Cardinals of countable cofinality and eventual domination
type: Item
instance of: scholarly article
title: Cardinals of countable cofinality and eventual domination
zbMATH Open document ID: 0817.06003
DOI: 10.1007/BF02115813
author: Marion Scheepers
published in: Order
publication date: 1995-07-27T00:00:00Z
review text: Let \((L,<)\) be a linearly ordered set, \(^ \omega L\) the set of all functions from the set \(\omega\) of natural numbers to \(L\). For \(f\) and \(g\) in \(^ \omega L\) the eventual domination order \(\prec\) is defined by \(f\prec g\) iff there exists an \(m\) such that \(f(n)< g(n)\) for each \(n\geq m\). For partially ordered sets \((P, <)\), \((Q, <)\) a function \(f: P\to Q\) is called an embedding if it is strictly isotone. This paper considers embeddability of partially ordered sets into others, in particular for \(\omega\)-sequences of ordinals below certain ordinals of countable cofinality; the simplest case is \(^ \omega\omega\). The first part of the paper establishes several theorems of an embeddability calculus. The main purpose of this calculus is to obtain from the embeddability of certain partially ordered sets the embeddability of more complex ones. In particular, strongly indecomposable ordinals \(\lambda> 1\) (i.e. ordinals satisfying: \(\alpha,\beta< \lambda\Rightarrow \alpha\cdot \beta< \lambda)\) which have countable cofinality (that means: are the upper limit of a countable sequence of ordinals) are studied. Typical results: Cor.10. If \(\mu\) is an infinite cardinal number, then \((^ \omega\alpha, \prec)\) embeds in \((^ \omega(\mu^ \omega),\prec)\) for each \(\alpha< \mu^ +\) (= the successor cardinal of \(\mu\)). Cor.11. If \(\lambda\) is a cardinal of countable cofinality, and if \(\sigma\) is \(\lambda^ +\), then \((^ \omega(\sigma^ n), \prec)\) embeds in \((^ \omega\lambda, \prec)\) for each natural \(n\). (Here, exponentiation is ordinal exponentiation).
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 720842
zbMATH Keywords: eventual domination order
zbMATH Keywords: embeddability
zbMATH Keywords: partially ordered sets
zbMATH Keywords: \(\omega\)-sequences of ordinals
zbMATH Keywords: strongly indecomposable ordinals
zbMATH Keywords: countable cofinality
reviewed by: Egbert Harzheim
MaRDI profile type: Publication
cites work: A new class of order types
cites work: Q3765764
cites work: Q3949052
cites work: A partition relation for partially ordered sets
cites work: Concerning n-Tactics in the Countable-Finite Game
full work available at URL: https://doi.org/10.1007/bf02115813
OpenAlex ID: W1984954712
description: scientific article; zbMATH DE number 720842