The complexity of a well-quasi-order (wqo) can be measured through three classical ordinal invariants: the width as a measure of antichains, the height as a measure of chains, and the maximal order type as a measure of bad sequences. This article considers the "finitary powerset" construction: the collection Pf(X) of finite subsets of a wqo X ordered with the Hoare embedding relation remains a wqo. The width, height and maximal order type of Pf(X) cannot be expressed as a function of the invariants of X, and we provide tight upper and lower bounds for the three invariants. The article also identifies an algebra of well-behaved wqos, that include finitary powersets as well as other more classical constructions, and for which the ordinal invariants can be computed compositionnally. This relies on a new ordinal invariant called the approximated maximal order type.

翻译：良拟序（wqo）的复杂度可通过三个经典序数不变量来度量：作为反链度量的宽度、作为链度量的高度，以及作为坏序列度量的最大序型。本文研究"有限幂集"构造：在良拟序X上赋予Hoare嵌入偏序后，其有限子集构成的集合Pf(X)仍保持良拟序性质。Pf(X)的宽度、高度和最大序型无法表示为X不变量的函数，我们为这三个不变量给出了紧致的上下界。本文还识别出一类具有良好性质的良拟序代数结构，其中包含有限幂集及其他经典构造，并能为这些结构组合式地计算序数不变量。这种计算依赖于名为近似最大序型的新型序数不变量。