P-Compatible Hypersubstitution and MP-Solid Varieties

The study of hyperidentities is a growing field of research. While hyperidentities hark back to before 1965 (cf. 1]), they have found a rebirth in the late seventies and early eighties (cf. 8], 9]). 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. This revised version was published online in June 2006 with corrections to the Cover Date.