Uniquely Representable Posets
Mai Gehrke · Annals of the New York Academy of Sciences · 1994
ABSTRACT: It is known that a partially ordered set is uniquely representable, that is, it is order isomorphic to the order component of a Priestley space of a unique bounded distributive lattice whenever this lattice is generated by its doubly irreducible elements. In this case the interval topology on the poset is Hausdorff. Priestley has conjectured that the interval topology is always Hausdorff when the poset is uniquely representable. We consider this problem in topological terms. This allows us to see how the conjecture is a natural one and to broaden the class of lattices that are known to have uniquely representable spectral sets. Furthermore, we see that the theory of generalized continuous lattices ends up playing a crucial role, and that the conjecture perhaps should be strengthened to assert that for uniquely representable posets, the entire interval of admissible topologies collapses.