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Partition numbers. (English) Zbl 0891.03019

For a partial order \(P\) let \({\mathfrak a}(P)\) denote the least cardinal such that there is no absolute partition of \(P\) of that size. In Section 1 the author investigates suborderings of \(({\mathcal P} (\omega)/ \text{fin})^\omega\). So let \(P_f\) and \(P_c\) denote the suborderings of \(({\mathcal P} (\omega)/ \text{fin})^\omega\) which consist of filtered elements and chains, respectively. The author shows that \({\mathfrak a}(P_f)= {\mathfrak a} (P_c)\), and that both numbers are greater than or equal to \({\mathfrak p}\) (the minimal cardinality of a filter on \({\mathcal P} (\omega)/ \text{fin}\) which has no lower bound).
For \(X\) an uncountable Polish space, let \({\mathfrak a} (X)\) be the minimal size of an uncountable partition of \(X\) into closed sets. In Section 2 it is shown that the dominating number is less than or equal to \({\mathfrak a} (X)\). Using Miller forcing he shows that the dominating number can be less than \({\mathfrak a} (X)\).
In Section 3 the author shows that the partition number for Mathias forcing is greater than or equal to the bounding number.
In Section 4 he deals with the lattice of partitions of \(\omega\). Let \((\omega)^\omega\) be the set of infinite partitions of \(\omega\). The author shows that it is consistent with ZFC that the distributivity number of \((\omega)^\omega\) is less than the distributivity number of \({\mathcal P} (\omega)/ \text{fin}\).
Reviewer: M.Weese (Potsdam)

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
06A07 Combinatorics of partially ordered sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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