The spectral theory of distributive continuous lattices

Authors:
Karl H. Hofmann and Jimmie D. Lawson

Journal:
Trans. Amer. Math. Soc. **246** (1978), 285-310

MSC:
Primary 54H12; Secondary 06D05, 22A26, 46L99

DOI:
https://doi.org/10.1090/S0002-9947-1978-0515540-7

MathSciNet review:
515540

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Abstract: In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special properties of their spectra are treated. The concept of the patch topology is extended from algebraic lattices to continuous lattices, and necessary and sufficient conditions for its compactness are given.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0515540-7

Keywords:
Continuous lattice,
spectrum,
hull-kernel topology,
locally quasicompact,
sober,
algebraic lattice,
patch topology,
prime

Article copyright:
© Copyright 1978
American Mathematical Society