Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maehara Interpolation in Extensions of R-mingle

Wesley Fussner, Krzysztof Krawczyk

TL;DR

The paper classifies which quasivarieties of Sugihara algebras possess the amalgamation property and links these algebraic properties to logical interpolation phenomena in extensions of RM without constants. By leveraging closure properties and the Rel Jónsson Lemma, it identifies exactly five AP quasivarieties: V(Z1), V(Z2), V(Z3), Q(E), and V(Z), and shows AP implies the relative congruence extension property and TIP, hence aligning AP with MIP for RM extensions. It then translates these algebraic results into a complete list of RM extensions with Maehara interpolation, tying them to Dugundji formulas δ_n, and proves that RP and MIP coincide in this setting. Additional decidability results are established: for finitely based, locally finite SA-quasivarieties, it is decidable whether they have AP, and whether finite RM extensions have MIP. The work thus provides a concrete, decidable map of the interpolation landscape surrounding R-mingle extensions.

Abstract

We show that there are exactly five quasivarieties of Sugihara algebras with the amalgamation property, and that all of these have the relative congruence extension property. As a consequence, we obtain that the amalgamation property and transferable injections property coincide for arbitrary quasivarieties of Sugihara algebras. These results provide a complete description of arbitrary (not merely axiomatic) extensions of the logic R-mingle that have the Maehara interpolation property, and further demonstrates that the Robinson property and Maehara interpolation property coincide for arbitrary extensions of R-mingle. Further, we show that the question of whether a given finitely based extension of R-mingle has the Maehara interpolation property is decidable.

Maehara Interpolation in Extensions of R-mingle

TL;DR

The paper classifies which quasivarieties of Sugihara algebras possess the amalgamation property and links these algebraic properties to logical interpolation phenomena in extensions of RM without constants. By leveraging closure properties and the Rel Jónsson Lemma, it identifies exactly five AP quasivarieties: V(Z1), V(Z2), V(Z3), Q(E), and V(Z), and shows AP implies the relative congruence extension property and TIP, hence aligning AP with MIP for RM extensions. It then translates these algebraic results into a complete list of RM extensions with Maehara interpolation, tying them to Dugundji formulas δ_n, and proves that RP and MIP coincide in this setting. Additional decidability results are established: for finitely based, locally finite SA-quasivarieties, it is decidable whether they have AP, and whether finite RM extensions have MIP. The work thus provides a concrete, decidable map of the interpolation landscape surrounding R-mingle extensions.

Abstract

We show that there are exactly five quasivarieties of Sugihara algebras with the amalgamation property, and that all of these have the relative congruence extension property. As a consequence, we obtain that the amalgamation property and transferable injections property coincide for arbitrary quasivarieties of Sugihara algebras. These results provide a complete description of arbitrary (not merely axiomatic) extensions of the logic R-mingle that have the Maehara interpolation property, and further demonstrates that the Robinson property and Maehara interpolation property coincide for arbitrary extensions of R-mingle. Further, we show that the question of whether a given finitely based extension of R-mingle has the Maehara interpolation property is decidable.

Paper Structure

This paper contains 6 sections, 27 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Algebraic preliminaries and the basics of Sugihara algebras
  3. Algebra and amalgamation
  4. Sugihara algebras and Sugihara monoids
  5. Amalgamation in Sugihara algebras
  6. From amalgamation to interpolation

Key Result

Lemma 1

Let ${\sf K}$ be any class of similar algebras. Then every nontrivial member of ${\mathbb Q}({\sf K})_\mathrm{RFSI}$ belongs to ${\mathbb I}{\mathbb S}{\mathbb P}_\mathsf{U}({\sf K})$.

Figures (2)

  • Figure 1: Commutative diagrams for the AP (left) and TIP (right).
  • Figure 2: The poset $\Omega({\sf SA})$ of quasivarieties of Sugihara algebras with the amalgamation property.

Theorems & Definitions (39)

  • Lemma 1: see, e.g., Czel1990
  • Lemma 2: Czel1990
  • Lemma 3: Czel1990
  • Lemma 4: Fussner2024
  • Lemma 5: Fussner2024
  • Lemma 6: see, e.g., FontPerez1992
  • Lemma 7: Blok1986
  • Lemma 8: see FGSugiharaCabPri2020
  • Lemma 9
  • proof
  • ...and 29 more