On cycles of length three

TL;DR

This paper investigates the structure of irreducible morphisms between indecomposable modules over string algebras, establishing bounds on their compositions and extending results to arbitrary lengths.

Contribution

It provides new bounds on compositions of irreducible morphisms in string algebras and generalizes the existence of specific morphism chains for any length.

Findings

No three irreducible morphisms compose to an element in 6^6 ackslash 6^7 with certain conditions.

Existence of n irreducible morphisms with compositions in 6^{n+4} ackslash 6^{n+5} for any n  3.

Abstract

We prove that if $A$ is a string algebra then there are not three irreducible morphisms between indecomposable $A$-modules such that its composition belongs to $\Re^{6} \backslash \Re^{7}$, whenever the compositions of two of them are not in $\Re^{3}$. Moreover, for any positive integer $n \geq 3$, we show that there are $n$ irreducible morphisms such that their composition is in $\Re^{n+4} \backslash \Re^{n+5}$.