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

Key Points

• This research aims to establish a fixed-point theorem related to face maps and explore the properties of random finite sets under deletion.
• Developed a fixed-point theorem for face maps that delete specific entries from ordered sets.
• Analyzed properties of random finite sets of integers under deletion conditions.
• Examined implications for monoids of order-preserving transformations.
• Established that random finite sets of integers can remain almost invariant under specific deletions.
• Identified connections between these findings and various order-preserving transformations of the natural numbers.