Details
| Original language | English |
|---|---|
| Pages (from-to) | 41-63 |
| Number of pages | 23 |
| Journal | Applied categorical structures |
| Volume | 9 |
| Issue number | 1 |
| Publication status | Published - 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
- Mathematics(all)
- Theoretical Computer Science
- Computer Science(all)
- General Computer Science
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In: Applied categorical structures, Vol. 9, No. 1, 01.2001, p. 41-63.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Z-Join Spectra of Z-Supercompactly Generated Lattices
AU - Erné, Marcel
AU - Zhao, Dongsheng
PY - 2001/1
Y1 - 2001/1
N2 - 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.
AB - 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.
KW - (script z sign-)below relation
KW - (script z sign-)continuous posets
KW - (script z sign-join) ideal completion
KW - (script z sign-super)compact
KW - (script z sign-super)sober space
KW - Completely distributive lattice
KW - Spectrum
UR - http://www.scopus.com/inward/record.url?scp=0043193877&partnerID=8YFLogxK
U2 - 10.1023/A:1008758815245
DO - 10.1023/A:1008758815245
M3 - Article
AN - SCOPUS:0043193877
VL - 9
SP - 41
EP - 63
JO - Applied categorical structures
JF - Applied categorical structures
SN - 0927-2852
IS - 1
ER -