ON FILTERS IN THE LATTICE OF QUASIVARIETIES OF GROUPS
A. I. Budkin · Mathematics of the USSR-Izvestiya · 1989
Suppose is a quasivariety of groups. Assume there exist finitely presented groups and such that , , and is not contained in the quasivariety generated by . 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 -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 are true, where and are terms of group signature, does not coincide with the class of all groups. Bibliography: 12 titles.