This article is cited in 2 scientific papers (total in 2 papers)
On filters in the lattice of quasivarieties of groups
A. I. Budkin
Suppose $\mathfrak M$ is a quasivariety of groups. Assume there exist finitely presented groups $A$ and $B$ such that $A\notin\mathfrak M$, $B\in\mathfrak M$, and $B$ is not contained in the quasivariety generated by $A$. It is proved that the principal filter generated in the lattice of quasivarieties of groups by has the cardinality of the continuum. In particular, the principal filters generated by a) the quasivariety generated by all proper varieties of groups, b) the quasivariety of all $RN$-groups and c) the quasivariety generated by all periodic groups, have the cardinality of the continuum. It is shown that the smallest quasivariety of groups containing all proper quasivarieties of groups in which nontrivial quasi-identities of the form
(\forall x_1)…(\forall x_n)(f(x_1,…,x_n)=1\to g(x_1,…,x_n)=1),
are true, where $f$ and $g$ are terms of group signature, does not coincide with the class of all groups.
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Mathematics of the USSR-Izvestiya, 1989, 33:1, 201–207
A. I. Budkin, “On filters in the lattice of quasivarieties of groups”, Izv. Akad. Nauk SSSR Ser. Mat., 52:4 (1988), 875–881; Math. USSR-Izv., 33:1 (1989), 201–207
Citation in format AMSBIB
\paper On~filters in the lattice of quasivarieties of groups
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\jour Math. USSR-Izv.
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A. I. Budkin, “Semivarieties of nilpotent groups”, Algebra and Logic, 49:5 (2010), 389–399
A. V. Kravchenko, A. M. Nurakunov, M. V. Schwidefsky, “Structure of Quasivariety Lattices. I. Independent Axiomatizability”, Algebra and Logic, 57:6 (2019), 445–462
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