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

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. Another class of semigroups forming medial magmas are the normal bands.[2] 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

is itself an endomorphism.

[edit] Bruck-Toyoda theorem

The Bruck-Toyoda theorem provides the following characterization of medial quasigroups. Given an abelian group A and two commuting automorphisms φ and ψ of A, define an operation ∗ on A by x ∗ y = φ(x) + ψ(y) + c

where c some fixed element of A. It is not hard to prove that A forms a medial quasigroup under this operation. The Bruck-Toyoda theorem states that every medial quasigroup is of this form, i.e. is isomorphic to a quasigroup defined from an abelian group in this way.[3] In particular, every medial quasigroup is isotopic to an abelian group. [edit] Generalizations

The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic...

(1)