Medial magma

In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that 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 ${\displaystyle xy\cdot uv=xu\cdot yv}$

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.[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 normal bands.[2] Medial magmas need not be associative: for any nontrivial abelian group with operation + and integers m ≠ n, the new binary operation defined by ${\displaystyle x\cdot y=mx+ny}$ yields a medial magma which in general is neither associative nor commutative.

Using the categorical definition of product, for a magma M, 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 mapping from M × M to 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

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

is itself an endomorphism. It follows that the set End(M) of all endomorphisms of a medial magma M is itself a medial magma.