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Finitely based monoids

We present a method for proving that a semigroup is finitely based and find some new sufficient conditions under which a monoid is finitely based. As an application, we find a class of finite 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 xtxysy = xtyxsy and xytxsy = yxtxsy is finitely based. Also, the method gives an alternative proof of the theorem of F. Blanchet-Sadri that a pseudovariety of n-testable languages is finitely based if and only if n < 4.