ABSTRACT
In Chapter 4 we generalized different variations of the concept of a hyperiden tity and a solid variety into one pattern, which we called M-hyperidentities and M-solidity. For any monoid M of hypersubstitutions, one may define Mhyperidentities and hence M-solid varieties by restricting the set of term oper ations, which must lead to identities when substituted for operation symbols, to come from M. Besides encompassing several examples into one framework, this generalized approach gives a new way to study lattices of varieties. Instead of the one complete sublattice of solid varieties of type r , for the lattice C(r) we now have a range of complete sublattices, the lattices of M-solid varieties of type r, as M varies over all submonoids of hypersubstitutions. By studying these sublattices, we may learn more about the whole lattice. This is especially true when we consider the lattice of varieties of semigroups and its sublattices of M-solid varieties, since more is known about these varieties.
