Mesh-comparable components of the Auslander-Reiten quiver

TL;DR

This paper introduces mesh-comparable components of the Auslander-Reiten quiver, enabling the study of irreducible morphism compositions without coverings, thus broadening the understanding of their structure and properties.

Contribution

It defines and investigates mesh-comparable components, expanding the applicability of Riedtmann functors in the analysis of Auslander-Reiten quivers.

Findings

Mesh-comparable components admit Riedtmann functors without coverings.

Properties of these components are characterized.

Implications for compositions of irreducible morphisms are explored.

Abstract

The idea of using Riedtmann's well-behaved functors to study compositions of irreducible morphisms has been explored in a number of articles. Here we introduce the concept of mesh-comparable components of the Auslander-Reiten quiver, which are components for which a Riedtmann functor exists without the necessity of taking a covering, such as the universal or the generic one. We show properties of this type of component, and study the problem of compositions of irreducible morphisms in this context.