Medial
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: ''This article is about ''medial'' in mathematics. For other uses, see [[medial (disambiguation)]].''
In [[abstract algebra]], a '''medial [[Magma_(algebra)|magma]]''' (or '''medial groupoid''') is a set with a [[binary operation]] which satisfies the [[identity (mathematics)|identity]]
:<math>(x \cdot y) \cdot (u \cdot z) = (x \cdot u) \cdot (y \cdot z)</math>, or more simply, <math>xy\cdot uz = xu\cdot yz</math>
using the convention that juxtaposition has higher precedence. This
identity has been variously called '''medial''', abelian, alternation, transposition, bi-commutative, bisymmetric, surcommutative, entropic, etc.<ref name=Jezek>[http://adela.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>
Any [[commutative]] [[semigroup]] is a medial magma, and a medial magma has an [[identity element]] if and only if it is a commutative [[monoid]]. An elementary example of a nonassociative medial [[quasigroup]] can be constructed as follows: take an [[abelian group]] except the group of order 2 (written additively) and define a new operation by ''x * y = (− x) + (− y)''.
A magma ''M'' is medial if and only if its binary operation is a [[homomorphism]] from the [[Cartesian square]] ''M x M'' to ''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 [[Category_(mathematics)|category]] with a [[Cartesian_product#Category_theory|cartesian product]]. (See the discussion in [[auto magma object]].)
If ''f'' and ''g'' are [[endomorphism]]s of a medial magma, then the mapping ''f.g'' defined by pointwise multiplication
:<math>(f\cdot g)(x) = f(x)\cdot g(x)</math>
is itself an endomorphism.
==See also==
* [[Medial category]]
==External links==
<references/>
[[Category:Nonassociative algebra]]
[[es:Medial (álgebra)]]