# 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.

• There appear to be a lot of them; Maple says there are $2$ of order $2$, $10$ of order $3$, $94$ of order $4$, $945$ of order $5$ and $14136$ of order $6$ (up to isomorphism). Surely, some of those are "interesting". Dec 6, 2013 at 21:44
• You might look at $\operatorname{Hom}( M, S )$, where $S$ is a semigroup of the sort you are interested in and $M$ is some magma. It will automatically inherit associativity and mediality, and there might be interesting ways to ensure it is non-commutative if $S$ is. Dec 6, 2013 at 21:55
• I forgot to mention that the operation on $\operatorname{Hom}(M,S)$ is defined pointwise. Dec 7, 2013 at 0:04

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).