# Topological Degree of Shift Spaces on Monoids

## TL;DR

This paper investigates the topological degree of G-shifts of finite type on nonabelian monoids, providing an algorithm to compute the degree and exploring its spectrum in relation to matrix representations of the monoid.

## Contribution

It introduces an algorithm for computing the topological degree of G-shifts on nonabelian monoids and characterizes the degree spectrum via matrix relations.

## Key findings

- The degree spectrum is finite and related to matrices with bounded row sums.
- The characteristic polynomial coefficients correspond to the number of children in the Cayley graph.
- The algorithm extends to monoids with finite free-followers.

## Abstract

This paper considers the topological degree of $G$-shifts of finite type for the case where $G$ is a nonabelian monoid. Whenever the Cayley graph of $G$ has a finite representation and the relationships among the generators of $G$ are determined by a matrix $A$, the coefficients of the characteristic polynomial of $A$ are revealed as the number of children of the graph. After introducing an algorithm for the computation of the degree, the degree spectrum, which is finite, relates to a collection of matrices in which the sum of each row of every matrix is bounded by the number of children of the graph. Furthermore, the algorithm extends to $G$ of finite free-followers.