Medial

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.