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 𝒦0 of torsion-free groups, (b) do not have an independent basis of quasi-identities in 𝒦0, and (c) the intersection of all quasivarieties in ℛ has an independent quasi-identity basis in 𝒦0. The collection of such sets ℛ has the cardinality of the continuum.

中文翻译：

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无扭群拟变数的ω-独立基

证明存在一组 ℛ 无扭群的拟变异，其中 (a) 在无扭群的类 𝒦0 中具有 ω 独立的拟恒等基，(b) 不具有𝒦0 中的准恒等式，以及 (c) ℛ 中所有准恒等式的交集在 𝒦0 中具有独立的准恒等基。此类集合 ℛ 的集合具有连续统的基数。