Subprincipal closed ideals in \(\beta N\)
type: Item
instance of: scholarly article
title: Subprincipal closed ideals in \(\beta N\)
zbMATH Open document ID: 0659.22002
DOI: 10.1007/BF02575018
author: Neil Hindman
author: Dennis E. Davenport
published in: Semigroup Forum
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134898
review text: Let \((N,+)\) be the semigroup of positive integers with usual addition and \((\beta N,+)\) be the Stone-Cech compactification of \((N,+)\), where the addition in \(\beta N\) is the usual left-continuous extension of the ordinary addition on N. The algebraic structure of \(\beta N\) was studied by \textit{N. Hindman} in 1980 [Semigroup Forum 21, 221-255 (1980; Zbl 0448.22004) and 25, 291-310 (1985; Zbl 0499.22002)] and has proved somewhat useful in solving problems in Ramsey Theory). In this paper, subprincipal closed ideals of \(\beta N\) generated by N are studied. The upper, lower and Banach density of a subset \(A\subset N\) are defined to be \(\overline{d(A)}=\limsup_{n\to \infty}\{(| A\cap \{1,2,...,k\}|)/k\); \(k\geq n\}\); \underbar{d(A)}\(=\lim_{n\to \infty}\inf \{(| A\cap \{1,2,...,k\}|)/k\); \(k\geq n\}\) and \(d^*(A)=\sup \{\alpha \in Real:\) \(\exists\) increasing sequences \(<t_ n>^{\infty}_{n=1}\) and \(<x_ n>^{\infty}_{n=1}\) in N such that, for each \(n\in N\), \(| A\cap \{t_ n+1,t_ n+2,...t_ n+x_ n\}| \geq \alpha x_ n\}\) respectively. Also, define \(\Delta =\{p\in \beta N:\) for all \(A\in p\), \(\bar d(A)>0\}\); \(\Delta^*=\{p\in \beta N:\) for all \(A\in p\), \(d^*(A)>0\}\) and \(\Delta_ 1=\{p\in \beta N:\) for all \(A\in p\) there exists \(k\in N\) with \(d^*(\cup^{k}_{t=1}A- t)=1\}\). The authors show that any closed ideal of \(N^*=\beta N\setminus N\) which is not equal to \(\Delta\), has non-empty intersection with \(\Delta^*\setminus \Delta\). Also, they point out that it is possible to find \(p\in \Delta \setminus \Delta_ 1\), such that \(\Delta_ 1\subsetneqq cl(N^*+p)\subsetneqq \Delta^*\) and \(\Delta\) is the intersection of the subprincipal closed ideals lying strictly between \(\Delta_ 1\) and \(\Delta^*\).
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 11B99
zbMATH DE Number: 4077580
zbMATH Keywords: Stone-Cech compactification
zbMATH Keywords: Ramsey Theory
zbMATH Keywords: subprincipal closed ideals
zbMATH Keywords: upper, lower and Banach density
reviewed by: K. P. Shum
MaRDI profile type: Publication
cites work: Q3784209
cites work: Filters and the Weak Almost Periodic Compactification of a Discrete Semigroup
cites work: Compact right topological semigroups and generalizations of almost periodicity
cites work: Partitions and sums of \((m,p,c)\)-sets
cites work: Q3909268
cites work: Minimal ideals and cancellation in beta(N)
cites work: On density, translates, and pairwise sums of integers
cites work: Ramsey's theorem for sums, products, and arithmetic progressions
cites work: Sums equal to products in \(\beta{\mathbb N}\)
cites work: The ideal structure of the space of \(\kappa\)-uniform ultrafilters on a discrete semigroup
cites work: Set theory. An introduction to independence proofs
cites work: In a left-topological semigroup with dense center the closure of any left ideal is an ideal
cites work: On sets of integers containing k elements in arithmetic progression
Recommended article: Compact subsemigroups of (βℕ, +) containing the idempotents
Recommended article: Ideals and commutativity in \(\beta\mathbb{N}\)
Recommended article: Prime properties of the smallest ideal of \(\beta\mathbb{N}\)
Recommended article: The minimal ideal of compact subsemigroups of \(\beta S\)
Recommended article: Q4856892
Recommended article: Free groups and semigroups in \(\beta\) N
Recommended article: The minimal ideals of a multiplicative and additive subsemigroup of \(\beta\) \({\mathbb{N}}\)
Recommended article: Ultrafilters on a discrete set with two binary operations
Recommended article: Large rectangular semigroups in Stone-Cech compactifications
Recommended article: Abelian semigroups whose Stone-Čech compactifications have left ideal decompositions
description: scientific article; zbMATH DE number 4077580