The complexity of quasivariety lattices for the classes of differential groupoids

Abstract

In this work we explore the complexity of structure of (relative) quasivariety lattices for the classes of differential groupoids. A question, what is the complexity of quasivariety lattice and what quasivariety lattices are complex according to this or that measure of complexity and which are not, studied by many authors. Two complexity measures of the structure of quasivariety lattices are known: unreasonability (non-computability the set of all finite sublattices of the quasivariety lattices) and Q-universality. Unreasonability of the quasivariety lattice means that there is no algorithm which would determine by the given finite lattice, this lattice is embeddable into the considered quasivariety lattice or not. Other complexity measure of the structure of quasivariety lattices is expressed by the concept of Q-universality. It means that the quasivariety lattice for any quasivariety of finite signature is a homomorphic image of some sublattice of the considered Q-universal lattice of quasivarieties. Two years ago it was established a connection between unreasonability and Q-universality; it was also posed the following problem. Does any Q-universal class of algebraic structures of a fixed signature contain a unreasonable subclass? Is there a class of algebraic structures which is not Q-universal but which is nevertheless unreasonable? We find a non-trivial identity holding in quasivariety lattices for the classes of differential groupoids. It is proved that there are continuum many unreasonable classes of the differential groupoids which are not Q-universal.