**Medial**

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

- , or more simply,

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 the normal bands. Medial magmas need not be associative: for any nontrivial abelian group and integers *m* ≠ *n*, replacing the group operation with the binary operation 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*) = (*x*∙*u*,*y*∙*v*) .

The binary operation ∙ of `M`, considered as a function on *M* × *M*, maps (*x*, *y*) to `x`∙`y`, (*u*, *v*) to `u`∙`v`, and (*x*∙*u*, *y*∙*v*) to (*x*∙*u*)∙(*y*∙*v*) . 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 *f*∙*g* defined by pointwise multiplication

is itself an endomorphism.

Read more about Medial: Bruck–Toyoda Theorem, Generalizations