Medial magma

In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) which satisfies the identity
for all "x", "y", "u" and "v", 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. 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 formula_3 yields a medial magma which 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. (See the discussion in auto magma object.)
If and are endomorphisms of a medial magma, then the mapping   defined by pointwise multiplication 
is itself an endomorphism. It follows that the set End() 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 φ and ψ 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–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 D.C. Murdoch and K. Toyoda. It was then rediscovered by Bruck in 1944.
Generalizations.
The term "medial" or (more commonly) "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 "f" and "g" be operations of arity "m" and "n", respectively. Then "f" and "g" are required to satisfy