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class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> </div> </div> <h1 id="binary_operations">Binary operations</h1> <div class='maruku_toc'> <ul> <li><a href='#definitions'>Definitions</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#square'>Square function</a></li> <li><a href='#magmas_as_actions'>Magmas as actions</a></li> <li><a href='#opposite_magmas'>Opposite magmas</a></li> </ul> <li><a href='#free_magmas'>Free magmas</a></li> <li><a href='#extensional_magmas'>Extensional magmas</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#literature'>Literature</a></li> </ul> </div> <h2 id="definitions">Definitions</h2> <p>A <strong>binary operation</strong> on a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">(-)\cdot (-) \colon S \times S \to S</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>×</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">S \times S</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. A <strong>magma</strong> (or binary algebraic structure, or, alternatively, a mono-binary algebra) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S,\cdot)</annotation></semantics></math> is a set equipped with a binary operation on it.</p> <p>A magma is called</p> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/unital+magma">unital</a></em> if it has a <a class="existingWikiWord" href="/nlab/show/identity+element">neutral element</a>; that is, an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">1 \in S</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>⋅</mo><mi>x</mi><mo>=</mo><mi>x</mi><mo>=</mo><mi>x</mi><mo>⋅</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \cdot x = x = x \cdot 1</annotation></semantics></math>. Some authors mean by ‘magma’ what we call a unital magma (cf. <a class="existingWikiWord" href="/nlab/show/Borceux-Bourn">Borceux-Bourn</a> Def. 1.2.1). One can consider one-sided unital elements separately: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>L</mi></msub><mo>⋅</mo><mi>x</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">1_L \cdot x = x</annotation></semantics></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>x</mi><mo>⋅</mo><msub><mn>1</mn> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">x = x \cdot 1_R</annotation></semantics></math>. Note that units may be far from unique.</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/commutative+magma">commutative</a></em> if the binary operation takes the same value when its two arguments are interchanged: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⋅</mo><mi>y</mi><mo>=</mo><mi>y</mi><mo>⋅</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x \cdot y = y \cdot x</annotation></semantics></math>.</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/associative+magma">associative</a></em> if the binary operation satisfies the <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>⋅</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>z</mi><mo>=</mo><mi>x</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⋅</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x \cdot y) \cdot z = x \cdot (y \cdot z)</annotation></semantics></math>.</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/invertible+magma">invertible</a></em> if it has an <a class="existingWikiWord" href="/nlab/show/inverse+element">inverse element</a>.</p> </li> <li> <p>an <em><a class="existingWikiWord" href="/nlab/show/absorption+magma">absorption magma</a></em> if it has an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">0 \in S</annotation></semantics></math> such that the binary operation satisfies the absorption condition: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>⋅</mo><mi>x</mi><mo>=</mo><mi>x</mi><mo>⋅</mo><mn>0</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \cdot x = x \cdot 0 = 0</annotation></semantics></math>.</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></em> if one-sided multiplication by any element is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>.</p> </li> </ul> <p>A magma has a <a class="existingWikiWord" href="/nlab/show/square+root">square root</a> <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mo lspace="verythinmathspace" rspace="0em">−</mo></msqrt><mo>:</mo><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\sqrt{-}:S \to S</annotation></semantics></math> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">x \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><mi>x</mi><mo>⋅</mo><mi>x</mi></mrow></msqrt><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sqrt{x \cdot x} = x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mi>x</mi></msqrt><mo>⋅</mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sqrt{x} \cdot \sqrt{x} = x</annotation></semantics></math>.</p> <p>The term ‘magma’ is from <a class="existingWikiWord" href="/nlab/show/Bourbaki">Bourbaki</a> and is intended to suggest the fluidity of the concept; special cases include <a class="existingWikiWord" href="/nlab/show/unital+magmas">unital magmas</a>, <a class="existingWikiWord" href="/nlab/show/semigroups">semigroups</a>/<a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>, <a class="existingWikiWord" href="/nlab/show/quasigroups">quasigroups</a>, <a class="existingWikiWord" href="/nlab/show/groups">groups</a>, and so on. The term ‘groupoid’ is also used, but here that word means <a class="existingWikiWord" href="/nlab/show/groupoid">something else</a> (see also related discussion at <a class="existingWikiWord" href="/nlab/show/historical+notes+on+quasigroups">historical notes on quasigroups</a>).</p> <p>More generally, in any <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, a <strong>magma object</strong> or <strong>magma in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></strong> is an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/multimorphism">multimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">m: X, X \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. Here the multimorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <strong>binary operation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></strong>. In particular, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, a magma structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">m\colon X \otimes X \to X</annotation></semantics></math>, and in a <a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a>, a magma structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">m\colon X \to [X, X]</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="square">Square function</h3> <p>Every magma <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M,\cdot)</annotation></semantics></math> has a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>:</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">(-)^2:M \to M</annotation></semantics></math> called the <strong>square</strong> and defined as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>:</mo><mo>=</mo><mi>x</mi><mo>⋅</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x^2 := x \cdot x</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">x \in M</annotation></semantics></math>.</p> <h3 id="magmas_as_actions">Magmas as actions</h3> <p>An <a class="existingWikiWord" href="/nlab/show/action">action</a> of a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> on another set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo>:</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">act:A \times B \to B</annotation></semantics></math>. So this means that a magma is just an action of a set on itself.</p> <h3 id="opposite_magmas">Opposite magmas</h3> <p>There exists a function on the binary operation set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>:</mo><mo stretchy="false">(</mo><mi>M</mi><mo>×</mo><mi>M</mi><mo>→</mo><mi>M</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>M</mi><mo>×</mo><mi>M</mi><mo>→</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B:(M\times M\to M)\to (M\times M\to M)</annotation></semantics></math> called the <strong><a class="existingWikiWord" href="/nlab/show/braiding">braiding</a></strong> that takes every binary operation on the set to its opposite binary operation, where for every magma operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>M</mi><mo>×</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">m:M\times M\to M</annotation></semantics></math> and for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a,b</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(m)(a,b) = m(b,a)</annotation></semantics></math>. The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> with the opposite binary operation is the <a class="existingWikiWord" href="/nlab/show/opposite+magma">opposite magma</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/involution">involution</a>; the opposite of an opposite magma is the original magma itself; this follows from the fact that <a class="existingWikiWord" href="/nlab/show/Set">Set</a> is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>. Fixed points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> are called <strong><a class="existingWikiWord" href="/nlab/show/commutative+magma">commutative</a></strong> binary operations.</p> <h2 id="free_magmas">Free magmas</h2> <p>The free magma on one generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mo>⋅</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M, \cdot, 1)</annotation></semantics></math> is a model of nesting parentheses around a magma, and is important in the study of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> as <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a> and <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-categories">(infinity,n)-categories</a> are not <a class="existingWikiWord" href="/nlab/show/associative">associative</a>, but rather satisfy <a class="existingWikiWord" href="/nlab/show/coherence+laws">coherence laws</a> such as the <a class="existingWikiWord" href="/nlab/show/pentagon+identity">pentagon identity</a> which relate the various ways to nest parentheses around the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> magma in an <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-category">(infinity,n)-category</a> with respect to <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>. Many other higher categorical objects have <a class="existingWikiWord" href="/nlab/show/coherence+theorems">coherence theorems</a> which also deal with nested parentheses around a binary operation.</p> <h2 id="extensional_magmas">Extensional magmas</h2> <p>In <a class="existingWikiWord" href="/nlab/show/foundations+of+mathematics">foundations of mathematics</a> such as <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional</a> Martin-Loef <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, where <a class="existingWikiWord" href="/nlab/show/functions">functions</a> do not <a class="existingWikiWord" href="/nlab/show/function+extensionality">preserve equality</a>, one could distinguish between ordinary magmas as defined above, and <strong>extensional magmas</strong>, whose binary operation preserves equality.</p> <p>For a magma <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, \cdot)</annotation></semantics></math> and for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a, b, c:A</annotation></semantics></math>, a magma is <strong>left extensional</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a = c</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>=</mo><mi>c</mi><mo>⋅</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \cdot b = c \cdot b</annotation></semantics></math>, and a magma is <strong>right extensional</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a = c</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>⋅</mo><mi>a</mi><mo>=</mo><mi>b</mi><mo>⋅</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">b \cdot a = b \cdot c</annotation></semantics></math>. A magma is <strong>extensional</strong> if it is both left and right extensional.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/magmoidal+groupoid">magmoidal groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magmoidal+category">magmoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-ary+operation">n-ary operation</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> <a class="existingWikiWord" href="/nlab/show/mathematical+structure">structure</a></th><th><a class="existingWikiWord" href="/nlab/show/oidification">oidification</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magma">magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magmoid">magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pointed+set">pointed</a> <a class="existingWikiWord" href="/nlab/show/magma">magma</a> with an <a class="existingWikiWord" href="/nlab/show/endofunction">endofunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/setoid">setoid</a>/<a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magma">unital magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroupoid">quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loop+%28algebra%29">loop</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loopoid">loopoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category">category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involutive</a> <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dagger+category">dagger category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroup">associative quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroupoid">associative quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/group">group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magma">flexible magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magma">alternative magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+category">absorption category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+category">cancellative category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rig">rig</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/CMon">CMon</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+ring">nonassociative ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring">ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ringoid">ringoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+algebra">nonunital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+category">C-star category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebra">differential algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebroid">differential algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+algebra">flexible algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+algebra">alternative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+poset">monoidal poset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a></td></tr> <tr><td style="text-align: left;"><span class="newWikiWord">strict monoidal groupoid<a href="/nlab/new/strict+monoidal+groupoid">?</a></span></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+%282%2C1%29-category">strict (2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-groupoid">strict 2-groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+monoidal+category">strict monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+groupoid">monoidal groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>/<a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></td></tr> </tbody></table> </div> <h2 id="literature">Literature</h2> <p>On the history of the notion:</p> <ul> <li id="Hollings09">Christopher Hollings, <em>The Early Development of the Algebraic Theory of Semigroups</em>, Archive for History of Exact Sciences <strong>63</strong> (2009) 497–536 [<a href="https://doi.org/10.1007/s00407-009-0044-3">doi:10.1007/s00407-009-0044-3</a>]</li> </ul> <p>The wikipedia entry <a href="http://en.wikipedia.org/wiki/Magma_%28algebra%29">magma</a> is quite useful, having a list and a table of various subclasses of magmas, hence of binary algebraic structures.</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/eom">eom</a>: <a href="http://www.encyclopediaofmath.org/index.php?title=m/m110040">magma</a></li> <li>R.H. Bruck, <em>A survey of binary systems</em>, Springer-Verlag 1958</li> </ul> <p>Formalization of magmas as <a class="existingWikiWord" href="/nlab/show/mathematical+structures">mathematical structures</a> in <a class="existingWikiWord" href="/nlab/show/proof+assistants">proof assistants</a>:</p> <p>in a context of plain <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mart%C3%ADn+Escard%C3%B3">Martín Escardó</a>, <em><a href="https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#magmasandmonoids">The Types of Magmas and Monoids</a></em>, §3.4 in: <em>Introduction to Univalent Foundations of Mathematics with Agda</em> [<a href="https://arxiv.org/abs/1911.00580">arXiv:1911.00580</a>, <a href="https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html">webpage</a>]</li> </ul> <p>in a context of <a class="existingWikiWord" href="/nlab/show/cubical+type+theory">cubical</a> <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/1lab">1lab</a>: <em><a href="https://1lab.dev/Algebra.Magma.html">Algebra.Magma</a></em></li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/algebra">algebra</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on August 21, 2024 at 02:20:21. See the <a href="/nlab/history/magma" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/magma" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12759/#Item_9">Discuss</a><span class="backintime"><a href="/nlab/revision/magma/30" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/magma" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/magma" accesskey="S" class="navlink" id="history" rel="nofollow">History (30 revisions)</a> <a href="/nlab/show/magma/cite" style="color: black">Cite</a> <a href="/nlab/print/magma" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/magma" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>