# The order type of scattered context-free orderings of rank one is computable

## TL;DR

This paper proves that determining whether a context-free ordering has rank at most one is decidable, and provides an effective method to compute its order type if so, addressing a specific case of the isomorphism problem.

## Contribution

It introduces a decision procedure for the rank of scattered context-free orderings and an effective computation method for their order type when the rank is at most one.

## Key findings

- Decidability of rank at most one for scattered context-free orderings
- Effective computation of order type for rank one orderings
- Resolution of a specific case in the isomorphism problem

## Abstract

A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that the isomorphism problem of scattered context-free orderings is undecidable, if one of them has a rank at least two. In this paper we show that it is decidable whether a context-free ordering has rank at most one, and if so, its order type is effectively computable.