# Medial magma

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

${\displaystyle (x\cdot y)\cdot (u\cdot v)=(x\cdot u)\cdot (y\cdot v)}$, or more simply, ${\displaystyle xy\cdot uv=xu\cdot yv}$

using the convention that juxtaposition denotes the same operation but has higher precedence. A magma or groupoid is an algebraic structure that generalizes a group. This identity has been variously called medial, abelian, alternation, transposition, interchange, 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] Medial magmas need not be associative: for any nontrivial abelian group and integers mn, replacing the group operation ${\displaystyle x+y}$ with the binary operation ${\displaystyle x\cdot y=mx+ny}$ yields a medial magma which in general is neither associative nor commutative.

Using the categorial definition of the product, one may define the Cartesian square magma M × M with the operation

(x, y) ∙ (u, v) = (xu, yv) .

The binary operation of M, considered as a function on M × M, maps (x, y) to xy, (u, v) to uv, and (xu, yv)  to (xu) ∙ (yv) . Hence, a magma M is medial if and only if its binary operation is a magma homomorphism from M × 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 fg defined by pointwise multiplication

${\displaystyle (f\cdot g)(x)=f(x)\cdot g(x)}$

is itself an endomorphism.

## Bruck–Murdoch–Toyoda theorem

The Bruck–Murdoch-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.

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[4] 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

${\displaystyle f(g(x_{11},\ldots ,x_{1n}),\ldots ,g(x_{m1},\ldots ,x_{mn}))=g(f(x_{11},\ldots ,x_{m1}),\ldots ,f(x_{1n},\ldots ,x_{mn})).}$