Details
| Original language | English |
|---|---|
| Pages (from-to) | 163-184 |
| Number of pages | 22 |
| Journal | Applied categorical structures |
| Volume | 15 |
| Issue number | 1-2 |
| Publication status | Published - 14 Dec 2006 |
Abstract
From the work of Simmons about nuclei in frames it follows that a topological space X is scattered if and only if each congruence Θ on the frame of open sets is induced by a unique subspace A so that Θ = { (U,V) | U∩ A = V∩ A}, and that the same holds without the uniqueness requirement iff X is weakly scattered (corrupt). We prove a seemingly similar but substantially different result about quasidiscrete topologies (in which arbitrary intersections of open sets are open): each complete congruence on such a topology is induced by a subspace if and only if the corresponding poset is (order) scattered, i.e. contains no dense chain. More questions concerning relations between frame, complete, spatial, induced and open congruences are discussed as well.
Keywords
- (Complete) congruence, Alexandroff topology, Frame, Quasidiscrete, Scattered, Spatial, Superalgebraic, Supercontinuous
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. 15, No. 1-2, 14.12.2006, p. 163-184.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Complete congruences on topologies and down-set lattices
AU - Erné, Marcel
AU - Gehrke, Mai
AU - Pultr, Aleš
PY - 2006/12/14
Y1 - 2006/12/14
N2 - From the work of Simmons about nuclei in frames it follows that a topological space X is scattered if and only if each congruence Θ on the frame of open sets is induced by a unique subspace A so that Θ = { (U,V) | U∩ A = V∩ A}, and that the same holds without the uniqueness requirement iff X is weakly scattered (corrupt). We prove a seemingly similar but substantially different result about quasidiscrete topologies (in which arbitrary intersections of open sets are open): each complete congruence on such a topology is induced by a subspace if and only if the corresponding poset is (order) scattered, i.e. contains no dense chain. More questions concerning relations between frame, complete, spatial, induced and open congruences are discussed as well.
AB - From the work of Simmons about nuclei in frames it follows that a topological space X is scattered if and only if each congruence Θ on the frame of open sets is induced by a unique subspace A so that Θ = { (U,V) | U∩ A = V∩ A}, and that the same holds without the uniqueness requirement iff X is weakly scattered (corrupt). We prove a seemingly similar but substantially different result about quasidiscrete topologies (in which arbitrary intersections of open sets are open): each complete congruence on such a topology is induced by a subspace if and only if the corresponding poset is (order) scattered, i.e. contains no dense chain. More questions concerning relations between frame, complete, spatial, induced and open congruences are discussed as well.
KW - (Complete) congruence
KW - Alexandroff topology
KW - Frame
KW - Quasidiscrete
KW - Scattered
KW - Spatial
KW - Superalgebraic
KW - Supercontinuous
UR - http://www.scopus.com/inward/record.url?scp=34248374431&partnerID=8YFLogxK
U2 - 10.1007/s10485-006-9054-3
DO - 10.1007/s10485-006-9054-3
M3 - Article
AN - SCOPUS:34248374431
VL - 15
SP - 163
EP - 184
JO - Applied categorical structures
JF - Applied categorical structures
SN - 0927-2852
IS - 1-2
ER -