ω-Independent Bases for Quasivarieites of Torsion-Free Groups

Abstract

It is proved that there exists a set ℛ of quasivarieties of torsion-free groups which (a) have an ω-independent basis of quasi-identities in the class ????₀ of torsion-free groups, (b) do not have an independent basis of quasi-identities in ????₀, and (c) the intersection of all quasivarieties in ℛ has an independent quasi-identity basis in ????₀. The collection of such sets ℛ has the cardinality of the continuum.