Binomial ideals and congruences on $\mathbb{N}^n$

Abstract: A \emph{congruence} on $\mathbb{N}^n$ is an equivalence relation on $\mathbb{N}^n$ that is compatible with the additive structure. If $\Bbbk$ is a field, and $I$ is a \emph{binomial ideal} in $\Bbbk[X_1,\dots,X_n]$ (that is, an ideal generated by polynomials with at most two terms), then $I$ induces a congruence on $\mathbb{N}^n$ by declaring $\mathbf{u}$ and $\mathbf{v}$ to be equivalent if there is a linear combination with nonzero coefficients of $\mathbf{X}^{\mathbf{u}}$ and $\mathbf{X}^{\mathbf{v}}$ that belongs to $I$. While every congruence on $\mathbb{N}^n$ arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on $\mathbb{N}^n$ are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly [Kahle and Miller, Algebra Number Theory 8(6):1297-1364, 2014] with an eye on [Eisenbud and Sturmfels. Duke Math J 84(1):1-45, 1996] and [Ojeda and Piedra S\'anchez, J. Symbolic Comput 30(4):383-400, 2000].