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Medial magma
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{{for|the triple product|Median algebra}} In [[abstract algebra]], a '''medial magma''' or '''medial groupoid''' is a [[Magma (algebra)|magma]] or [[Magma (algebra)#History_and_terminology|groupoid]] (that is, a [[Set (mathematics)|set]] with a [[binary operation]]) that satisfies the [[identity (mathematics)|identity]] : {{math|1=(''x'' • ''y'') • (''u'' • ''v'') = (''x'' • ''u'') • (''y'' • ''v'')}}, or more simply, : {{math|1=''xy'' • ''uv'' = ''xu'' • ''yv''}} for all {{math|''x''}}, {{math|''y''}}, {{math|''u''}} and {{math|''v''}}, using the convention that juxtaposition denotes the same operation but has higher precedence. This identity has been variously called ''medial'', ''abelian'', ''alternation'', ''transposition'', ''interchange'', ''bi-commutative'', ''bisymmetric'', ''surcommutative'', [[#Generalizations|''entropic'']], etc.{{sfn|ps=|Ježek|Kepka|1983}} Any [[Semigroup|commutative semigroup]] is a medial magma, and a medial magma has an [[identity element]] if and only if it is a [[Monoid#Commutative_monoid|commutative]] [[monoid]]. The "only if" direction is the [[Eckmann–Hilton argument]]. Another class of semigroups forming medial magmas are [[Band (mathematics)|normal bands]].{{sfn|ps=|Yamada|1971}} Medial magmas need not be associative: for any nontrivial [[abelian group]] with operation {{math|+}} and [[integer]]s {{math|''m'' ≠ ''n''}}, the new binary operation defined by {{math|1=''x'' • ''y'' = ''mx'' + ''ny''}} yields a medial magma that in general is neither associative nor commutative. Using the [[category theory|categorical]] definition of [[product (category theory)|product]], for a magma {{math|''M''}}, one may define the [[Cartesian square]] magma {{math|''M'' × ''M''}} with the operation : {{math|1=(''x'', ''y'') • (''u'', ''v'') = (''x'' • ''u'', ''y'' • ''v'')}}. The binary operation {{math|•}} of {{math|''M''}}, considered as a mapping from {{math|''M'' × ''M''}} to {{math|''M''}}, maps {{math|(''x'', ''y'')}} to {{math|''x'' • ''y''}}, {{math|(''u'', ''v'')}} to {{math|''u'' • ''v''}}, and {{math|(''x'' • ''u'', ''y'' • ''v'') }} to {{math|(''x'' • ''u'') • (''y'' • ''v'') }}. Hence, a magma {{math|''M''}} is medial if and only if its binary operation is a magma [[homomorphism]] from {{math|''M'' × ''M''}} to {{math|''M''}}. This can easily be expressed in terms of a [[commutative diagram]], and thus leads to the notion of a '''medial magma object''' in a [[Cartesian closed category|category with a Cartesian product]]. (See the discussion in auto magma object.) If {{math|''f''}} and {{math|''g''}} are [[endomorphism]]s of a medial magma, then the mapping {{math|''f'' • ''g''}} defined by pointwise multiplication : {{math|1=(''f'' • ''g'')(''x'') = ''f''(''x'') • ''g''(''x'')}} is itself an endomorphism. It follows that the set {{math|End(''M'')}} of all endomorphisms of a medial magma {{math|''M''}} is itself a medial magma. == Bruck–Murdoch–Toyoda theorem == The '''Bruck–Murdoch–Toyoda theorem''' provides the following characterization of medial [[quasigroup]]s. Given an abelian group {{math|''A''}} and two commuting [[group automorphism|automorphisms]] {{math|''φ''}} and {{math|''ψ''}} of {{math|''A''}}, define an operation {{math|•}} on {{math|''A''}} by : {{math|1=''x'' • ''y'' = ''φ''(''x'') + ''ψ''(''y'') + ''c''}}, where {{math|''c''}} some fixed element of {{math|''A''}}. It is not hard to prove that {{math|''A''}} forms a medial quasigroup under this operation. The Bruck–Murdoch-Toyoda theorem states that every medial quasigroup is of this form, i.e. is [[isomorphic]] to a quasigroup defined from an abelian group in this way.{{sfn|ps=|Kuzʹmin|Shestakov|1995}} In particular, every medial quasigroup is [[isotopy of loops|isotopic]] to an abelian group. The result was obtained independently in 1941 by Murdoch and Toyoda.{{sfn|ps=|Murdoch|1941}}{{sfn|ps=|Toyoda|1941}} It was then rediscovered by Bruck in 1944.{{sfn|ps=|Bruck|1944}} == Generalizations == The term ''medial'' or (more commonly) ''entropic'' is also used for a generalization to multiple operations. An [[algebraic structure]] is an entropic algebra{{sfn|ps=|Davey|Davis|1985}} if every two operations satisfy a generalization of the medial identity. Let {{math|''f''}} and {{math|''g''}} be operations of [[arity]] {{math|''m''}} and {{math|''n''}}, respectively. Then {{math|''f''}} and {{math|''g''}} are required to satisfy : <math>f(g(x_{11}, \ldots, x_{1n}), \ldots, g(x_{m1}, \ldots, x_{mn})) = g(f(x_{11}, \ldots, x_{m1}), \ldots, f(x_{1n}, \ldots, x_{mn})).</math> == Nonassociative examples == A particularly natural example of a nonassociative medial magma is given by collinear points on [[elliptic curves]]. The operation {{math|1=''x'' • ''y'' = −(''x'' + ''y'')}} for points on the curve, corresponding to drawing a line between x and y and defining {{math|''x'' • ''y''}} as the third intersection point of the line with the elliptic curve, is a (commutative) medial magma which is isotopic to the operation of elliptic curve addition. Unlike elliptic curve addition, {{math|''x'' • ''y''}} is independent of the choice of a neutral element on the curve, and further satisfies the identities {{math|1=''x'' • (''x'' • ''y'') = ''y''}}. This property is commonly used in purely geometric proofs that elliptic curve addition is associative. == Citations == {{reflist}} == References == {{refbegin}} * {{citation |last=Murdoch |first=D.C. |title=Structure of abelian quasi-groups |journal=Trans. Amer. Math. Soc. |volume=49 |issue=3 |pages=392–409 |date=May 1941 |jstor=1989940 |doi=10.1090/s0002-9947-1941-0003427-2 |doi-access=free }} * {{citation |last=Toyoda |first=K. |year=1941 |title=On axioms of linear functions |journal=Proc. Imp. Acad. Tokyo |volume=17 |issue=7 |pages=221–227 |url=https://www.jstage.jst.go.jp/article/pjab1912/17/7/17_7_221/_article |doi=10.3792/pia/1195578751 |doi-access=free }} * {{citation |last=Bruck |first=R.H. |date=January 1944 |title=Some results in the theory of quasigroups |journal=Trans. Amer. Math. Soc. |volume=55 |issue=1 |pages=19–52 |jstor=1990138 |doi=10.1090/s0002-9947-1944-0009963-x |doi-access=free }} * {{citation |last=Yamada |first=Miyuki |year=1971 |title=Note on exclusive semigroups |journal=[[Semigroup Forum]] |volume=3 |issue=1 |pages=160–167 |doi=10.1007/BF02572956 }} * {{cite journal |first1=J. |last1=Ježek |first2=T. |last2=Kepka |year=1983 |title=Medial groupoids |journal=Rozpravy Československé Akad. Věd Řada Mat. Přírod. Věd |volume=93 |issue=2 |pages=93pp |url=http://www.karlin.mff.cuni.cz/~jezek/medial/03.jpg |archive-url=https://web.archive.org/web/20110718093325/http://www.karlin.mff.cuni.cz/~jezek/medial/03.jpg |archive-date=2011-07-18 }} * {{cite journal |last1=Davey |first1=B. A. |last2=Davis |first2=G. |year=1985 |title=Tensor products and entropic varieties |journal=Algebra Universalis |volume=21 |pages=68–88 |doi=10.1007/BF01187558 }} * {{cite book |last1=Kuzʹmin |first1=E. N. |last2=Shestakov |first2= I. P. |year=1995 |title=Algebra VI |location=Berlin, New York |publisher=[[Springer-Verlag]] |series=Encyclopaedia of Mathematical Sciences |volume=6 |chapter=Non-associative structures |pages=197–280 |isbn=978-3-540-54699-3 }} {{refend}} [[Category:Non-associative algebra]]
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