Canonical extensions: an algebraic approach to Stone duality

Gehrke, Mai

doi:10.1007/s00012-018-0544-6

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2020

2022

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Cited by 3 publications

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“…This has its origins in the 1951-52 papers of Jónsson and Tarski [13]. We refer to Gehrke and Vosmaer [10] for a survey of the theory of canonical extensions for lattice-based algebras and, for further background, to recent papers by Gehrke [8] and Goldblatt [11] and the references there, in particular to the first section of [11] called "A biography of canonical extension".…”

Section: Introductionmentioning

confidence: 99%

Canonical extensions of lattices are more than perfect

Craig¹,

Gouveia²,

Haviar³

2020

Preprint

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In [2] we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions of lattices. In this continuation of [2] we answer Problem 2 from there by characterising the perfect lattices that are dual to TiRS frames (and hence TiRS graphs). We introduce a new subclass of perfect lattices called PTi lattices and show that the canonical extensions of lattices are PTi lattices, and so are 'more' than just perfect lattices. We introduce morphisms of TiRS structures and put our correspondence between TiRS graphs and TiRS frames from [2] into a full categorical framework. We illustrate our correspondences between classes of perfects lattices and classes of TiRS graphs by examples.

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