PROPER P-COMPATIBLE HYPERSUBSTITUTIONS

Abstract

An identity s ≈ t is called a hyperidentity in a variety V if by substituting terms of appropriate arity for the operation symbols in s ≈ t, one obtains an identity satisfied in V . If every identity in V is a hyperidentity, the variety V is called solid. All solid varieties of a given type τ form a complete sublattice S(τ) of the lattice L(τ) of all varieties of type τ . The concept of an M-solid variety generalizes that of a solid variety. An equation s ≈ t of terms of type τ is called P-compatible where P is a partition of the set F = {fi|i ∈ I} of operation symbols of type τ if it has the form xi ≈ xi or fi(t1,...,tni) ≈ fj(t1,...,t nj) with fj ∈ [fi]P , where [ fj]P is the block of P containing fj. A variety is called P-compatible if it contains only P-compatible identities. All Pcompatible varieties of type τ form also a sublattice of the lattice of all varieties of type τ. We ask for the intersection of both lattices, i.e. we want to characterize solid varieties which are P-compatible or M-solid varieties which are P-compatible.