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

Abstract

A bi-Heyting algebra validates the Gödel-Dummett axiom $(p\to q)\vee (q\to p)$ iff the poset of its prime filters is a disjoint union of co-trees. Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety $\mathsf{bi}-\mathsf{GA}$ that algebraizes the extension $\mathsf{bi}-\mathsf{GD}$ of bi-intuitionistic logic axiomatized by the Gödel-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-Gödel algebras has shown that $\mathsf{bi}-\mathsf{GA}$ has continuum many subvarieties, among which the locally finite ones coincide with the subvarieties of the ${\mathsf{V}}_n:=\{\mathbf{A}\in \mathsf{bi}-\mathsf{GA}:\mathbf{A}\models \beta ({\mathfrak{C}}_n)\}$ (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 $\mathsf{bi}-\mathsf{GD}$, the main result follows. We also provide an informative depiction of the lattice of varieties of bi-Gödel algebras.