Changing the depth of an ordered set by decomposition
E. C. Milner, Karel Prikry · Transactions of the American Mathematical Society · 1985
The depth of a partially ordered set ⟨ P , > ⟩ \langle P, > \rangle is the smallest ordinal γ \gamma such that ⟨ P , > ⟩ \langle P, > \rangle does not embed γ ∗ {\gamma ^\ast } . The width of ⟨ P , > ⟩ \langle P, > \rangle is the smallest cardinal number μ \mu such that there is no antichain of size μ + 1 \mu + 1 in P P . We show that if γ > ω \gamma > \omega and γ \gamma is not an infinite successor cardinal, then any partially ordered set of depth γ \gamma can be decomposed into cf ( | γ | )