Quasivarieties are classes of algebras of the same type, which are defined by sets of implications. Equivalently, a class of algebras of the same type is a quasivariety if it is isomorphically and multiplicatively closed, closed under ultraproducts, hereditary and contains a trivial algebra. Quasivarieties became much more important when it turned out that they are the appropriate classes of algebras to algebraize logical calculi. The study of all subquasivarieties of a given variety is a challenging problem. Unfortunately, little is known about the lattice of all subquasivarieties of a given quasivariety. The theory of hyperidentities (hyperquasi-identities) and hypersubstitutions, to which the author of this monograph contributed a lot in the past twenty years, is offering such methods. For each submonoid M of the monoid of all hypersubstitutions of a given type, M-solid varieties and M-solid quasivarieties were invented in order to consider a complete sublattice of the lattice of all subvarieties and subquasivarieties, respectively. The concept of an M-derived algebra and an M-derived quasivariety is a useful tool. Some of the concepts were invented by the author and collaborators. The Birkhoff-type characterization of M-hypervarieties and M-hyperquasivarieties is presented. Some newer ideas concern the description of the derivation concepts of M-hyperequational and M-hyperquasi-equational logic. The solution of the M-hyperbasis and M-hyperquasi-basis is given. Results of an M-dimension of a variety are also presented.
