Lattice in toric geometry ($N\cong {\mathbb{Z}}^n,M=\mathrm{Hom}(N,\mathbb{Z})$)

A toric lattice is a finite-rank free abelian group $N$, embedded as a discrete full-rank subgroup of $N_{\mathbb{R}}=N{\otimes }_{\mathbb{Z}}\mathbb{R}$. Its dual is $M=\mathrm{Hom}(N,\mathbb{Z})$. This lattice of integral vectors is distinct from an order-theoretic lattice. It supplies the integral structure for toric cones and their dual monomials.

A lattice basis $v_1,\dots ,v_n$ expresses each vector of the toric lattice uniquely as ${\sum }_i{a}_i{v}_i$ with $a_i\in \mathbb{Z}$. Relative to any fixed integral basis, vectors form another lattice basis exactly when their determinant is $\pm 1$. A primitive vector is one whose integer span is saturated; it can be completed to such a basis by the Euclidean algorithm.