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Piggyback dualities revisited

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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-forone 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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, La Trobe University, Victoria, 3086, Australia

    B. A. Davey

  2. Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01, Banská Bystrica, Slovak Republic

    M. Haviar

  3. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, OX2 6GG, United Kingdom

    H. A. Priestley

Authors
  1. B. A. Davey
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  2. M. Haviar
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  3. H. A. Priestley
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Corresponding author

Correspondence to B. A. Davey.

Additional information

Presented by M. Ploscica.

Dedicated to the memory of Ervin Fried and Jiří Sichler

The second author acknowledges support from Slovak grants VEGA 1/0212/13 and APVV-0223-10.

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Davey, B.A., Haviar, M. & Priestley, H.A. Piggyback dualities revisited. Algebra Univers. 76, 245–285 (2016). https://doi.org/10.1007/s00012-016-0395-y

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