## 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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### References:

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