# Minimal right determiners of irreducible morphisms in string algebras

## TL;DR

This paper provides a formula for counting minimal right determiners of irreducible morphisms in string algebras with a tree quiver, linking algebraic properties to graph-theoretic parameters.

## Contribution

It introduces a precise formula for the number of minimal right determiners in string algebras based on quiver structure and vertex ideals.

## Key findings

- The number of minimal right determiners is given by 2n - p - q - 1.
- The formula relates algebraic invariants to quiver topology.
- The result applies to string algebras with tree-shaped quivers.

## Abstract

Let $\Lambda$ be a finite dimensional string algebra over a field with the quiver $Q$ such that the underlying graph of $Q$ is a tree, and let $|\Det(\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between indecomposable left $\Lambda$-modules. Then we have $$|\Det(\Lambda)|=2n-p-q-1,$$ where $n$ is the number of vertices in $Q$, $p=|\{i\mid i$ is a source in $Q$ with two neighbours$\}|$ and $q$ is the number of non-zero vertex ideals of $\Lambda$.