P-compatible identities are built up from terms with a special structure. We investigate a variety defined by a set ofP-compatible hybrid identities and answer the question whether a variety defined by a set ofP-compatible hyperidentities can be solid.

Hybrid identities are sentences in a special second order language with identity. The model classes of sets of hybrid identities are called hybrid solid varieties. We give a Birkhoff-type-characterization of hybrid solid varieties and develop a hybrid equational logic.

The study of hyperidentities is a growing field of research. While hyperidentities hark back to before 1965, they have found a rebirth in the late seventies and early eighties. It is being expanded in several directions, from connections with clone theory, to finite basis problems, to semigroup theory, to classification of M-solid varieties. Applications to digital logic, formal languages, and hypertext systems have been suggested. The concept of a P-compatible equation, where P is a partition on the set of operation (...) symbols, is a good tool to study the structure of identities. In [4] we asked for P-compatible hyperidentities. In this paper we will consider hypersubstitutions which are compatible with the partition P and will develop a generalized equational theory for certain P-compatible hyperidentities. (shrink)