Cardinals of countable cofinality and eventual domination
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).
scientific article; zbMATH DE number 720842