There are only countably many locally tabular bi-intermediate logics of co-trees

Abstract

A bi-Heyting algebra validates the G\"odel-Dummett axiom $(p \to q) \lor (q \to p)$ iff the poset of its prime filters is a disjoint union of co-trees. Bi-Heyting algebras of this kind are called bi-G\"odel algebras and form a variety $\operatorname{\mathsf{bi-GA}}$ that algebraizes the extension $\operatorname{\mathsf{bi-GD}}$ of bi-intuitionistic logic axiomatized by the G\"odel-Dummett axiom. In this paper we show that there are only countably many locally tabular bi-intermediate logics of co-trees, all of which are finitely axiomatizable. The theory of canonical formulas of bi-G\"odel algebras has shown that $\operatorname{\mathsf{bi-GA}}$ has continuum many subvarieties, among which the locally finite ones coincide with the subvarieties of the $\mathsf{V}_n \coloneqq \{\mathbf{A} \in \operatorname{\mathsf{bi-GA}} \colon \mathbf{A} \models \beta(\mathfrak{C}_n)\}$ (where $\beta(\mathfrak{C}_n)$ is the subframe formula of the $n$-comb). We identify the multiset projectivity relation (a binary relation that, when defined on the set of finite multisets of a better partial order, is necessarily a better partial order) and use it to prove that every $\mathsf{V}_n$ is a Specht variety, hence has only countably many subvarieties, all of which are finitely axiomatizable. By the algebraizability of $\operatorname{\mathsf{bi-GD}}$, the main result follows. We also provide an informative depiction of the lattice of varieties of bi-G\"odel algebras.