Changing the depth of an ordered set by decomposition

The depth of a partially ordered set (P, >) is the smallest ordinal γ such that (P, >) does not embed γ^*. The width of (P, γ) is the smallest cardinal number μ such that there is no antichain of size μ+1 in P. We show that if γ > ω and γ is not an infinite successor cardinal, then any partially ordered set of depth γ can be decomposed into cf(γ) parts so that the depth of each part is strictly less than γ. If γ = ω or if γ is an infinite successor cardinal, then for any infinite cardinal ⋋ there is a linearly ordered set of depth γ such that for any ⋋-decomposition one of the parts has the same depth γ. 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 ω which cannot be split into ⋋ parts of finite width. We prove that, for any cardinal ⋋ and any infinite cardinal ν there is a partial order of width ν⁺ which cannot be split into ⋋ parts of smaller width.