Algebra universalis

, Volume 64, Issue 1–2, pp 69–100

# 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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