Representable Forests and Diamond Systems
Damiano Fornasiere
Abstract:
We study the classical problem of representing partially ordered sets as prime spectra. A poset is said to be Priestley (resp. Esakia) representable if it is isomorphic to the prime spectrum of a bounded distributive lattice (resp. Heyting algebra). We study this problem by restricting the attention to two classes of posets: forests, i.e., disjoint union of trees, and diamond systems, a class that includes the order duals of forests. This class has been introduced recently in order to characterize the varieties of Heyting algebras whose profinite members are profinite completions.
We provide a characterization of Priestley and Esakia representable diamond systems. As Priestley representable posets are closed under order duals, this yields a new proof of Lewisâ€™ description of Priestley representable forests. While a classification of arbitrary Esakia representable forests remains open, the main result of this thesis gives a full description of the well-ordered ones. Moreover, we investigate the Esakia representability of countable forests and provide two forbidden configurations of Esakia representable countable forests. We also prove a number of facts about Priestley and Esakia topologies on arbitrary posets. In particular, we identify some properties of Priestley (resp. Esakia) topologies that revolve around infinite chains.