Using coloured ordered sets to study finite-level full dualities

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• 作者：Brian A. Davey ; Miroslav Haviar and Jane G. Pitkethly
• 关键词：Natural duality ; full duality ; Priestley duality
• 刊名：Algebra Universalis
• 出版时间：October 2010
• 出版年：2010
• 期刊代码：8_0002-5240
• 类别：mat
• 卷：64
• 期：1-2
• 页码：69-100
• 数据来源：sp

摘要

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 F₃{\mathcal{F}_{\underline{3}}}. Using Priestley duality, we establish a correspondence between the elements of F₃{\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 F₃{\mathcal{F}_{\underline{3}}} has cardinality 2脌₀{2^{\aleph_{0}}} and is non-modular. This is the first investigation into the structure of an infinite lattice of finite-level full dualities.