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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.

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    $\begingroup$ 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". $\endgroup$ – James Dec 6 '13 at 21:44
  • $\begingroup$ 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. $\endgroup$ – James Dec 6 '13 at 21:55
  • $\begingroup$ I forgot to mention that the operation on $\operatorname{Hom}(M,S)$ is defined pointwise. $\endgroup$ – James Dec 7 '13 at 0:04
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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].

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