Transfinite tree quivers and their representations

TL;DR

This paper characterizes all indecomposable injective representations of arbitrary-sized tree quivers, revealing the role of 'vertices at infinity' and determining the structure of injective representations in noetherian trees.

Contribution

It formalizes the concept of 'vertices at infinity' and provides a complete description of indecomposable injective representations for tree quivers of any size.

Findings

Structured all indecomposable injective representations of arbitrary cardinality tree quivers.

Determined the structure of injective representations in noetherian κ-trees.

Explored conditions under which trees are source injective representation quivers.

Abstract

The idea of "vertex at the infinity" naturally appears when studying indecomposable injective representations of tree quivers. In this paper we formalize this behavior and find the structure of all the indecomposable injective representations of a tree quiver of size an arbitrary cardinal $\kappa$. As a consequence the structure of injective representations of noetherian $\kappa$-trees is completely determined. In the second part we will consider the problem whether arbitrary trees are source injective representation quivers or not.