Patchworks

Abstract

In this note, we want to introduce a concept that - as we shall see - exhibits an interesting relationship with hierarchies: We define a collection C⊆P(X) of subsets of a set X to be a patchwork if ≠C′⊆C and ∩C∈C′C≠ implies ∩C∈C′C∈C and ∪C∈C′C∈C. In this note, we will investigate patchworks C that contain a maximal hierarchy. We will show that this holds if and only if (i) the empty subset , all one-element subsets x (x∈X) of X, and the set X itself belong to C, and (ii) the patchwork is ample, that is, A, B∈C and #C∈CA⊆C⊆B=2 implies B-A∈C. © 2001 Academic Press.