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A Spectral-style Duality for Distributive Posets

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Abstract

In this paper, we present a topological duality for a category of partially ordered sets that satisfy a distributivity condition studied by David and Erné. We call these posets mo-distributive. Our duality extends a duality given by David and Erné because our category of spaces has the same objects as theirs but the class of morphisms that we consider strictly includes their morphisms. As a consequence of our duality, the duality of David and Erné easily follows. Using the dual spaces of the mo-distributive posets we prove the existence of a particular Δ1-completion for mo-distributive posets that might be different from the canonical extension. This allows us to show that the canonical extension of a distributive meet-semilattice is a completely distributive algebraic lattice.

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Acknowledgements

We would like to thank the referee for his/her suggestions and comments that allowed to improve the presentation of the paper.

The first author was partially supported by Universidad Nacional de La Pampa (FCEyN) under the grant P. I. 64, Resol. 141/15; and also by Consejo Nacional de Investigaciones Científicas y Técnicas under the grant PIP 112-20150-100412CO, all from the goverment of Argentina.

The second author was partially supported by the research grant 2014 SGR 788 from the government of Catalonia; and also by the research projects MTM2011-25747 and MTM2016-74892-P (which include feder funds from the European Union) and the project MDM-2014-044 of the María de Maeztu program, all from the government of Spain.

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Authors and Affiliations

  1. Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa, Uruguay 151, 6300, Santa Rosa, Argentina

    Luciano J. González

  2. Department de Filosofia, IMUB and BGSMath, Universitat de Barcelona, Montalegre 6, 08001, Barcelona, Spain

    Ramon Jansana

Authors
  1. Luciano J. González
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  2. Ramon Jansana
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Correspondence to Luciano J. González.

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González, L.J., Jansana, R. A Spectral-style Duality for Distributive Posets. Order 35, 321–347 (2018). https://doi.org/10.1007/s11083-017-9435-2

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  • DOI: https://doi.org/10.1007/s11083-017-9435-2

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