On the commutative equivalence of semi-linear sets of NkNk

Given two subsets S1,S2S1,S2 of NkNk, we say that S1S1 is commutatively equivalent to S2S2 if there exists a bijection f:S1⟶S2f:S1⟶S2 from S1S1 onto S2S2 such that, for every v∈S1v∈S1, |v|=|f(v)||v|=|f(v)|, where |v||v| denotes the sum of the components of v. We prove that every semi-linear set of NkNk is commutatively equivalent to a recognizable subset of NkNk.