On the computation of order types of hammocks for domestic string algebras

TL;DR

This paper develops an algorithm to compute the order types of intervals in hammocks associated with domestic string algebras, expanding understanding of their structure through a modified bridge quiver.

Contribution

It introduces a new variation called the arch bridge quiver and uses it to algorithmically determine order types of hammocks in domestic string algebras.

Findings

Provides an algorithm for computing order types of hammock intervals.

Characterizes the class of order types as bounded discrete linear orders.

Identifies the class as the smallest containing finite orders and closed under specific operations.

Abstract

For the representation-theoretic study of domestic string algebras, Schr\"{o}er introduced a version of hammocks that are bounded discrete linear orders. He introduced a finite combinatorial gadget called the bridge quiver, which we modified in the prequel of this paper to get a variation called the arch bridge quiver. Here we use it as a tool to provide an algorithm to compute the order type of an arbitrary closed interval in such hammocks. Moreover, we characterize the class of order types of these hammocks as the bounded discrete ones amongst the class of finitely presented linear orders--the smallest class of linear orders containing finite linear orders as well as $\omega$, and that is closed under isomorphisms, order reversal, finite order sums and lexicographic products.