Details
| Original language | English |
|---|---|
| Pages (from-to) | 31-70 |
| Number of pages | 40 |
| Journal | Applied categorical structures |
| Volume | 7 |
| Issue number | 1-2 |
| Publication status | Published - Jun 1999 |
Abstract
A subset selection script Z sign assigns to each partially ordered set P a certain collection script Z signP of subsets. The theory of topological and of algebraic (i.e. finitary) closure spaces extends to the general script Z sign-level, by replacing finite or directed sets, respectively, with arbitrary 'script Z sign-sets'. This leads to a theory of script Z sign-union completeness, script Z sign-arity, script Z sign-soberness etc. Order-theoretical notions such as complete distributivity and continuity of lattices or posets extend to the general script Z sign-setting as well. For example, we characterize script Z sign-distributive posets and script Z sign-continuous posets by certain homomorphism properties and adjunctions. It turns out that for arbitrary subset selections script Z sign, a poset P is strongly script Z sign-continuous iff its script Z sign-join ideal completion script Z signv P is script Z sign-ary and completely distributive. Using that characterization, we show that the category of strongly script Z sign-continuous posets (with interpolation) is concretely isomorphic to the category of script Z sign-ary script Z sign-complete core spaces. For suitable subset selections script y sign, and script Z sign, these are precisely the script y sign-sober core spaces.
Keywords
- Closure space, Compact, Completely distributive, Completion, Continuous poset, Core, Sober space
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Mathematics(all)
- Algebra and Number Theory
- Computer Science(all)
- General Computer Science
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In: Applied categorical structures, Vol. 7, No. 1-2, 06.1999, p. 31-70.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Script Z sign-Continuous Posets and Their Topological Manifestation
AU - Erné, Marcel
PY - 1999/6
Y1 - 1999/6
N2 - A subset selection script Z sign assigns to each partially ordered set P a certain collection script Z signP of subsets. The theory of topological and of algebraic (i.e. finitary) closure spaces extends to the general script Z sign-level, by replacing finite or directed sets, respectively, with arbitrary 'script Z sign-sets'. This leads to a theory of script Z sign-union completeness, script Z sign-arity, script Z sign-soberness etc. Order-theoretical notions such as complete distributivity and continuity of lattices or posets extend to the general script Z sign-setting as well. For example, we characterize script Z sign-distributive posets and script Z sign-continuous posets by certain homomorphism properties and adjunctions. It turns out that for arbitrary subset selections script Z sign, a poset P is strongly script Z sign-continuous iff its script Z sign-join ideal completion script Z signv P is script Z sign-ary and completely distributive. Using that characterization, we show that the category of strongly script Z sign-continuous posets (with interpolation) is concretely isomorphic to the category of script Z sign-ary script Z sign-complete core spaces. For suitable subset selections script y sign, and script Z sign, these are precisely the script y sign-sober core spaces.
AB - A subset selection script Z sign assigns to each partially ordered set P a certain collection script Z signP of subsets. The theory of topological and of algebraic (i.e. finitary) closure spaces extends to the general script Z sign-level, by replacing finite or directed sets, respectively, with arbitrary 'script Z sign-sets'. This leads to a theory of script Z sign-union completeness, script Z sign-arity, script Z sign-soberness etc. Order-theoretical notions such as complete distributivity and continuity of lattices or posets extend to the general script Z sign-setting as well. For example, we characterize script Z sign-distributive posets and script Z sign-continuous posets by certain homomorphism properties and adjunctions. It turns out that for arbitrary subset selections script Z sign, a poset P is strongly script Z sign-continuous iff its script Z sign-join ideal completion script Z signv P is script Z sign-ary and completely distributive. Using that characterization, we show that the category of strongly script Z sign-continuous posets (with interpolation) is concretely isomorphic to the category of script Z sign-ary script Z sign-complete core spaces. For suitable subset selections script y sign, and script Z sign, these are precisely the script y sign-sober core spaces.
KW - Closure space
KW - Compact
KW - Completely distributive
KW - Completion
KW - Continuous poset
KW - Core
KW - Sober space
UR - http://www.scopus.com/inward/record.url?scp=0007468993&partnerID=8YFLogxK
U2 - 10.1023/a:1008657800278
DO - 10.1023/a:1008657800278
M3 - Article
AN - SCOPUS:0007468993
VL - 7
SP - 31
EP - 70
JO - Applied categorical structures
JF - Applied categorical structures
SN - 0927-2852
IS - 1-2
ER -