A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras

Abstract

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.