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

TL;DR

This paper demonstrates the existence of numerous bi-Heyting algebra varieties not generated by complete members, leading to many topologically incomplete extensions of Heyting-Brouwer logic and providing a negative answer to Kuznetsov's problem.

Contribution

It introduces a negative solution to Kuznetsov's problem by showing many such algebraic varieties are not generated by their complete members.

Findings

Existence of continuum many varieties of bi-Heyting algebras not generated by complete members

Many extensions of Heyting-Brouwer logic are topologically incomplete

Provides a negative solution to a long-standing open problem in algebraic logic

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 $\mathsf{HB}$ 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 intermediate logics to extensions of $\mathsf{HB}$.