In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) that satisfies the identity or more simply, 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.
Given an abelian group A and two commuting automorphisms φ and ψ of A, define an operation • on A by where c some fixed element of A.
Then f and g are required to satisfy A particularly natural example of a nonassociative medial magma is given by collinear points on elliptic curves.
This property is commonly used in purely geometric proofs that elliptic curve addition is associative.