Universality of the lattice of transformation monoids
type: Item
instance of: scholarly article
title: Universality of the lattice of transformation monoids
zbMATH Open document ID: 1286.06009
DOI: 10.1090/S0002-9939-2013-11566-2
author: Michael Pinsker
author: S. Shelah
published in: Proceedings of the American Mathematical Society
publication date: 2013-08-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.3946
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 20M20
zbMATH DE Number: 6203428
zbMATH Keywords: algebraic lattice
zbMATH Keywords: transformation monoid
zbMATH Keywords: submonoid
zbMATH Keywords: closed sublattice
MaRDI profile type: Publication
arXiv ID: 1107.3946
cites work: Representations of Lattices by Sets
cites work: A survey of clones on infinite sets
cites work: Q4399900
cites work: Set Theory
cites work: Algebraic lattices are complete sublattices of the clone lattice over an infinite set
cites work: Intervals in subgroup lattices of infinite groups
cites work: Lattices, equivalence relations, and subgroups
review text: The authors solve an open problem in [\textit{M. Goldstern} and \textit{M. Pinsker}, Algebra Univers. 59, No. 3--4, 365--403 (2008; Zbl 1201.08003)]:NEWLINENEWLINENEWLINETheorem 1.1: Mon(\(\lambda\)) is \textit{universal} for complete algebraic lattices with at most \(2^{\lambda}\) compact elements with respect to closed sublattices; i.e., the closed sublattices on Mon(\(\lambda\)) are, up to isomorphism, precisely the complete algebraic lattices with at most \(2^{\lambda}\) compact elements.NEWLINENEWLINEAs a corollary, they also prove that if \(L\) is an algebraic lattice with at most \(2^{\lambda}\) compact elements, then it is even isomorphic to a closed sublattice of Mon(\(\lambda\)) via an isomorphism which preserves the smallest element.NEWLINENEWLINELet \(\lambda\) be a fixed infinite set of cardinality \(\lambda\) (the set is identified with its cardinality). A subset of \(\lambda^{\lambda}\) which is closed under composition and contains the identity function is called a transformation monoid on \(\lambda\) and denoted by Mon(\(\lambda\)).
reviewed by: Michiro Kondo
description: scientific article; zbMATH DE number 6203428