Non-finitely based monoids

We present a general method for proving that a semigroup is non-finitely based (NFB). The method is strong enough to cover a majority of existing non-finite basis arguments for a periodic semigroups and allows to generalize several previous results and to simplify their proofs. The method also allows to remove one of the requirements on the "special system of identities" used by Perkins in 1968 to find the first two examples of finite NFB semigroups. We use our method to prove eleven new sufficient conditions under which a monoid is NFB. As an application, we find infinitely many new examples of finite finitely based aperiodic monoids whose direct product is NFB.