On linearly ordered sets and permutation groups of countable degree
type: Item
instance of: scholarly article
title: On linearly ordered sets and permutation groups of countable degree
zbMATH Open document ID: 0681.03030
DOI: 10.1007/BF01620766
author: Hans Läuchli
author: Peter M. Neumann
published in: Archive for Mathematical Logic
publication date: 1988-01-01T00:00:00Z
review text: The following problem is partially solved: Does there exist a permutation group G on \(\Omega\) that has precisely m orbits in the power set \({\mathcal P}(\Omega)?\) And can this question be settled within ZFC? More exactly it is proved that there are precisely \(\aleph_ 1\) equivalence classes of countable linearly ordered sets under piecewise order-preserving bijections, and the group of such permutations of \({\mathbb{Q}}\) has precisely \(\aleph_ 1\) orbits in \({\mathcal P}({\mathbb{Q}})\).
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 06A05
Mathematics Subject Classification ID: 20B27
zbMATH DE Number: 4114614
zbMATH Keywords: number of orbits in the power set
zbMATH Keywords: permutation group
zbMATH Keywords: countable linearly ordered sets
reviewed by: Y. Ventsov
MaRDI profile type: Publication
cites work: Arithmetic of ordinals with applications to the theory of ordered Abelian groups
cites work: On Fraissé's order type conjecture
cites work: Über Mischsummen von Ordnungszahlen
cites work: Implausible Subgroups of Infinite Symmetric Groups
full work available at URL: https://doi.org/10.1007/bf01620766
OpenAlex ID: W2046425313
description: scientific article; zbMATH DE number 4114614