Medial magma

In abstract algebra, a medial magma or medial groupoid is a magma or groupoid that satisfies the identity
or more simply,
for all,, and, using the convention that juxtaposition denotes the same operation but has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, interchange, 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. The "only if" direction is the Eckmann–Hilton argument. Another class of semigroups forming medial magmas are normal bands. Medial magmas need not be associative: for any nontrivial abelian group with operation and integers, the new binary operation defined by yields a medial magma that in general is neither associative nor commutative.
Using the categorical definition of product, for a magma, one may define the Cartesian square magma with the operation
The binary operation of, considered as a mapping from to, maps to, to, and to.
Hence, a magma is medial if and only if its binary operation is a magma homomorphism from to . 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.
If and are endomorphisms of a medial magma, then the mapping defined by pointwise multiplication
is itself an endomorphism. It follows that the set of all endomorphisms of a medial magma is itself a medial magma.

Bruck–Murdoch–Toyoda theorem

The Bruck–Murdoch–Toyoda theorem provides the following characterization of medial quasigroups. Given an abelian group and two commuting automorphisms and of, define an operation on by
where some fixed element of . It is not hard to prove that forms a medial quasigroup under this operation. The Bruck–Murdoch-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. In particular, every medial quasigroup is isotopic to an abelian group.
The result was obtained independently in 1941 by Murdoch and Toyoda. It was then rediscovered by Bruck in 1944.

Generalizations

The term medial or entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebra if every two operations satisfy a generalization of the medial identity. Let and be operations of arity and, respectively. Then and are required to satisfy

Nonassociative examples

A particularly natural example of a nonassociative medial magma is given by collinear points on elliptic curves. The operation for points on the curve, corresponding to drawing a line between x and y and defining as the third intersection point of the line with the elliptic curve, is a medial magma which is isotopic to the operation of elliptic curve addition.
Unlike elliptic curve addition, is independent of the choice of a neutral element on the curve, and further satisfies the identities. This property is commonly used in purely geometric proofs that elliptic curve addition is associative.