Partially ordered permutation groups

Abstract

In this note we discuss permutation groups (G, Ω) in which the set Ω admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of Ω such that α<β implies αg<βg, for all α and β in Ω andg inG. If the set Ω admits aG-partial order which is a total order, then (G, Ω) is an O-permutation group (orderable permutation group).