A natural characterization of semilattices of rectangular bands and groups of exponent two

Abstract

In a recent paper, Monzo characterized semilattices of rectangular bands and groups of exponent $2$ as the semigroups that satisfy the following conditions: $x = x^{3}$ and $xyx \in \{xy^{2}x, y^{2}xy^{2}\}$. However, this definition does not seem to point directly to the properties of rectangular bands and groups of exponent $2$ (namely, idempotency and commutativity). So, in order to provide a more natural characterization of the class of semigroups under consideration we prove the following theorem: Main Theorem In a semigroup $S$, the following are equivalent:

• $S$ is a semilattice of rectangular bands and groups of exponent $2$;

• for all $x,y \in S$, we have $x = x^{3} and xy \in \{yx, (xy)^{2}\}$.

The paper ends with a list of problems.