We have seen that combinatorics is useful to understand decompositions of binomial ideals, but one can also use decompositions of binomial ideals to understand combinatorics.

Consider $\mathbb{N}^{n}$, the set of lattice points in the non-negative orthant. Any finite set $\mathcal{B} \subset \mathbb{Z}^{n}$ defines a graph $G_{\mathcal{B}}$ on $\mathbb{N}^{n}$ with edge set given by all translates of $\mathcal{B}$. Binomial algebra can be used to study connectivity of this graph, a common task in algebraic statistics, combinatorial game theory, integer programming, and other areas working with lattice points.

- Consider the polynomial ring $\newcommand{\<}{\langle}\renewcommand{\>}{\rangle}\newcommand{\NN}{\mathbb{N}}\newcommand{\kk}{k}\newcommand{\ZZ}{\mathbb{Z}}\kk[x_{1},\dots,x_{n}]$ and the binomial ideal $I_{\mathcal{B}} := \<x^{b^{+}} - x^{b^{-}} : b \in \mathcal{B}\>$ where[[ $b^{\pm}_{i} = \max\{\pm b_{i}, 0\}$]] such that $b = b^{+} - b^{-}$
- Convince yourself that $u,v\in \NN^{n}$ are connected by moves from $\mathcal{B}$ (without leaving $\NN^{n}$!) if and only if $x^{u} - x^{v} \in I_{\mathcal{B}}$.
- What kind of statements can you make about the connectivity of $G_{\mathcal{B}}$ if $I_{\mathcal{B}} = J_{1}\cap \dots \cap J_{r}$ is a primary decomposition? What about a mesoprimary decomposition?
- Describe the graphs on $\NN^{2}$ defined by $\mathcal{B}_{1} = \{(2,-2), (3,-3)\}$ and $\mathcal{B}_{2} = \{(1,1), (2,-1)\}$. Draw the graph on paper to check any algebraic results you get.