A Classification of Countable Lower 1-transitive Linear Orders

This paper contains a classification of countable lower 1-transitive linear orders. This is the first step in the classification of countable 1-transitive trees given in Chicot and Truss (2009): the notion of lower 1-transitivity generalises that of 1-transitivity for linear orders, and it is essential for the structure theory of 1-transitive trees. The classification is given in terms of coding trees, which 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 that a coding tree can be constructed from a lower 1-transitive linear order (X, ⩽) by examining all the invariant partitions on X. Then we show that a lower 1-transitive linear order can be recovered from a coding tree up to isomorphism.