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Priestley Rings and Priestley Order-Compactifications

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Abstract

We introduce Priestley rings of upsets (of a poset) and prove that inequivalent Priestley ring representations of a bounded distributive lattice L are in 1-1 correspondence with dense subspaces of the Priestley space of L. This generalizes a 1955 result of Bauer that inequivalent reduced field representations of a Boolean algebra B are in 1-1 correspondence with dense subspaces of the Stone space of B. We also introduce Priestley order-compactifications and Priestley bases of an ordered topological space, and show that they are in 1-1 correspondence. This generalizes a 1961 result of Dwinger that zero-dimensional compactifications of a topological space are in 1-1 correspondence with its Boolean bases.

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Authors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003-8001, USA

    Guram Bezhanishvili & Patrick J. Morandi

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  1. Guram Bezhanishvili
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  2. Patrick J. Morandi
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Correspondence to Patrick J. Morandi.

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Bezhanishvili, G., Morandi, P.J. Priestley Rings and Priestley Order-Compactifications. Order 28, 399–413 (2011). https://doi.org/10.1007/s11083-010-9180-2

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  • DOI: https://doi.org/10.1007/s11083-010-9180-2

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