Finitely based monoids

Abstract

We present a method for proving that a semigroup is finitely based (FB) and find some new sufficient conditions under which a monoid is FB. As an application, we find a class of finite aperiodic monoids where the finite basis property behaves in a complicated way with respect to the lattice operations but can be recognized by a simple algorithm. The method results in a short proof of the theorem of E. Lee that every monoid that satisfies $$xt_1xyt_2y \approx xt_1yxt_2y$$ and $$xyt_1xt_2y \approx yxt_1xt_2y$$ is FB. Also, the method gives an alternative proof of the theorem of F. Blanchet-Sadri that a pseudovariety of $$n$$ -testable languages is FB if and only if $$n \le 3$$ .