Poset limits can be totally ordered

TL;DR

This paper proves that every poset limit can be represented as a kernel on the unit interval with the standard order, resolving an open question and providing new proofs including a Szemeredi-type Regularity Lemma for posets.

Contribution

It establishes that all poset limits can be represented as kernels on the unit interval, answering an open question and introducing a new combinatorial proof with a Regularity Lemma for posets.

Findings

Poset limits can be represented as kernels on the unit interval.

A Szemeredi-type Regularity Lemma for posets is developed.

Every atomless ordered probability space admits a measure-preserving, almost order-preserving map.

Abstract

S.Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529--563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The combinatorial proof is based on a Szemeredi-type Regularity Lemma for posets which may be of independent interest. Also, as a by-product of the analytic proof, we show that every atomless ordered probability space admits a measure-preserving and almost order-preserving map to the unit interval.