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.

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.