Medial 502296 191989595 2008-02-17T03:34:45Z Kilva 1424543 : ''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 = (&minus; x) + (&minus; 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)]]