Groups of automorphisms of linearly ordered sets

J.L. Hickman
1976 Bulletin of the Australian Mathematical Society  
I show that a group of order-automorphisms of a linearly ordered set can be expressed as an unrestricted direct product in which each factor is either the infinite cyclic group or else a group of order-automorphisms of a densely ordered set. From this a couple of simple group embedding theorems can be derived. The technique used to obtain the main result of this paper was motivated by the Erdos-Hajnal inductive classification of scattered sets. Unless the contrary is either obvious or stated
more » ... licitly, throughout this paper we shall take "set" to mean "linearly ordered set". The ordering will always be denoted by "<" , any ambiguity being resolved by context. Sets and elements of sets will be denoted by upper and lower case Latin letters respectively, with the exception that "f, "g", "h" will denote functions and "i", "j", "k", "m", "n" integers. Lower case Greek letters will denote order-types, with "UJ" always being reserved for the first transfinite ordinal, and the order-type of a set S will sometimes be denoted by "o(5)" . Given a set S , we can define a new ordering <* on S by setting s <* t whenever t < s . The resulting ordered set is called the "converse" of S , and is denoted by "S*" . If n = o(S) , then we denote o(S*) by "n*" • Given two sets R, S , we can form their ordered union as follows. By
doi:10.1017/s000497270003673x fatcat:z32frux5hrbqdhpahl7ic53fty