Homogeneous finitely presented monoids of linear growth

TL;DR

This paper characterizes finitely generated monoids with linear growth, showing they decompose into specific subsets called sandwiches, and introduces a numerical invariant based on free sandwiches.

Contribution

It provides a structural decomposition of monoids with linear growth and introduces a new invariant related to free sandwiches in the decomposition.

Findings

Monoids with linear growth decompose into sandwiches.

Decomposition can be refined into singletons or free sandwiches.

The number of free sandwiches is a key invariant.

Abstract

If a finitely generated monoid M is defined by a finite number of degree-preserving relations, then it has linear growth if and only if it can be decomposed into a finite disjoint union of subsets (which we call "sandwiches") of the form $a<w>b$, where $a,b,w$ are elements of $M$ and $<w>$ denotes the monogenic semigroup generated by $w$. Moreover, the decomposition can be chosen in such a way that the sandwiches are either singletons or "free" ones (meaning that all elements $a w^n b$ in each sandwich are pairwise different). So, the minimal number of free sandwiches in such a decomposition is a numerical invariant of a homogeneous (and conjecturally, non-homogeneous) finitely presented monoid of linear growth.