Intervals in subgroup lattices of infinite groups
type: Item
instance of: scholarly article
title: Intervals in subgroup lattices of infinite groups
zbMATH Open document ID: 0679.20024
DOI: 10.1016/0021-8693(89)90171-3
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: An outstanding problem of the lattice theory asserts: is every finite lattice isomorphic to the congruence lattice of a finite algebra? \textit{P. P. Pálfy} and \textit{P. Pudlák} [Algebra Univers. 11, 22-27 (1980; Zbl 0386.06002)] proved that every finite lattice is isomorphic to the congruence lattice of a finite algebra if and only if every finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. The author deals with the infinite version of this theorem. The main result of the paper asserts: Every algebraic lattice is isomorphic to an interval in the subgroup lattice of an infinite group. The proof gives a new proof of the Grätzer-Schmidt theorem which characterizes the congruence lattices of algebras.
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 4112870
zbMATH Keywords: intervals of subgroup lattices
zbMATH Keywords: algebraic lattices
zbMATH Keywords: congruence lattices of algebras
author: Jiři Tůma
reviewed by: Tamás Schmidt
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90171-3
OpenAlex ID: W2160805110
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description: scientific article; zbMATH DE number 4112870