Non-finitely based monoids

We present a general method for proving that a semigroup is non-finitely based. The method is strong enough to cover the non-finite basis arguments in articles [1,3,4,5,7,8, 11,14,16,21,27,31,36,37]. In particular, the method allows to generalize the results in [1,8,36,37] and to simplify their proofs. The method also allows to remove one of the requirements on the "special system of identities" used by P. Perkins in [16] to find the first two examples of finite non-finitely based semigroups. We use our method to prove eleven new sufficient conditions under which a monoid is non-finitely based. As an application, we find infinitely many new examples of finite finitely based aperiodic monoids whose direct product is non-finitely based.