Towards a classification of countable 1-transitive trees: countable lower 1-transitive linear orders

Abstract

This paper contains a classification of countable lower 1-transitive linear orders. The notion of lower 1-transitivity generalises that of 1-transitivity for linear orders, and is essential for the structure theory of 1-transitive trees. The classification is given in terms of 'coding trees'. These describe how a linear order is fabricated from simpler pieces using concatenations, lexicographic products and other kinds of construction. We define coding trees and show how they encode lower 1-transitive linear orders. Then we show that a coding tree can be recovered from a lower 1-transitive linear order $(X, \leq)$ by examining all the invariant partitions on $X$.