Complexity of Quasivariety Lattices

• DOI: 10.1007/s10469-015-9344-7
• Publication Date: 2015-10-22T05:47:06Z
• Authors: M. V. Schwidefsky
• Keywords: Undecidable problem; Lattice (music); Chatterjee

If a quasivariety A of algebraic systems of finite signature satisfies some generalization of a sufficient condition for Q-universality treated by M. E. Adams and W. A. Dziobiak, then, for any at most countable set {Si | i ∈ I} of finite semilattices, the lattice $$ {\displaystyle \prod_{i\in I}\mathrm{S}\mathrm{u}\mathrm{b}\left({S}_i\right)} $$ is a homomorphic image of some sublattice of a quasivariety lattice Lq(A). Specifically, there exists a subclass K ⊆ A such that the problem of embedding a finite lattice in a lattice Lq(K) of K -quasivarieties is undecidable. This, in particular, implies a recent result of A. M. Nurakunov.