Narrow Results

A quasivariety is a universal Horn class of algebraic structures containing the trivial structure. The set of all subquasivarieties of a quasivariety forms a complete lattice under inclusion. A lattice isomorphic to for some quasivariety is called a lattice of quasivarieties or a quasivariety lattice. The Birkhoff–Maltsev Problem asks which lattices are isomorphic to lattices of quasivarieties. A lattice L is called unreasonable if the set of all finite sublattices of L is not computable, that is, there is no algorithm for deciding whether a finite lattice is a sublattice of L. The main result of this paper states that for any signature σ containing at least one non-constant operation, there is a quasivariety of signature σ such that the quasivariety lattice is unreasonable. Moreover, there are uncountable unreasonable lattices of quasivarieties. We also present some corollaries of the main result.

We prove that the class K(σ) of all algebraic structures of signature σ is Q-universal if and only if there is a class K ⊆ K(σ) such that the problem whether a finite lattice embeds into the lattice of K-quasivarieties is undecidable.

We find sufficient conditions guaranteeing that for a quasivariety $M$ of structures of finite type containing a $\mathrm{\text{B}}$-class with respect to $M$, there exists a subquasivariety $K\subseteq M$ and a structure ${𝒜}\in K$ such that the problems whether a finite lattice embeds into the lattice $\mathrm{\text{Lv}}(K)$ of $K$-varieties and into the lattice ${Con}_K{𝒜}$ are undecidable.