Z-Join Spectra of Z-Supercompactly Generated Lattices

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Marcel Erné
  • Dongsheng Zhao

External Research Organisations

  • Nanyang Technological University (NTU)
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Details

Original languageEnglish
Pages (from-to)41-63
Number of pages23
JournalApplied categorical structures
Volume9
Issue number1
Publication statusPublished - Jan 2001

Abstract

The main result of this paper is a generalization of the classical equivalence between the category of continuous posets and the category of completely distributive lattices, based on the fact that the continuous posets are precisely the spectra of completely distributive lattices. Here we show that for so-called hereditary and union complete subset selections script Z sign, the category of script Z sign-continuous posets is equivalent (via a suitable spectrum functor) to the category of script Z sign-supercompactly generated lattices; these are completely distributive lattices with a join-dense subset of certain script Z sign-hypercompact elements. By appropriate change of the morphisms, these equivalences turn into dualities. We present two different approaches: the first one directly uses the script Z sign-join ideal completion and the script Z sign-below relation; the other combines two known equivalence theorems, namely a topological representation of script Z sign-continuous posets and a general lattice theoretical representation of closure spaces.

Keywords

    (script z sign-)below relation, (script z sign-)continuous posets, (script z sign-join) ideal completion, (script z sign-super)compact, (script z sign-super)sober space, Completely distributive lattice, Spectrum

ASJC Scopus subject areas

Cite this

Z-Join Spectra of Z-Supercompactly Generated Lattices. / Erné, Marcel; Zhao, Dongsheng.
In: Applied categorical structures, Vol. 9, No. 1, 01.2001, p. 41-63.

Research output: Contribution to journalArticleResearchpeer review

Erné M, Zhao D. Z-Join Spectra of Z-Supercompactly Generated Lattices. Applied categorical structures. 2001 Jan;9(1):41-63. doi: 10.1023/A:1008758815245
Erné, Marcel ; Zhao, Dongsheng. / Z-Join Spectra of Z-Supercompactly Generated Lattices. In: Applied categorical structures. 2001 ; Vol. 9, No. 1. pp. 41-63.
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