On the commutative equivalence of semi-linear sets of ${\mathbb{N}}^k$

Abstract

Given two subsets $S_1,S_2$ of ${\mathbb{N}}^k$, we say that $S_1$ is commutatively equivalent to $S_2$ if there exists a bijection $f:S_1⟶S_2$ from $S_1$ onto $S_2$ such that, for every $\mathbf{v}\in S_1$, $|\mathbf{v}|=|f(\mathbf{v})|$, where $|\mathbf{v}|$ denotes the sum of the components of v. We prove that every semi-linear set of ${\mathbb{N}}^k$ is commutatively equivalent to a recognizable subset of ${\mathbb{N}}^k$.