On selected developments in the theory of natural dualities

TL;DR

This survey reviews key developments in the theory of natural dualities over 20 years, highlighting breakthroughs, open problems, and applications to various algebraic structures.

Contribution

The paper provides a comprehensive reflection on the author's contributions to natural dualities, including new insights and unresolved problems in the field.

Findings

Identified main open problems: Dualisability and Decidability.

Analyzed applications to entailment and endodualisability.

Discussed the Full versus Strong Duality problem.

Abstract

This is a survey on selected developments in the theory of natural dualities where the author had the opportunity to make with his foreign colleagues several breakthroughs and move the theory forward. It is aimed as author's reflection on his works on the natural dualities in Oxford and Melbourne over the period of twenty years 1993-2012 (before his attention with the colleagues in universal algebra and lattice theory has been fully focused on the theory of canonical extensions and the theory of bilattices). It is also meant as a remainder that the main problems of the theory of natural dualities, Dualisability Problem and Decidability Problem for Dualisability, remain still open. Theory of natural dualities is a general theory for quasi-varieties of algebras that generalizes `classical' dualities such as Stone duality for Boolean algebras, Pontryagin duality for abelian groups, Priestley duality for distributive lattices, and Hofmann-Mislove-Stralka duality for semilattices. We present a brief background of the theory and then illustrate its applications on our study of Entailment Problem, Problem of Endodualisability versus Endoprimality and then a famous Full versus Strong Problem with related developments.