The finite basis property is often connected with the finite rank property. The two properties are not, in general, equivalent. Examples of varieties of semigroups which have finite rank and infinitely based are well known. Yet, no previous examples of pseudovarieties with such a property seem to be available in the literature. Indeed, for a non-locally finite pseudovariety V, the only technique so far available to prove that V is infinitely based is to show V has infinite rank. Thus, to find such examples, new proof techniques need to be devised. As a joint work with Almeida, we construct an example which shows that they are not equivalent in the context of pseudovarieties of semigroups. We submitted this work at international journal paper.

For a semigroup S with elements x,y,z_{1},z_{2}, \ldots one recursively defines two sequences \lambda_n=\lambda_{n}(x,y,z_{1},\ldots, z_{n}) and  \rho_n=\rho_{n}(x,y,z_{1},\ldots, z_{n}) by \lambda_{0}=x, \rho_{0}=y and \lambda_{n+1}=\lambda_{n} z_{n+1} \rho_{n}, \rho_{n+1}=\rho_{n} z_{n+1} \lambda_{n}. A semigroup is said to be nilpotent (MN, in the sense of Mal'cev) if there exists a positive integer n such that \lambda_{n}(a,b,c_{1},\ldots, c_{n}) = \rho_{n}(a,b,c_{1},\ldots, c_{n}) for all a,b in S and c_{1}, \ldots, c_{n} in S^{1}. The smallest such n is called the nilpotency class of S. Clearly, null semigroups are nilpotent in the sense of Mal'cev. As a joint work with Almeida and Kufleitner, for the elements x_1,\ldots,x_t\in S and z_{1}, \ldots , z_{n}\in S^{1}, by inspiration from the sequences \lambda_{n} and \rho_n, we define sequences \lambda_{n,i}(x_1,\ldots,x_t;z_{1},\ldots, z_{n}), for every 1\leq i\leq t. We called the semigroup S is nilpotent', if there exists a positive integer n such that \lambda_{n,1}(x_1,\ldots,x_t;z_{1},\ldots, z_{n})=\ldots=\lambda_{n,t}(x_1,\ldots,x_t;z_{1},\ldots, z_{n}) for all x_1,\ldots,x_t in S and z_{1}, \ldots , z_{n} in S^{1}. The classify of the nilpotent' semigroups constitutes a pseudovariety and we denote it by MN'. We show that G_{nil}\subset MN'\subset MN when G_{nil} is the pseudovariety of all finite nilpotent groups. Higgins and Margolis showed that \langle A\cap Inv\rangle\subset A\cap\langle Inv\rangle. We consider the relation of pseudovarieties MN' and \langle A\cap Inv \rangle and prove that \langle A \cap Inv}\rangle is strictly subset of A \cap MN'. At another joint work with Almeida, we proved that MN=\llbracket \phi^{\omega}(x)=\phi^{\omega}(y) \rrbracket where \phi is the continuous endomorphism of the free profinite semigroup on \{x,y,z,t\} such that \phi(x)=xzytyzx, \phi(y)=yzxtxzy, \phi(z)=z, and \phi(t)=t. Then, the pseudovariety MN has finite rank and is finitely based. In this work, we show the pseudovarieties MN and MN' is the intersection of the pseudovariety BG_{nil} and a pseudovariety defines by some \omega-identities. We also show that the pseudovarieties MN \cap A and MN' \cap A can be described by some \omega-identities. Also, we prove that the pseudovariety MN' has infinite rank and, therefore, it is non-finitely based. At the end of the work, we investigate the nilpotency and nilpotency' of the Sch\"utzenberger product of monoids. The work is in process.