A fixed-point theorem for face maps, or deletion-tolerant random finite sets

Abstract

We establish a fixed-point theorem for the face maps that consist in deleting the $i$th entry of an ordered set. Furthermore, we show that there exists random finite sets of integers that are almost invariant under such deletions. Consequences for various monoids of order-preserving transformations of $\mathbf{N}$ are discussed in an appendix.