Details
| Original language | English |
|---|---|
| Pages (from-to) | 95-113 |
| Number of pages | 19 |
| Journal | Topology and its applications |
| Volume | 44 |
| Issue number | 1-3 |
| Publication status | Published - 22 May 1992 |
Abstract
A quasiordered set Q (qoset for short) is ideal-distributive iff QI, the lattice of ideals in the sense of Frink, is distributive. The principal ideal embedding of Q in QI is characterized by certain density properties, by extremal conditions and by a universal property. The reflector I from the category of qosets and ideal-continuous maps (where inverse images of ideals are ideals) to the category of algebraic lattices and join-preserving maps has several interesting subreflectors, for example, from the category of certain ideal-distributive qosets (including all bounded distributive lattices) to the subcategory of algebraic frames. Generalizing the classical Stone duality for distributive (semi-)lattices, we establish a dual equivalence between the category of ideal-distributive posets with so-called ∧-stable ideal-continuous maps and the category of pairs (X,B) where B is a base of a sober topology on X and B is meet-dense in the collection of all compact open sets; morphisms in this category preserve the distinguished bases under inverse images. We also study a self-dual notion of distributivity for qosets, compare it with ideal-distributivity and determine the corresponding Stone representation.
Keywords
- completion, duality, ideal, ideal-continuous, ideal-distributive, join- and meet-dense, join- and meet-stable, poset, Qoset, reflective subcategory, Stone space
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
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In: Topology and its applications, Vol. 44, No. 1-3, 22.05.1992, p. 95-113.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Ideal completion and Stone representation of ideal-distributive ordered sets
AU - David, Elias
AU - Erné, Marcel
PY - 1992/5/22
Y1 - 1992/5/22
N2 - A quasiordered set Q (qoset for short) is ideal-distributive iff QI, the lattice of ideals in the sense of Frink, is distributive. The principal ideal embedding of Q in QI is characterized by certain density properties, by extremal conditions and by a universal property. The reflector I from the category of qosets and ideal-continuous maps (where inverse images of ideals are ideals) to the category of algebraic lattices and join-preserving maps has several interesting subreflectors, for example, from the category of certain ideal-distributive qosets (including all bounded distributive lattices) to the subcategory of algebraic frames. Generalizing the classical Stone duality for distributive (semi-)lattices, we establish a dual equivalence between the category of ideal-distributive posets with so-called ∧-stable ideal-continuous maps and the category of pairs (X,B) where B is a base of a sober topology on X and B is meet-dense in the collection of all compact open sets; morphisms in this category preserve the distinguished bases under inverse images. We also study a self-dual notion of distributivity for qosets, compare it with ideal-distributivity and determine the corresponding Stone representation.
AB - A quasiordered set Q (qoset for short) is ideal-distributive iff QI, the lattice of ideals in the sense of Frink, is distributive. The principal ideal embedding of Q in QI is characterized by certain density properties, by extremal conditions and by a universal property. The reflector I from the category of qosets and ideal-continuous maps (where inverse images of ideals are ideals) to the category of algebraic lattices and join-preserving maps has several interesting subreflectors, for example, from the category of certain ideal-distributive qosets (including all bounded distributive lattices) to the subcategory of algebraic frames. Generalizing the classical Stone duality for distributive (semi-)lattices, we establish a dual equivalence between the category of ideal-distributive posets with so-called ∧-stable ideal-continuous maps and the category of pairs (X,B) where B is a base of a sober topology on X and B is meet-dense in the collection of all compact open sets; morphisms in this category preserve the distinguished bases under inverse images. We also study a self-dual notion of distributivity for qosets, compare it with ideal-distributivity and determine the corresponding Stone representation.
KW - completion
KW - duality
KW - ideal
KW - ideal-continuous
KW - ideal-distributive
KW - join- and meet-dense
KW - join- and meet-stable
KW - poset
KW - Qoset
KW - reflective subcategory
KW - Stone space
UR - http://www.scopus.com/inward/record.url?scp=38249012946&partnerID=8YFLogxK
U2 - 10.1016/0166-8641(92)90083-C
DO - 10.1016/0166-8641(92)90083-C
M3 - Article
AN - SCOPUS:38249012946
VL - 44
SP - 95
EP - 113
JO - Topology and its applications
JF - Topology and its applications
SN - 0166-8641
IS - 1-3
ER -