High Quality Content by WIKIPEDIA articles! In abstract algebra, a medial magma (or medial groupoid) is a set with a binary operation which satisfies the identity (x.y) . (u . v) = (x . u) . (y . u), or more simply, xy . uv = xu . yu using the convention that juxtaposition has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, bi-commutative, bisymmetric, surcommutative, entropic, etc. 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).

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