# Sugihara algebras and monoids: multisorted dualities

## TL;DR

This paper develops a unified multisorted duality framework for Sugihara algebras and monoids, enabling better analysis of their algebraic properties and applications in relevance logic.

## Contribution

It introduces a comprehensive suite of multisorted duality theorems for Sugihara algebras and monoids, extending previous dualities to include both odd and even cases.

## Key findings

- Unified duality theorems for Sugihara algebras and monoids
- Enhanced representations of free algebras
- Foundation for further applications in logic and algebra

## Abstract

The authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by Cabrer et al., give access to a viable method for studying admissible rules within relevance logic, specifically for extensions of the deductive system $R$-mingle. This paper builds on the work already done on the theory of natural dualities for Sugihara algebras. Its purpose is to provide an integrated suite of multisorted duality theorems of a uniform type, encompassing finitely generated quasivarieties and varieties of both Sugihara algebras and Sugihara monoids, and embracing both the odd and the even cases. The overarching theoretical framework of multisorted duality theory developed here leads on to amenable representations of free algebras. More widely, it provides a springboard to further applications.