Abstract
We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic subsemilattices are those omitting certain types. (3) Compact clopen pseudobasic posets are characterized by separativity. We also show how to obtain the tight spectrum of a poset as the Stone space of a generalized Boolean algebra that is universal for tight representations.
Highlights
BackgroundA number of dualities exist between classes of lattices and topological spaces
We show how to obtain the tight spectrum of a poset as the Stone space of a generalized Boolean algebra that is universal for tight representations
The only issue here is that R-lattices represent relatively compact basic sublattices of RO(X)(= regular open subsets of X ) rather than O(X)
Summary
A number of dualities exist between classes of lattices and topological spaces. Those most relevant to the present paper are summarized below. The only issue here is that R-lattices represent relatively compact basic (ie forming a basis in the usual topological sense) sublattices of RO(X)(= regular open subsets of X ) rather than O(X). Our first goal is to modify the axioms of R-lattices (see Definition 1.1) so as to axiomatize relatively compact basic sublattices of O(X) instead We use a well known set theoretic construction to define a generalized Boolean algebra from any p0set that is universal for tight representations (see Theorem 7.4) This allows us to identify the tight spectrum of the p0set with the Stone space of the algebra, providing a different take on some of the theory from Exel [6]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
