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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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
Keywords
- Ordered topological space
- Order-compactification
- Priestley space
- Stone space
- Distributive lattice
- Boolean algebra