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| Formato: | Preprint |
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2026
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| Acceso en línea: | https://arxiv.org/abs/2602.21960 |
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| _version_ | 1866915816193130496 |
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| author | Martins, Miguel |
| author_facet | Martins, Miguel |
| contents | A bi-Heyting algebra validates the Gödel-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-Gödel algebras and form a variety $\operatorname{\mathsf{bi-GA}}$ that algebraizes the extension $\operatorname{\mathsf{bi-GD}}$ 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 $\operatorname{\mathsf{bi-GA}}$ has continuum many subvarieties, among which the locally finite ones coincide with the subvarieties of the $\mathsf{V}_n \coloneqq \{\mathbf{A} \in \operatorname{\mathsf{bi-GA}} \colon \mathbf{A} \models β(\mathfrak{C}_n)\}$ (where $β(\mathfrak{C}_n)$ 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-Gödel algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21960 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | There are only countably many locally tabular bi-intermediate logics of co-trees Martins, Miguel Logic A bi-Heyting algebra validates the Gödel-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-Gödel algebras and form a variety $\operatorname{\mathsf{bi-GA}}$ that algebraizes the extension $\operatorname{\mathsf{bi-GD}}$ 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 $\operatorname{\mathsf{bi-GA}}$ has continuum many subvarieties, among which the locally finite ones coincide with the subvarieties of the $\mathsf{V}_n \coloneqq \{\mathbf{A} \in \operatorname{\mathsf{bi-GA}} \colon \mathbf{A} \models β(\mathfrak{C}_n)\}$ (where $β(\mathfrak{C}_n)$ 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-Gödel algebras. |
| title | There are only countably many locally tabular bi-intermediate logics of co-trees |
| topic | Logic |
| url | https://arxiv.org/abs/2602.21960 |