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.
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(M. V. Schwidefsky) The work is supported by Russian Science Foundation (project 14-21-00065).
Translated from Algebra i Logika, Vol. 54, No. 3, pp. 381–398, May-June, 2015.
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Schwidefsky, M.V. Complexity of Quasivariety Lattices. Algebra Logic 54, 245–257 (2015). https://doi.org/10.1007/s10469-015-9344-7
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DOI: https://doi.org/10.1007/s10469-015-9344-7