A Fra\"iss\'e theory for partial orders of a fixed finite dimension

TL;DR

This paper develops a Fra"iss"e theory for finite partial orders of fixed dimension, characterizes their limits, and explores their automorphism groups and Ramsey properties.

Contribution

It introduces a new Fra"iss"e class for finite n-dimensional partial orders with linear extensions, providing a finite axiomatization and analyzing automorphism groups.

Findings

The class of finite n-dimensional partial orders with linear extensions forms a Fra"iss"e class.

The Fra"iss"e limit is uniquely characterized by a finite set of axioms.

The automorphism group of the limit is extremely amenable, and the universal minimal flow is identified.

Abstract

For each $n\geq 2$, we show that the class of all finite $n$-dimensional partial orders, when expanded with $n$ linear orders which realize the partial order, forms a Fra\"iss\'e class and identify its Fra\"iss\'e limit $(D_n,<,<_1,\ldots,<_n)$. We give a finite axiomatization of this limit which specifies it uniquely up to isomorphism among countable structures. We then show that the aforementioned class of finite structures satisfies the Ramsey property and conclude, by the Kechris-Pestov-Todor\v{c}evi\'{c} correspondence, that the automorphism group of its Fra\"iss\'e limit is extremely amenable. Finally, we identify the universal minimal flow of the automorphism group of the reduct $(D_n,<)$.