Permutations on the Random Permutation

Michael Pinsker,Julie Linman

doi:10.37236/4910

Abstract

The random permutation is the Fraïssé limit of the class of finite structures with two linear orders. Answering a problem stated by Peter Cameron in 2002, we use a recent Ramsey-theoretic technique to show that there exist precisely 39 closed supergroups of the automorphism group of the random permutation, and thereby expose all symmetries of this structure. Equivalently, we classify all structures which have a first-order definition in the random permutation.

Highlights

1.1 Homogeneous permutations and the random permutation.In a paper in 2002, Peter Cameron regarded finite permutations as two linear orders on a finite set, thereby taking a more “passive” perspective on permutations than the one which views them as bijections [10]
Answering a problem stated by Peter Cameron in 2002, we use a recent Ramsey-theoretic technique to show that there exist precisely 39 closed supergroups of the automorphism group of the random permutation, and thereby expose all symmetries of this structure
In [10], Cameron listed 37 closed supergroups of Aut(Π). His count included the 25 groups which arise as intersections of closed supergroups of Aut(D;

Summary

Introduction

1.1 Homogeneous permutations and the random permutation. In a paper in 2002, Peter Cameron regarded finite permutations as two linear orders on a finite set, thereby taking a more “passive” perspective on permutations than the one which views them as bijections [10]. [11] for all standard model-theoretic notions and theorems) that if we consider two reducts equivalent iff they are reducts of one another, the reducts of an ω-categorical structure ∆ correspond precisely to the closed supergroups of the automorphism group Aut(∆). Based on so-called canonical functions, this method turned out to be very effective in reduct classifications of homogeneous structures with a Ramsey expansion First applied to this kind of problem in 2011 to determine the reducts of the random partial order [13], it has since served to find the reducts of the Kn-free graphs with a constant [14] and the random ordered graph [6].

The Reducts of Π

The Proof

Discussion