In this paper, we show that there exist (continuum many) varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist (continuum many) extensions of the Heyting–Brouwer logic $HB$ that are topologically incomplete. This result provides further insight into the long-standing open problem of Kuznetsov by yielding a negative solution of the reformulation of the problem from extensions of $IPC$ to extensions of $HB$.

中文翻译：

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库兹涅佐夫问题的双海廷代数簇的负解

在本文中，我们表明存在（连续统许多）双 Heyting 代数的变体，这些代数不是由其完整成员生成的。由此可见，存在（连续许多）Heyting–Brouwer 逻辑的扩展$H乙$在拓扑上是不完整的。这一结果通过从$我PC$扩展到$H乙$.