Definitions

# Medial

[mee-dee-uhl]
In abstract algebra, a medial magma (or medial groupoid) is a set with a binary operation which satisfies the identity

$\left(x cdot y\right) cdot \left(u cdot z\right) = \left(x cdot u\right) cdot \left(y cdot z\right)$, or more simply, $xycdot uz = xucdot yz$

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

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 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

$\left(fcdot g\right)\left(x\right) = f\left(x\right)cdot g\left(x\right)$

is itself an endomorphism.