This article is cited in 13 papers

Mathematical logic, algebra and number theory

Lattices of subclasses. III

A. Basheyeva^a, A. Nurakunov^b, M. Schwidefsky^cd, A. Zamojska-Dzienio<sup>e

a</sup> The L.N. Gumilyov Eurasian National University, Satpaev str. 2, 010000 Astana, Kazakhstan
^b Institute of Mathematics of the National Academy of Sciences, Chui prosp. 265a, 720071 Bishkek, Kyrgyzstan
^c Sobolev Institute of Mathematics of the Siberian Branch RAS, Acad. Koptyug prosp. 4, 630090 Novosibirsk, Russia
^d Novosibirsk State University, Pirogova str. 1, 630090 Novosibirsk, Russia
^e Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa str. 75, 00-662 Warsaw, Poland

Abstract: We prove that for certain $Q$-universal quasivarieties $\mathbf{K}$, the lattice of $\mathbf{K}$-quasivarieties contains continuum many subquasivarieties with the undecidable quasi-equational theory and for which the finite membership problem is also undecidable. Moreover, we prove that certain $Q$-universal quasivarieties have continuum many subquasivarieties with no independent quasi-equational basis.

Keywords: Abelian group, differential groupoid, finite membership problem, graph, independent basis, quasi-identity, quasi-equational theory, quasivariety, $Q$-universal, undecidable theory.

UDC: 512.56, 512.57

MSC: 06B15, 08C15

Received February 17, 2017, published March 24, 2017

Language: English

DOI: 10.17377/semi.2017.14.023

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