The structure of generic automorphisms of the random poset

TL;DR

This paper characterizes generic automorphisms of the random poset by associating them with a new auxiliary structure that encodes their action, revealing properties like ultrahomogeneity and universality, and exploring model-theoretic aspects.

Contribution

It introduces a novel auxiliary structure for the random poset that explicitly characterizes generic automorphisms and investigates its model-theoretic properties.

Findings

The auxiliary structure is not saturated.

Its theory is neither $oldsymbol{ extomega}$-categorical nor admits quantifier-elimination.

The paper introduces new orders on orbitals to describe automorphisms.

Abstract

We examine properties of generic automorphisms of the random poset, with the goal of explicitly characterizing them. We associate to each automorphism an auxiliary first-order structure, consisting of the random poset equipped with an infinite sequence of binary relations which encode the action of the automorphism. We then explicitly characterize generic automorphisms in terms of properties of this structure. Two notable such properties are ultrahomogeneity, and universality for a certain class of finite structures in this language. As this auxiliary structure seems to be new, we also address some model-theoretic questions. In particular, this structure fails to be saturated, and its theory neither is $\omega$-categorical nor admits quantifier-elimination, in contrast to many known ultrahomogeneous structures in finite languages. We also examine orbitals -- order-convex hulls of orbits -- and their use in describing automorphisms. In particular, we introduce and use new orders on the space of orbitals.