Let $X$ be a set and denote $\mathbb{Z}^{(X)}$ for the free abelian group on $X$. That is, $\mathbb{Z}^{(X)}$ consists of all the finite sums $\sum_{x \in X}n_xx$, with coefficients $n_x \in \mathbb{Z}$. Similarly, denote free commutative monoid on $X$ by $\mathbb{N}^{(X)}$ (so $\mathbb{N}^{(X)}$ consists of all the finite sums $\sum_{x \in X}n_xx$, with $n_x \in \mathbb{N}$). One can define a left and right invariant partial order on $\mathbb{Z}^{(X)}$ as follows:
$$a \leq b \Leftrightarrow b-a \in \mathbb{N}^{(X)},$$
for $a,b \in \mathbb{Z}^{(X)}$. How does one obtain a lattice structure on $\mathbb{Z}^{(X)}$?