Hammocks for non-domestic string algebras

TL;DR

This paper characterizes the order types of hammocks in non-domestic string algebras, showing they belong to a specific class of finite description linear orders, and introduces tools to analyze their structure.

Contribution

It establishes that the order type of hammocks in non-domestic string algebras is within a well-defined class of linear orders and introduces finite subsets of bands for structural analysis.

Findings

Order types of hammocks are in the class of finite description linear orders.

Introduction of finite subsets of bands to describe string locations.

Use of condensation to analyze linear order structures.

Abstract

We show that the order type of the simplest version of a hammock for string algebras lies in the class of finite description linear orders--the smallest class of linear orders containing $\mathbf 0$, $\mathbf 1$, and that is closed under isomorphisms, finite order sum, anti-lexicographic product with $\omega$ and $\omega^*$, and shuffle of finite subsets--using condensation (localization) of linear orders as a tool. We also introduce two finite subsets of the set of bands and use them to describe the location of left $\mathbb N$-strings in the completion of hammocks.