Identities and quasi-identities in lattices of pseudovarieties (Q1901895)

A class of finite universal algebras (all of the same type) is called a pseudovariety if it is closed with respect to evaluation of homomorphic images, subalgebras, and finite direct products. Theories of pseudovarieties go back to S. Eilenberg and M. P. Schützenberger. In recent years it turned into a subject of active investigations, and one of its main directions is the study of lattices of pseudovarieties. Even the definition of a pseudovariety permits to suppose a priori that the theory of pseudovarieties is closely connected with the theory of varieties. Concerning the lattices of pseudovarieties, this conjecture is confirmed by results of \textit{P. Agliano} and \textit{J. B. Nation} [J. Aust. Math. Soc., Ser. A 46, No. 2, 177-183 (1989; Zbl 0671.08007)]. On the other hand, there were obtained a number of results concerning identities and quasi-identities in lattices of varieties. In the present article we obtain a number of results concerning identities and quasi-identities in lattices of pseudovarieties by means of these two circumstances.