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In abstract algebra, a medial magma or medial groupoid is a magma or groupoid which satisfies the identity
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.
Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid. The "only if" direction is the Eckmann–Hilton argument. Another class of semigroups forming medial magmas are normal bands. Medial magmas need not be associative: for any nontrivial abelian group with operation + and integers m ≠ n, the new binary operation defined by x ⋅ y = m x + n y {\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