In a recent paper, Monzo characterized semilattices of rectangular bands and groups of exponent $$2$$ 2 as the semigroups that satisfy the following conditions: $$x = x^{3}$$ x = x 3 and $$xyx \in \{xy^{2}x, y^{2}xy^{2}\}$$ x y x ∈ { x y 2 x , y 2 x y 2 } . However, this definition does not seem to point directly to the properties of rectangular bands and groups of exponent $$2$$ 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 $$S$$ S , the following are equivalent:
$$S$$ S is a semilattice of rectangular bands and groups of exponent $$2$$ 2 ;
for all $$x,y \in S$$ x , y ∈ S , we have $$x = x^{3} and xy \in \{yx, (xy)^{2}\}$$ x = x 3 a n d x y ∈ { y x , ( x y ) 2 } .
The paper ends with a list of problems.