## 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

Zbl 0564.04005
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