitehead</cite></div></blockquote></dd></dl> <p>Whitehead, bagaimanapun, tidak memiliki hasil yang bersifat umum. Pekerjaan pada subjek itu minimal sampai awal 1930-an, ketika <a href="/w/index.php?title=Garrett_Birkhoff&action=edit&redlink=1" class="new" title="Garrett Birkhoff (halaman belum tersedia)">Garrett Birkhoff</a> dan <a href="/w/index.php?title=%C3%98ystein_Ore&action=edit&redlink=1" class="new" title="Øystein Ore (halaman belum tersedia)">Øystein Ore</a> mulai menerbitkan aljabar universal. Perkembangan dalam <a href="/w/index.php?title=Metamathematics&action=edit&redlink=1" class="new" title="Metamathematics (halaman belum tersedia)">metamathematics</a> dan <a href="/wiki/Teori_kategori" title="Teori kategori">teori kategori</a> pada 1940-an dan 1950-an memajukan bidang tersebut, terutama karya <a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a>, <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, <a href="/w/index.php?title=Andrzej_Mostowski&action=edit&redlink=1" class="new" title="Andrzej Mostowski (halaman belum tersedia)">Andrzej Mostowski</a>, dan siswa lain.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Pada periode antara 1935 dan 1950, sebagian besar makalah ditulis sesuai dengan yang disarankan oleh makalah Birkhoff, berhubungan dengan <a href="/wiki/Objek_bebas" title="Objek bebas">aljabar bebas</a>, kesesuaian dan kisi subaljabar, dan teorema homomorfisme. Meskipun perkembangan logika matematika telah memungkinkan penerapan aljabar, hal itu terjadi dengan lambat; hasil yang diterbitkan oleh <a href="/w/index.php?title=Anatoly_Maltsev&action=edit&redlink=1" class="new" title="Anatoly Maltsev (halaman belum tersedia)">Anatoly Maltsev</a> pada tahun 1940-an tidak diketahui karena perang. Kuliah Tarski di tahun 1950 <a href="/w/index.php?title=Kongres_Internasional_Matematikawan&action=edit&redlink=1" class="new" title="Kongres Internasional Matematikawan (halaman belum tersedia)">Kongres Internasional Matematikawan</a> di Cambridge mengantarkan periode baru di mana aspek model-teori dikembangkan, terutama oleh Tarski sendiri, serta C.C. Chang, <a href="/w/index.php?title=Leon_Henkin&action=edit&redlink=1" class="new" title="Leon Henkin (halaman belum tersedia)">Leon Henkin</a>, <a href="/wiki/Bjarni_J%C3%B3nsson" title="Bjarni Jónsson">Bjarni Jónsson</a>, <a href="/w/index.php?title=Roger_Lyndon&action=edit&redlink=1" class="new" title="Roger Lyndon (halaman belum tersedia)">Roger Lyndon</a>, dan lainnya. </p><p>Pada akhir 1950-an, <a href="/w/index.php?title=Edward_Marczewski&action=edit&redlink=1" class="new" title="Edward Marczewski (halaman belum tersedia)">Edward Marczewski</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> menekankan pentingnya aljabar bebas, yang mengarah ke publikasi lebih dari 50 makalah tentang teori aljabar aljabar bebas oleh Marczewski sendiri, bersama dengan <a href="/w/index.php?title=Jan_Mycielski&action=edit&redlink=1" class="new" title="Jan Mycielski (halaman belum tersedia)">Jan Mycielski</a>, Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski, <a href="/w/index.php?title=Kazimierz_Urbanik&action=edit&redlink=1" class="new" title="Kazimierz Urbanik (halaman belum tersedia)">K. Urbanik</a>, dan lainnya. </p><p>Dimulai dengan tesis <a href="/w/index.php?title=William_Lawvere&action=edit&redlink=1" class="new" title="William Lawvere (halaman belum tersedia)">William Lawvere</a> pada tahun 1963, teknik dari teori kategori telah menjadi penting dalam aljabar universal.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=12" title="Sunting bagian: Lihat pula" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=12" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span 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portalborder tright"> <ul> <li><span><span class="noviewer" typeof="mw:File"><a href="/wiki/Berkas:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span><a href="/wiki/Portal:Matematika" title="Portal:Matematika">Portal Matematika </a></span></li></ul></div> <ul><li><a href="/w/index.php?title=Aljabar_grafik&action=edit&redlink=1" class="new" title="Aljabar grafik (halaman belum tersedia)">Aljabar grafik</a></li> <li><a href="/w/index.php?title=Aljabar_istilah&action=edit&redlink=1" class="new" title="Aljabar istilah (halaman belum tersedia)">Aljabar istilah</a></li> <li><a href="/w/index.php?title=Klon_(aljabar)&action=edit&redlink=1" class="new" title="Klon (aljabar) (halaman belum tersedia)">Klon</a></li> <li><a href="/w/index.php?title=Geometri_aljabar_universal&action=edit&redlink=1" class="new" title="Geometri aljabar universal (halaman belum tersedia)">Geometri aljabar universal</a></li> <li><a href="/w/index.php?title=Aljabar_universal_sederhana&action=edit&redlink=1" class="new" title="Aljabar universal sederhana (halaman belum tersedia)">Aljabar universal sederhana</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Catatan_kaki">Catatan kaki</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=13" title="Sunting bagian: Catatan kaki" 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.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"><span class="citation Journal">Zhuk, Dmitriy (2017). "The Proof of CSP Dichotomy Conjecture".  <i><a href="/wiki/ArXiv" title="ArXiv">arΧiv</a>:<a href="https://arxiv.org/abs/1704.01914" class="extiw" title="arxiv:1704.01914">1704.01914</a></i> [cs.cc].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=The+Proof+of+CSP+Dichotomy+Conjecture&rft.atitle=&rft.aulast=Zhuk&rft.aufirst=Dmitriy&rft.au=Zhuk%2C%26%2332%3BDmitriy&rft.date=2017&rfr_id=info:sid/en.wikipedia.org:Aljabar_universal"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-Gratzer.1968-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gratzer.1968_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gratzer.1968_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">George Grätzer (1968). M.H. Stone and L. Nirenberg and S.S. Chern, ed. <i>Universal Algebra</i> (edisi ke-1st). Van Nostrand Co., Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Universal+Algebra&rft.edition=1st&rft.pub=Van+Nostrand+Co.%2C+Inc&rft.date=1968&rft.au=George+Gr%C3%A4tzer&rfr_id=info%3Asid%2Fid.wikipedia.org%3AAljabar+universal" class="Z3988"><span style="display:none;"> </span></span></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"><a href="/w/index.php?title=Alexander_Macfarlane&action=edit&redlink=1" class="new" title="Alexander Macfarlane (halaman belum tersedia)">Alexander Macfarlane</a> (1899) <a rel="nofollow" class="external text" href="https://archive.org/details/jstor-1626993">Review:<i>A Treatise on Universal Algebra</i> (pdf)</a>, <a href="/wiki/Science_(journal)" class="mw-redirect" title="Science (journal)">Science</a> 9: 324–8 via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Brainerd, Barron (Aug–Sep 1967) "Review of <i>Universal Algebra</i> by <a href="/w/index.php?title=P._M._Cohn&action=edit&redlink=1" class="new" title="P. M. Cohn (halaman belum tersedia)">P. M. Cohn</a>", <a href="/w/index.php?title=American_Mathematical_Monthly&action=edit&redlink=1" class="new" title="American Mathematical Monthly (halaman belum tersedia)">American Mathematical Monthly</a> 74(7): 878–880.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Marczewski, E. "A general scheme of the notions of independence in mathematics." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. <b>6</b> (1958), 731–736.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFLawvere1964" class="citation"><a href="/w/index.php?title=William_Lawvere&action=edit&redlink=1" class="new" title="William Lawvere (halaman belum tersedia)">Lawvere, William F.</a> (1964), <a rel="nofollow" class="external text" href="http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html"><i>Functorial Semantics of Algebraic Theories (PhD Thesis)</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functorial+Semantics+of+Algebraic+Theories+%28PhD+Thesis%29&rft.date=1964&rft.aulast=Lawvere&rft.aufirst=William+F.&rft_id=http%3A%2F%2Fwww.tac.mta.ca%2Ftac%2Freprints%2Farticles%2F5%2Ftr5abs.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3AAljabar+universal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Referensi">Referensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=14" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=14" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Bergman, George M., 1998. <i><a rel="nofollow" class="external text" href="http://math.berkeley.edu/~gbergman/245/">An Invitation to General Algebra and Universal Constructions</a></i> (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 398 pp. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-9655211-4-1" title="Istimewa:Sumber buku/0-9655211-4-1">0-9655211-4-1</a>.</li> <li>Birkhoff, Garrett, 1946. Universal algebra. <i>Comptes Rendus du Premier Congrès Canadien de Mathématiques</i>, University of Toronto Press, Toronto, pp. 310–326.</li> <li>Burris, Stanley N., and H.P. Sankappanavar, 1981. <i><a rel="nofollow" class="external text" href="http://www.thoralf.uwaterloo.ca/htdocs/ualg.html">A Course in Universal Algebra</a></i> Springer-Verlag. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/3-540-90578-2" title="Istimewa:Sumber buku/3-540-90578-2">3-540-90578-2</a> <i>Free online edition</i>.</li> <li>Cohn, Paul Moritz, 1981. <i>Universal Algebra</i>. Dordrecht, Netherlands: D.Reidel Publishing. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/90-277-1213-1" title="Istimewa:Sumber buku/90-277-1213-1">90-277-1213-1</a> <i>(First published in 1965 by Harper & Row)</i></li> <li>Freese, Ralph, and Ralph McKenzie, 1987. <i><a rel="nofollow" class="external text" href="http://www.math.hawaii.edu/~ralph/Commutator">Commutator Theory for Congruence Modular Varieties</a>, 1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-521-34832-3" title="Istimewa:Sumber buku/0-521-34832-3">0-521-34832-3</a>. Free online second edition</i>.</li> <li>Grätzer, George, 1968. <i>Universal Algebra</i> D. Van Nostrand Company, Inc.</li> <li>Higgins, P. J. <a rel="nofollow" class="external text" href="https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-6.3.366">Groups with multiple operators</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220307143342/https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-6.3.366">Diarsipkan</a> 2022-03-07 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.. Proc. London Math. Soc. (3) 6 (1956), 366–416.</li> <li>Higgins, P.J., Algebras with a scheme of operators. <i><a href="/w/index.php?title=Mathematische_Nachrichten&action=edit&redlink=1" class="new" title="Mathematische Nachrichten (halaman belum tersedia)">Mathematische Nachrichten</a></i> (27) (1963) 115–132.</li> <li>Hobby, David, and Ralph McKenzie, 1988. <i><a rel="nofollow" class="external text" href="http://www.ams.org/books/conm/076/">The Structure of Finite Algebras</a></i> American Mathematical Society. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-8218-3400-2" title="Istimewa:Sumber buku/0-8218-3400-2">0-8218-3400-2</a>. <i>Free online edition.</i></li> <li>Jipsen, Peter, and Henry Rose, 1992. <i><a rel="nofollow" class="external text" href="http://www1.chapman.edu/~jipsen/JipsenRoseVoL.html">Varieties of Lattices</a></i>, Lecture Notes in Mathematics 1533. Springer Verlag. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-387-56314-8" title="Istimewa:Sumber buku/0-387-56314-8">0-387-56314-8</a>. <i>Free online edition</i>.</li> <li>Pigozzi, Don. <i><a rel="nofollow" class="external text" href="http://people.math.sc.edu/mcnulty/alglatvar/pigozzinotes.pdf">General Theory of Algebras</a></i>. <i>Free online edition.</i></li> <li>Smith, J.D.H., 1976. <i>Mal'cev Varieties</i>, Springer-Verlag.</li> <li><a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Whitehead, Alfred North</a>, 1898. <i><a rel="nofollow" class="external text" href="http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=01950001&seq=5">A Treatise on Universal Algebra</a></i>, Cambridge. (<i>Mainly of historical interest.</i>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Pranala_luar">Pranala luar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Aljabar_universal&veaction=edit&section=15" title="Sunting bagian: Pranala luar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Aljabar_universal&action=edit&section=15" title="Sunting kode sumber bagian: Pranala luar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.springer.com/birkhauser/mathematics/journal/12"><i>Algebra Universalis</i></a>—jurnal yang didedikasikan untuk Aljabar Universal.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Diperoleh dari "<a dir="ltr" href="https://id.wikipedia.org/w/index.php?title=Aljabar_universal&oldid=22427891">https://id.wikipedia.org/w/index.php?title=Aljabar_universal&oldid=22427891</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Istimewa:Daftar_kategori" title="Istimewa:Daftar kategori">Kategori</a>: <ul><li><a href="/wiki/Kategori:Aljabar_universal" title="Kategori:Aljabar universal">Aljabar universal</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Kategori tersembunyi: <ul><li><a href="/wiki/Kategori:Artikel_dengan_pernyataan_yang_tidak_disertai_rujukan" title="Kategori:Artikel dengan pernyataan yang tidak disertai rujukan">Artikel dengan pernyataan yang tidak disertai rujukan</a></li><li><a href="/wiki/Kategori:Artikel_dengan_pernyataan_yang_tidak_disertai_rujukan_Desember_2022" title="Kategori:Artikel dengan pernyataan yang tidak disertai rujukan Desember 2022">Artikel dengan pernyataan yang tidak disertai rujukan Desember 2022</a></li><li><a href="/wiki/Kategori:Artikel_yang_tidak_memiliki_referensi_Desember_2022" title="Kategori:Artikel yang tidak memiliki referensi Desember 2022">Artikel yang tidak memiliki referensi Desember 2022</a></li><li><a href="/wiki/Kategori:Templat_webarchive_tautan_wayback" title="Kategori:Templat webarchive tautan wayback">Templat webarchive tautan wayback</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Halaman ini terakhir diubah pada 22 Desember 2022, pukul 05.17.</li> <li id="footer-info-copyright">Teks tersedia di bawah <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Lisensi Atribusi-BerbagiSerupa Creative Commons</a>; ketentuan tambahan mungkin berlaku. 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