  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].

Commutative Algebra