Are there interesting examples of medial non-commutative semigroups?

Question

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.

Answer 1

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].