A new operation on partially ordered sets

TL;DR

This paper introduces rotations for finite posets, explores their properties, and shows how they relate to the automorphism group of the random poset, extending concepts from graph theory to posets.

Contribution

It defines rotations for finite posets, analyzes their properties, and connects them to the automorphism group of the random poset, providing new insights into poset symmetries.

Findings

Rotations for finite posets are well-defined and have interesting combinatorial properties.

Two finite posets are rotation-equivalent if connected by a sequence of rotations.

The group of rotating permutations of the random poset is the automorphism group of a homogeneous structure.

Abstract

Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and the second type are permutations induced by so-called rotations. In this paper we introduce rotations for finite posets, which can be seen as the poset counterpart of Seidel-switch for finite graphs. We analyze some of their combinatorial properties, and investigate in particular the question of when two finite posets are rotation-equivalent. We moreover give an explicit combinatorial construction of a rotation of the random poset whose image is again isomorphic to the random poset. As an corollary of our results on rotations of finite posets, we obtain that the group of rotating permutations of the random poset is the automorphism group of a homogeneous structure in a finite language.