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Algebra universalis

, Volume 64, Issue 1–2, pp 69–100 | Cite as

Using coloured ordered sets to study finite-level full dualities

  • Brian A. Davey
  • Miroslav Haviar
  • Jane G. Pitkethly
Article

Abstract

We consider all the full dualities for the class of finite bounded distributive lattices that are based on the three-element chain 3. Under a natural quasi-order, these full dualities form a doubly algebraic lattice \({\mathcal{F}_{\underline{3}}}\). Using Priestley duality, we establish a correspondence between the elements of \({\mathcal{F}_{\underline{3}}}\) and special enriched ordered sets, which we call ‘coloured ordered sets’. We can then use combinatorial arguments to show that the lattice \({\mathcal{F}_{\underline{3}}}\) has cardinality \({2^{\aleph_{0}}}\) and is non-modular. This is the first investigation into the structure of an infinite lattice of finite-level full dualities.

2000 Mathematics Subject Classification

Primary: 06D50 Secondary: 06D05 06A07 

Keywords and phrases

Natural duality full duality Priestley duality 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Brian A. Davey
    • 1
  • Miroslav Haviar
    • 2
  • Jane G. Pitkethly
    • 1
  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityVictoriaAustralia
  2. 2.Research Institute, Matej Bel UniversityBanska BystricaSlovak Republic

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