Euclidean algorithm for a class of linear orders

TL;DR

This paper develops a Euclidean algorithm for finitely presented linear orders, classifies their isomorphism classes using 3-signed trees, and extends the correspondence between signed trees and linear orders.

Contribution

It introduces finitely presented linear orders and generalizes the Euclidean algorithm to classify their isomorphism classes.

Findings

Classified finitely presented linear orders via 3-signed trees.

Developed a Euclidean algorithm based on order width.

Established a correspondence between signed trees and linear orders.

Abstract

Borrowing inspiration from Marcone and Mont\'{a}lban's one-one correspondence between the class of signed trees and the equimorphism classes of indecomposable scattered linear orders, we find a subclass of signed trees which has an analogous correspondence with equimorphism classes of indecomposable finite rank discrete linear orders. We also introduce the class of \emph{finitely presented linear orders}-- the smallest subclass of finite rank linear orders containing $\mathbf 1$, $\omega$ and $\omega^*$ and closed under finite sums and lexicographic products. For this class we develop a generalization of the Euclidean algorithm where the \emph{width} of a linear order plays the role of the Euclidean norm. Using this as a tool we classify the isomorphism classes of finitely presented linear orders in terms of an equivalence relation on their presentations using \emph{3-signed trees}.