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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Changing the depth of an ordered set by decomposition


Authors: E. C. Milner and K. Prikry
Journal: Trans. Amer. Math. Soc. 290 (1985), 773-785
MSC: Primary 03E05; Secondary 04A20, 06A05, 06A10
MathSciNet review: 792827
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Abstract: The depth of a partially ordered set $ \langle P, < \rangle $ is the smallest ordinal $ \gamma $ such that $ \langle P, < \rangle $ does not embed $ {\gamma ^\ast}$. The width of $ \langle P, < \rangle $ is the smallest cardinal number $ \mu $ such that there is no antichain of size $ \mu + 1$ in $ P$. We show that if $ \gamma > \omega $ and $ \gamma $ is not an infinite successor cardinal, then any partially ordered set of depth $ \gamma $ can be decomposed into $ \operatorname{cf}(\vert\gamma \vert)$ parts so that the depth of each part is strictly less than $ \gamma $. If $ \gamma = \omega $ or if $ \gamma $ is an infinite successor cardinal, then for any infinite cardinal $ \lambda $ there is a linearly ordered set of depth $ \gamma $ such that for any $ \lambda $-decomposition one of the parts has the same depth $ \gamma $. These results are used to solve an analogous problem about width. It is well known that, for any cardinal $ \lambda $, there is a partial order of width $ \omega $ which cannot be split into $ \lambda $ parts of finite width. We prove that, for any cardinal $ \lambda $ and any infinite cardinal $ \nu $, there is a partial order of width $ {\nu ^ + }$ which cannot be split into $ \lambda $ parts of smaller width.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0792827-8
PII: S 0002-9947(1985)0792827-8
Keywords: Partial order, depth, width, extendable ordinal, partition relation, graph, category
Article copyright: © Copyright 1985 American Mathematical Society