There exist semigroups $S$ (written additively) such that
- $S$ is medial, meaning $(a+b)+(a'+b') = (a+a')+(b+b')$.
- $S$ is not commutative.
Example. The left (and right) zero semigroups are all medial, but those having two or more elements are non-commutative.
Soft question: Does anyone know of other, more "interesting" examples of medial non-commutative semigroups?
A few remarks:
- In an arbitrary semigroup, commutativity implies mediality.
- In a magma with an identity element $0$, mediality implies commutativity. Indeed:
$$a+b = (0+a)+(b+0)=(0+b)+(a+0) =b+a.$$
Thus, every medial non-commutative semigroup lacks an identity element.