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Short branches in the Rudin-Frolík order. (English) Zbl 0588.54005

For \(p,p'\in \omega^*\) we write \(p\sim P'\) if there is a permutation f of \(\omega\) such that the Stone extension \(\bar f:\) \(\beta\) \(\omega\) \(\to \beta \omega\) of f satisfies \(\bar f(\)p)\(=p'\). We set \([p]=\{p': p\sim p'\}\) and \(T=\{[p]: p\in \omega^*\}\). The Rudin-Frolík partial order \(\leq\) on T is defined as follows: \([p]<[q]\) if there is a one-to- one function \(f: \omega\) \(\to \omega^*\) with discrete range such that the Stone extension \(\bar f:\) \(\beta\) \(\omega\) \(\to \omega^*\) satisfies \(\bar f(\)p)\(=q\). We write \(P[q]=\{[p]:[p]<[q]\}\). It was proved nearly 20 years ago by Z. Frolík that each \(q\in \omega^*\) satisfies (a) P[q] is linearly ordered, (b) \(| P[q]| \leq {\mathfrak c}\), and (c) \(| \{[p]: [q]\in P[p]\}| =2^{{\mathfrak c}}\). It follows that each maximal linearly ordered subset B of T (i.e., each branch B of T) satisfies \({\mathfrak c}\leq | B| \leq {\mathfrak c}^+\). In earlier work [ibid. 24, 563-570 (1983; Zbl 0564.04005)] the author proved the existence in ZFC of branches B such that \(| B| ={\mathfrak c}^+\), but she left unsettled the question whether there are branches B with \(| B| ={\mathfrak c}\). In the present paper, responding positively to this question, she shows in ZFC the existence of an unbounded chain C in T order-isomorphic to the well-ordered set of countable ordinal numbers. From the theorems of Frolík cited above it then follows that the set \[ B=U\{P[q]: [q]\in C\} \] is a branch B with the desired property: \(| B| ={\mathfrak c}\).
Reviewer: W.W.Comfort

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E05 Other combinatorial set theory

Citations:

Zbl 0564.04005
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