Sugihara Algebras: Admissibility Algebras via the Test Spaces Method

TL;DR

This paper develops a duality-based method to identify minimal algebras for testing rule admissibility in Sugihara algebra quasivarieties, advancing the algebraic understanding of relevant logic R-mingle.

Contribution

It introduces the Test Spaces Method for Sugihara algebras, enabling the construction of admissibility algebras using natural duality theory.

Findings

Constructed strong dualities for Sugihara algebra quasivarieties.

Identified minimal admissibility algebras for these quasivarieties.

Provided a framework for analyzing admissibility in relevant logic R-mingle.

Abstract

This paper studies finitely generated quasivarieties of Sugihara algebras. These quasivarieties provide complete algebraic semantics for certain propositional logics associated with the relevant logic R-mingle. The motivation for the paper comes from the study of admissible rules. Recent earlier work by the present authors, jointly with Freisberg and Metcalfe, laid the theoretical foundations for a feasible approach to this problem for a range of logics---the Test Spaces Method. The method, based on natural duality theory, provides an algorithm to obtain the algebra of minimum size on which admissibility of sets of rules can be tested. (In the most general case a set of such algebras may be needed rather than just one.) The method enables us to identify this `admissibility algebra' for each quasivariety of Sugihara algebras which is generated by an algebra whose underlying lattice is a finite chain. To achieve our goals, it was first necessary to develop a (strong) duality for each of these quasivarieties. The dualities promise also to also provide a valuable new tool for studying the structure of Sugihara algebras more widely.