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 , the new binary operation defined by x \cdot y = mx+ny yields a medial magma which in general is neither associative nor commutative.
