Checking Admissibility Using Natural Dualities

TL;DR

This paper introduces a novel method leveraging natural dualities to efficiently verify admissibility in finitely generated quasivarieties, overcoming limitations of previous algebraic approaches.

Contribution

The paper develops a new duality-based technique for checking admissibility that handles smaller structures and surjective morphisms, enabling solutions for complex quasivarieties.

Findings

Successfully applied to MS-algebras, double Stone algebras, and involutive Stone algebras.

Allows checking admissibility using smaller algebras and surjective morphisms.

Solves cases previously infeasible with algebraic methods.

Abstract

This paper presents a new method for obtaining small algebras to check the admissibility-equivalently, validity in free algebras-of quasi-identities in a finitely generated quasivariety. Unlike a previous algebraic approach of Metcalfe and Rothlisberger that is feasible only when the relevant free algebra is not too large, this method exploits natural dualities for quasivarieties to work with structures of smaller cardinality and surjective rather than injective morphisms. A number of case studies are described here that could not be be solved using the algebraic approach, including (quasi)varieties of MS-algebras, double Stone algebras, and involutive Stone algebras.