The spectral theory of distributive continuous lattices

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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. © 1978 American Mathematical Society.

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Transactions of the American Mathematical Society

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