The subquasivariety lattice of a discriminator variety

Let V' be a discriminator variety such that the class A = {A is an element of V': A is simple and has no trivial subalgebra} is closed under ultraproducts. This property holds, for example, if V' is locally finite or if the language is finite. Let v(V') and q(V') denote the lattice of subvarieties and subquasivarieties of V', respectively. We prove that q(V') is modular iff q(V') is distributive iff v(V') satisfies a certain condition where the case in which the language has a constant symbol is "v(V') is a chain or q(V') = v(V')." We give an isomorphism between q(V') and a lattice constructed in terms of v(V'). Via this isomorphism we characterize the completely meet irreducible (prime) elements of q(V') in terms of the completely meet irreducible elements of v(V'). We conclude the paper with applications to the varieties of Boolean algebras, relatively complemented distributive lattices, Lukasiewicz algebras. Post algebras, complementary semigroups of rank k (x(n) approximate to x)-rings. R5 lattices (P-algebras, B-algebras), and monadic algebras. (C) 2001 Academic Press.