Medial

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This article is about medial in mathematics. For other uses, see medial (disambiguation).

In abstract algebra, a medial magma (or medial groupoid) is a set with a binary operation which satisfies the identity

$(x \cdot y) \cdot (u \cdot z) = (x \cdot u) \cdot (y \cdot z)$ , or more simply, $xy\cdot uz = xu\cdot yz$

using the convention that juxtaposition has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, bi-commutative, bisymmetric, surcommutative, entropic, etc.^[1]

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 (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 with a cartesian product. (See the discussion in auto magma object.)

If f and g are endomorphisms of a medial magma, then the mapping f.g defined by pointwise multiplication

$(f\cdot g)(x) = f(x)\cdot g(x)$