Algebraic Linear Orderings

TL;DR

This paper characterizes algebraic linear orderings as those representable by algebraic trees and deterministic context-free languages, establishing bounds on their complexity and connecting them to algebraic ordinals.

Contribution

It provides a new characterization of algebraic linear orderings using algebraic trees and context-free languages, and determines bounds on their Hausdorff rank and order types.

Findings

Algebraic linear orderings are exactly those isomorphic to leaves of algebraic trees.

A linear ordering is algebraic iff it can be represented as the lexicographic order of a deterministic context-free language.

The Hausdorff rank of scattered algebraic linear orderings is less than ω^ω, and algebraic ordinals are less than ω^{ω^ω}.

Abstract

An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wright type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelle's characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than $\omega^\omega$. It follows that the algebraic ordinals are exactly those less than $\omega^{\omega^\omega}$.