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Canonical extensions of lattices are more than perfect

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Abstract

In a paper published in 2015, we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions of lattices. Here, we firstly introduce morphisms of TiRS structures and put our correspondence between TiRS graphs and TiRS frames into a full categorical framework. We then answer Problem 2 from our 2015 paper 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 illustrate the correspondences between classes of our newly-described PTi lattices and classes of TiRS graphs by examples. We conclude by outlining a direction for future research.

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Acknowledgements

The first author acknowledges the hospitality of Matej Bel University during his visit in September 2017. The third author acknowledges the hospitality of the University of Lisbon during his visit in September 2019 and the position of Visiting Professor at the University of Johannesburg since June 1, 2020.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Johannesburg, PO Box 524,,  Auckland Park, 2006, South Africa

    Andrew P. K. Craig & Miroslav Haviar

  2. Faculdade de Ciências da Universidade de Lisboa, 1749-016, Lisbon, Portugal

    Maria J. Gouveia

  3. Department of Mathematics, Faculty of Natural Sciences, M. Bel University, Tajovského 40, 974 01, Banská Bystrica, Slovakia

    Miroslav Haviar

Authors
  1. Andrew P. K. Craig
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  2. Maria J. Gouveia
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  3. Miroslav Haviar
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Correspondence to Andrew P. K. Craig.

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Presented by P. Jipsen.

In memory of Bjarni Jónsson.

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A. P. K. Craig acknowledges the support of the NRF South Africa (Grant 127266) and M. Haviar acknowledges the support of Slovak Grant VEGA 1/0337/16.

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Craig, A.P.K., Gouveia, M.J. & Haviar, M. Canonical extensions of lattices are more than perfect. Algebra Univers. 83, 12 (2022). https://doi.org/10.1007/s00012-022-00769-2

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