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

TL;DR

This paper proves a fixed-point theorem for face maps involving deletion of entries in ordered sets and explores invariance properties of certain random finite sets under these deletions.

Contribution

It introduces a fixed-point theorem for face maps and demonstrates the existence of almost invariant random finite sets under deletion operations.

Findings

Fixed-point theorem for face maps involving deletions

Existence of random finite sets almost invariant under deletions

Implications for monoids of order-preserving transformations

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.