Medial magma

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 (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 and be operations of arity and , respectively. Then and are required to satisfy : f(g(x_{11}, \ldots, x_{1n}), \ldots, g(x_{m1}, \ldots, x_{mn})) = g(f(x_{11}, \ldots, x_{m1}), \ldots, f(x_{1n}, \ldots, x_{mn})). == 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 (commutative) 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. == Citations ==