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

DOI:
https://doi.org/10.1090/S0002-9947-1985-0792827-8

MathSciNet review:
792827

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Abstract: The depth of a partially ordered set is the smallest ordinal such that does not embed . The *width* of is the smallest cardinal number such that there is no antichain of size in . We show that if and is not an infinite successor cardinal, then any partially ordered set of depth can be decomposed into parts so that the depth of each part is strictly less than . If or if is an infinite successor cardinal, then for any infinite cardinal there is a linearly ordered set of depth such that for any -decomposition one of the parts has the same depth . These results are used to solve an analogous problem about width. It is well known that, for any cardinal , there is a partial order of width which cannot be split into parts of finite width. We prove that, for any cardinal and any infinite cardinal , there is a partial order of width which cannot be split into parts of smaller width.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0792827-8

Keywords:
Partial order,
depth,
width,
extendable ordinal,
partition relation,
graph,
category

Article copyright:
© Copyright 1985
American Mathematical Society