  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.
