Bases of identities for semigroups of bounded rank transformations of a set

We complete the series of results by M. V. Sapir, M. V. Volkov and the author solving the Finite Basis Problem for semigroups of rank a parts per thousand currency sign k transformations of a set, namely based on these results we prove that the semigroup T (k) (X) of rank a parts per thousand currency sign k transformations of a set X has no finite basis of identities if and only if k is a natural number and either k = 2 and |X| a A << 3, 4A >> or k a parts per thousand yen 3. A new method for constructing finite non-finitely based semigroups is developed. We prove that the semigroup of rank a parts per thousand currency sign 2 transformations of a 4-element set has no finite basis of identities but that the problem of checking its identities is tractable (polynomial).