Commuting upper triangular binary morphisms

TL;DR

This paper characterizes all commuting upper triangular binary morphisms, which are morphisms with an upper triangular matrix structure, from the free monoid into itself.

Contribution

It provides a complete characterization of all pairs of upper triangular binary morphisms that commute.

Findings

Identifies conditions under which two upper triangular binary morphisms commute.

Provides a classification of all such morphisms based on their matrix structure.

Abstract

A morphism $g$ from the free monoid $X^*$ into itself is called upper triangular if the matrix of $g$ is upper triangular. We characterize all upper triangular binary morphisms $g_1$ and $g_2$ such that $g_1g_2=g_2g_1$.