The automorphism group of the universal distributive lattice
type: Item
instance of: scholarly article
title: The automorphism group of the universal distributive lattice
zbMATH Open document ID: 1013.06009
author: Dugald Macpherson
author: Manfred Droste
published in: Algebra Universalis
publication date: 2003-04-23T00:00:00Z
review text: The normal subgroup lattices of the automorphism groups of the countable universal homogeneous distributive lattice \(\mathbb D\) and of the countable atomless generalized Boolean lattice \(\mathbb B\) are described. It is known that there exists a unique (up to isomorphism) countable distributive lattice \(\mathbb D\) which is homogeneous, i.e. any isomorphism between two finite sublattices extends to an automorphism of \(\mathbb D\), and universal, i.e. it embeds every countable distributive lattice. A generalized Boolean lattice is a distributive lattice with a smallest element 0 (but not necessarily with 1) such that each interval is a Boolean algebra. There is also a unique countable universal homogeneous Boolean lattice \(\mathbb B\). For any lattice \(\Omega \), let \(A(\Omega)\) stand for the automorphism group of \(\Omega \), \(L(\Omega)=\{ g\in A(\Omega); \exists \;x\in \Omega , \forall \;y\in \Omega : x\leq y \Rightarrow \;g(y)=y\}\), \(R(\Omega)=\{ g\in A(\Omega); \exists \;x\in \Omega , \forall \;y\in \Omega : y\leq x \Rightarrow \;g(y)=y\}\) and \(B(\Omega)=L(\Omega)\cap R(\Omega).\) For \(\Omega =(\mathbb Q,\leq)\) it was known that these are the only non-trivial normal subgroups of \(A({\mathbb Q})\). In this paper the following theorem is proved: Theorem. (a) \(L({\mathbb B})\) is the only proper non-trivial normal subgroup of \(A({\mathbb B})\), and \(L({\mathbb B})\) is simple. (b) \(R({\mathbb D})\), \(L({\mathbb D})\) and \(B({\mathbb D})\) are the only proper non-trivial normal subgroups of \(A({\mathbb D})\), and \(L(\mathbb D)/B(\mathbb D)\cong R(\mathbb D)/B(\mathbb D)\) and \(B(\mathbb D)\) are simple. If \(G\) is a permutation group on \(X\), and \(Y\subseteq X,\) then \(G_{\{ Y\}}, G_{(Y)}\) denote respectively the setwise and pointwise stabilizers of \(Y\) in \(G\). A countably infinite first-order structure \(M\) is \(\omega \)-categorical if it is determined among all countable structures by its first-order theory. An \(\omega \)-categ…
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 03C35
zbMATH DE Number: 1899940
zbMATH Keywords: automorphism group
zbMATH Keywords: distributive lattice
zbMATH Keywords: Boolean algebra
zbMATH Keywords: generalized Boolean lattice
MaRDI profile type: Publication
full work available at URL: s000120050160
OpenAlex ID: W2065302746
DOI: 10.1007/S000120050160
description: scientific article; zbMATH DE number 1899940