Commutative
<ol style="margin: 0px; padding: 0px 0px 0px 2.25em; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; background-image: none; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial; color: rgb(34, 34, 34);"><li style="margin: 0px 0px 10px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">(</span><span class="ib-content" style="margin: 0px; padding: 0px; border: 0px; font-style: italic; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><a href="https://en.m.wiktionary.org/wiki/mathematics" title="mathematics" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">mathematics</a><span class="ib-comma" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">,</span> of a <a href="https://en.m.wiktionary.org/wiki/binary" title="binary" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">binary</a> <a href="https://en.m.wiktionary.org/wiki/operation" title="operation" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">operation</a></span><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">)</span> Such that the <a href="https://en.m.wiktionary.org/wiki/order" title="order" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">order</a> in which the <a href="https://en.m.wiktionary.org/wiki/operand" title="operand" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">operands</a> are taken does not affect their <a href="https://en.m.wiktionary.org/wiki/image" title="image" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">image</a> under the operation.<dl style="margin: 0px 0px 0px 1em; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><dd style="margin: 0px 0px 0px 1em; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; overflow: auto;"><i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">Addition on the real numbers is <b style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">commutative</b> because for any real numbers <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,t}"><semantics><annotation encoding="application/x-tex">{\displaystyle s,t}</annotation></semantics></math></span><img width="0" height="0" class="mwe-math-fallback-image-inline image-lazy-loaded" alt="{\displaystyle s,t}" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ce81dd4a9013d5a5502724585368c58ace5971" srcset="" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: middle; background: none; display: inline-block; animation: 0.3s ease-in 0s 1 normal none running fadeInImage; width: 2.964ex; height: 2.343ex;"></span>, it is true that <span class="mwe-math-element" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; display: none; clip: rect(1px, 1px, 1px, 1px); overflow: hidden; position: absolute; width: 1px; height: 1px; opacity: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s+t=t+s}"><semantics><annotation encoding="application/x-tex">{\displaystyle s+t=t+s}</annotation></semantics></math></span><img width="0" height="0" class="mwe-math-fallback-image-inline image-lazy-loaded" alt="{\displaystyle s+t=t+s}" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c780c42cabed22ddd02361c8e93343ebccf2d6" srcset="" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: middle; background: none; display: inline-block; animation: 0.3s ease-in 0s 1 normal none running fadeInImage; width: 12.639ex; height: 2.176ex;"></span>.</i></dd><dd style="margin: 0px 0px 0px 1em; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; overflow: auto;"><i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">Addition and multiplication are <b style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">commutative</b> operations but subtraction and division are not.</i></dd></dl></li><li style="margin: 0px 0px 10px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">(</span><span class="ib-content" style="margin: 0px; padding: 0px; border: 0px; font-style: italic; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><a href="https://en.m.wiktionary.org/wiki/algebra" title="algebra" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">algebra</a><span class="ib-comma" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">,</span> of an <a href="https://en.m.wiktionary.org/wiki/algebraic_structure" title="algebraic structure" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">algebraic structure</a></span><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">)</span> Having a commutative <a href="https://en.m.wiktionary.org/wiki/operation" title="operation" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">operation</a>.</li><li style="margin: 0px 0px 10px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">(</span><span class="ib-content" style="margin: 0px; padding: 0px; border: 0px; font-style: italic; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><a href="https://en.m.wiktionary.org/wiki/mathematics" title="mathematics" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">mathematics</a><span class="ib-comma" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">,</span> of a <a href="https://en.m.wiktionary.org/wiki/diagram" title="diagram" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">diagram</a> of <a href="https://en.m.wiktionary.org/wiki/morphism" title="morphism" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">morphisms</a></span><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">)</span> Such that any two sequences of morphisms with the same initial and final positions <a href="https://en.m.wiktionary.org/wiki/compose" title="compose" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">compose</a> to the same morphism.</li><li style="margin-top: 0px; margin-right: 0px; margin-bottom: inherit; margin-left: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">Relating to <a href="https://en.m.wiktionary.org/wiki/exchange" title="exchange" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">exchange</a>; <a href="https://en.m.wiktionary.org/wiki/interchangeable" title="interchangeable" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">interchangeable</a>.</li></ol><h4 class="in-block" style="margin: 0px; padding: 0.5em 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: 1.3; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; background-image: none; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial; overflow-wrap: break-word; word-break: break-word; display: table; color: rgb(34, 34, 34);"><span class="mw-headline" id="Synonyms" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: middle; background: none; display: table-cell; width: 296px;">Synonyms</span><span class="mw-editsection" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: middle; background: none; display: table-cell;"><a href="https://en.m.wiktionary.org/w/index.php?title=commutative&action=edit&section=4" title="Edit section: Synonyms" data-section="4" class="mw-ui-icon mw-ui-icon-element mw-ui-icon-minerva-edit-enabled edit-page mw-ui-icon-flush-right" style="margin: 0px -0.75em 0px 0px; padding: 0.75em; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: 999px; font-family: inherit; font-size: initial; vertical-align: middle; background: none transparent; color: transparent; hyphens: none; position: relative; display: inline-block; width: 1.25em; height: 1.25em; flex-basis: 1.25em; overflow: hidden; -webkit-appearance: none; border-radius: 2px; box-sizing: content-box !important;">Edit</a></span></h4><ul style="margin: 0px; padding: 0px 0px 0px 1em; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; background-image: none; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial; list-style-position: initial; list-style-image: initial; color: rgb(34, 34, 34);"><li style="margin-top: 0px; margin-right: 0px; margin-bottom: inherit; margin-left: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="ib-brac qualifier-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">(</span><span class="ib-content qualifier-content" style="margin: 0px; padding: 0px; border: 0px; font-style: italic; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">of an algebraic structure</span><span class="ib-brac qualifier-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">)</span><span class="ib-colon sense-qualifier-colon" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">:</span> <span class="Latn" lang="en" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><a href="https://en.m.wiktionary.org/wiki/abelian#English" title="abelian" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">abelian</a></span>, <span class="Latn" lang="en" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><a href="https://en.m.wiktionary.org/wiki/Abelian#English" title="Abelian" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">Abelian</a></span></li></ul><h4 class="in-block" style="margin: 0px; padding: 0.5em 0px; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: 1.3; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; background-image: none; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial; overflow-wrap: break-word; word-break: break-word; display: table; color: rgb(34, 34, 34);"><span class="mw-headline" id="Antonyms" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: middle; background: none; display: table-cell; width: 296px;">Antonyms</span><span class="mw-editsection" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: middle; background: none; display: table-cell;"><a href="https://en.m.wiktionary.org/w/index.php?title=commutative&action=edit&section=5" title="Edit section: Antonyms" data-section="5" class="mw-ui-icon mw-ui-icon-element mw-ui-icon-minerva-edit-enabled edit-page mw-ui-icon-flush-right" style="margin: 0px -0.75em 0px 0px; padding: 0.75em; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: 999px; font-family: inherit; font-size: initial; vertical-align: middle; background: none transparent; color: transparent; hyphens: none; position: relative; display: inline-block; width: 1.25em; height: 1.25em; flex-basis: 1.25em; overflow: hidden; -webkit-appearance: none; border-radius: 2px; box-sizing: content-box !important;">Edit</a></span></h4><ul style="margin: 0px; padding: 0px 0px 0px 1em; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; background-image: none; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial; list-style-position: initial; list-style-image: initial; color: rgb(34, 34, 34);"><li style="margin-top: 0px; margin-right: 0px; margin-bottom: inherit; margin-left: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="Latn" lang="en" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><a href="https://en.m.wiktionary.org/wiki/anticommutative#English" title="anticommutative" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">anticommutative</a></span></li></ul>