Piggyback dualities revisited

TL;DR

This paper extends the piggyback duality technique in natural duality theory to a broader class of structures, providing new criteria for strong and two-for-one dualities, and unifying many existing dualities under a common framework.

Contribution

It generalizes piggyback duality theorems to prevarieties of structures, enabling the derivation of new dualities and criteria for strong and two-for-one dualities.

Findings

Derived piggyback duality theorems for prevarieties of structures.

Established criteria for strong and two-for-one dualities.

Unified existing dualities like Priestley and Hofmann--Mislove--Stralka under the extended framework.

Abstract

In natural duality theory, the piggybacking technique is a valuable tool for constructing dualities. As originally devised by Davey and Werner, and extended by Davey and Priestley, it can be applied to finitely generated quasivarieties of algebras having term-reducts in a quasivariety for which a well-behaved natural duality is already available. This paper presents a comprehensive study of the method in a much wider setting: piggyback duality theorems are obtained for suitable prevarieties of structures. For the first time, and within this extended framework, piggybacking is used to derive theorems giving criteria for establishing strong dualities and two-for-one dualities. The general theorems specialise in particular to the familiar situation in which we piggyback on Priestley duality for distributive lattices or Hofmann--Mislove--Stralka duality for semilattices, and many well-known dualities are thereby subsumed. A selection of new dualities is also presented.