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| {{for|the triple product|Median algebra}}
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| In [[abstract algebra]], a '''medial magma''', or '''medial groupoid''', is a set with a [[binary operation]] which satisfies the [[identity (mathematics)|identity]]
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| :<math>(x \cdot y) \cdot (u \cdot v) = (x \cdot u) \cdot (y \cdot v)</math>, or more simply, <math>xy\cdot uv = xu\cdot yv</math>
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| using the convention that juxtaposition denotes the same operation but has higher precedence. A [[magma (algebra)|magma]] or [[groupoid]] is an algebraic structure that generalizes a group. This identity has been variously called ''medial'', ''abelian'', ''alternation'', ''transposition'', ''interchange'', ''bi-commutative'', ''bisymmetric'', ''surcommutative'', [[#Generalizations|''entropic'']] etc.<ref name=Jezek>[http://www.karlin.mff.cuni.cz/~jezek/medial/03.jpg Historical comments] J.Jezek and T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir. ved 93/2 (1983), 93 pp</ref>
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| Any [[commutative]] [[semigroup]] is a medial magma, and a medial magma has an [[identity element]] if and only if it is a commutative [[monoid]]. Another class of semigroups forming medial magmas are the [[Band (mathematics)|normal bands]].<ref>{{citation
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| | last = Yamada | first = Miyuki
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| | doi = 10.1007/BF02572956
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| | issue = 1
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| | journal = Semigroup Forum
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| | pages = 160–167
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| | title = Note on exclusive semigroups
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| | volume = 3
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| | year = 1971}}.</ref> Medial magmas need not be associative: for any nontrivial [[abelian group]] and integers {{math|''m'' ≠ ''n''}}, replacing the group operation <math>x+y</math> with the binary operation <math>x \cdot y = mx+ny </math> yields a medial magma which in general is neither associative nor commutative.
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| Using the [[category theory|categorial]] definition of the [[product (category theory)|product]], one may define the [[Cartesian square]] magma {{math|''M'' × ''M''}} with the operation
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| : {{math|1=(''x'', ''y'') ∙ (''u'', ''v'') = (''x'' ∙ ''u'', ''y'' ∙ ''v'') }}.
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| The binary operation{{math| ∙ }} of {{mvar|M}}, considered as a function on {{math|''M'' × ''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'') }}.
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| Hence, a magma {{mvar|M}} is medial if and only if its binary operation is a magma [[homomorphism]] from {{math|''M'' × ''M''}} to {{mvar|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]].)
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| If {{mvar|f}} and {{mvar|g}} are [[endomorphism]]s of a medial magma, then the mapping {{math|''f''∙''g''}} defined by pointwise multiplication
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| :<math>(f\cdot g)(x) = f(x)\cdot g(x)</math> | |
| is itself an endomorphism.
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| == Bruck–Toyoda theorem ==
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| The '''Bruck–Toyoda theorem''' provides the following characterization of medial [[quasigroup]]s. Given an [[abelian group]] {{mvar|A}} and two commuting [[group automorphism|automorphisms]] φ and ψ of {{mvar|A}}, define an operation {{math|∗}} on {{mvar|A}} by
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| : {{math|1=''x'' ∗ ''y'' = φ(''x'') + ψ(''y'') + c}} | |
| where {{mvar|c}} some fixed element of {{mvar|A}}. It is not hard to prove that {{mvar|A}} forms a medial quasigroup under this operation. The Bruck–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.<ref>{{cite book | author = Kuzʹmin, E. N. and Shestakov, I. P. | title=Algebra VI | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Encyclopaedia of Mathematical Sciences | isbn=978-3-540-54699-3 | year=1995 | volume=6 | chapter=Non-associative structures | pages=197–280}}</ref> In particular, every medial quasigroup is [[isotopy of loops|isotopic]] to an abelian group.
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| == Generalizations ==
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| The term ''medial'' or (more commonly) ''entropic'' is also used for a generalization to multiple operations. An [[algebraic structure]] is an entropic algebra<ref name=generaliation>[http://www.latrobe.edu.au/mathstats/department/algebra-research-group/Papers/DD_TensorProducts.pdf] </ref> if every two operations satisfy a generalization of the medial identity. Let ''f'' and ''g'' be operations of arity ''m'' and ''n'', respectively. Then ''f'' and ''g'' are required to satisfy
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| :<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>
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| ==See also==
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| * [[Category of medial magmas]]
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| ==References==
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| <references/>
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| [[Category:Non-associative algebra]]
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