Are there interesting examples of medial non-commutative semigroups? 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: 1. In an arbitrary semigroup, commutativity implies mediality. 2. 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.

An example: semigroups with the identity $xy=x$ (semigroups of left zeroes).

Another exemple: Let $A,B$ be arbitrary sets, $S=A\times B$ with the multiplication $(a_1,b_1)(a_2,b_2)=(a_1,b_2)$ (a rectangular band).

Moreover, there is a description of medial semigroups (similarly to commutative ones): Every medial semigroup is a semilattice of medial archmedean semigroups. [A. Nagy, Special Classes of Semigroups. Springer, 2001; Theorem 9.3].