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

Miguel Martins

Abstract

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.

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

Abstract

A bi-Heyting algebra validates the Gödel-Dummett axiom 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 that algebraizes the extension 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 has continuum many subvarieties, among which the locally finite ones coincide with the subvarieties of the (where is the subframe formula of the -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 is a Specht variety, hence has only countably many subvarieties, all of which are finitely axiomatizable. By the algebraizability of , the main result follows. We also provide an informative depiction of the lattice of varieties of bi-Gödel algebras.
Paper Structure (4 sections, 24 theorems, 82 equations, 10 figures)

This paper contains 4 sections, 24 theorems, 82 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Better partial orders
  3. Bi-Gödel algebras and co-trees
  4. Counting the locally finite varieties of bi-Gödel algebras

Key Result

Theorem 1.1

Paper1 Let ${\vdash} \in \Lambda(\operatorname{\mathsf{bi-GD}})$ and $\mathsf{V}_\vdash$ be its variety of bi-Gödel algebras. The following conditions are equivalent:

Figures (10)

  • Figure 1: The $n$-comb $\mathfrak{C}_n$, where $n \in \mathbb{Z}^+$.
  • Figure 2: The co-trees $\mathfrak{F}_0$, $\mathfrak{F}_1$, $\mathfrak{F}_2,\text{ and }\mathfrak{F}_3$.
  • Figure 3: The lattice $\Lambda(\operatorname{\mathsf{bi-GA}})$ of nontrivial subvarieties of bi-Gödel algebras.
  • Figure 4: The $n$-hcomb $\mathfrak{C}_n'$.
  • Figure 5: Rado's Poset
  • ...and 5 more figures

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5: BPO
  • Theorem 2.6
  • Definition 2.7
  • ...and 29 more