On~filters in the lattice of quasivarieties of groups

A. I. Budkin

Izvestiya. Mathematics , Tome 33 (1989) no. 1, pp. 201-207

Résumé

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 $$ (\forall\,x_1)\dots(\forall\,x_n)(f(x_1,\dots,x_n)=1\to g(x_1,\dots,x_n)=1), $$ are true, where $f$ and $g$ are terms of group signature, does not coincide with the class of all groups. Bibliography: 12 titles.