Some examples of bands with two components

Abstract

A band is a semigroup whose elements are idempotents. The paper is motivated by some open problems concerning linear representations of bands. A counterexample to a conjecture stated in F. Cedó and J. Okniński, "Faithful linear representations of bands," Publ. Mat. 53 (1) (2009), 119–140 is given. On the other hand, it is proved that there exist linear bands with two components such that the associated commutative $K$-algebra constructed in Cedó and Okniński cannot be embedded into any Noetherian ring, even in the case where this algebra is reduced.