I've been trying to read a paper regarding the analogue of a GCD for lattices, but I'm not sure I understand how to decipher this notion. This is given in Section 3.1 when the author discusses the 'height' of a substitution.
Let $\Gamma=\mathbb{Z}^d$ be the standard lattice. Given two sublattices, $\Gamma_1,\Gamma_2\subseteq \Gamma$, the GCD of the two lattices is the smallest lattice $\hat{\Gamma}\subseteq \Gamma$ containing both $\Gamma_1$ and $\Gamma_2$.
From what I understood, when $\Gamma=\mathbb{Z}$, one can relate a number $n$ to the lattice by $$\Gamma(n):=\{ m\in \mathbb{Z}: m=k\cdot n \quad \text{for some} \; k\in \mathbb{Z} \}.$$
Then the GCD of the lattices $\Gamma(n_1)$ and $\Gamma(n_2)$, is the lattice $\Gamma\big( \gcd(n_1,n_2) \big)$.
However I understand relatively well how to find out what the gcd is for a pair of numbers by the Euclidean algorithm, but can\is there an analogue for this notion in lattices of full rank? Is there a way to find a GCD of two lattices by some basic manipulation of their basis?
For any lattice, we can generally define a canonical form for its basis in the Hermite normal form. It is an analogue of reduced echelon form for vectors over $\mathbb Z$. Then, finding a canonical basis for $\hat \Gamma$ seems to be the same as finding the Hermite normal form for the union of bases of $\Gamma_1$ and $\Gamma_2$.
Algorithmically, you can reduce any basis to the Hermite normal form in a way that is very similar to the Gauss method for matrices over $\mathbb R$, but you use Euclidean algorithm instead of division to get the GCD on the leading entries in every row.
So, if you have a two rows
$$ \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \end{pmatrix}, $$
you can perform the extended Euclidean algorithm on $a_{11}$ and $a_{21}$, repeating the operations on the whole rows, until one of them is equal to $\gcd(a_{11}, a_{21})$, and the other to $0$. After that you do the same for other columns, until the Hermite normal form is achieved. This approach allows to generalize Gauss method to any Euclidean domain.