Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice. There should be a basis matrix $B$ of $\Lambda$, s.t. $B$ is positive integer valued and upper triangular.. if so, are the diagonal entries of this matrix unique?
EDIT: More general, is this upper triangular matrix $B \in \mathfrak{ut}(\mathbb{Z},n)$ (now we allow integer values) unique up to the right action of $\mathfrak{ut}(\mathbb{Z},n) \cap GL(\mathbb{Z},n)$ (unimodular upper triangular matrices, i.e. $\pm 1$ on the diagonals)?