New proofs of results concerning bases of a lattice

Abstract

Applying basic facts of linear algebra, we present new simpler and much shorter proofs of results presented by Cherednik in the paper [The non-negative basis of a lattice, Diskret. Mat. 26(3) 2014, 127--135]. Recall, Cherednik proved that each lattice of dimension $ n $ in the linear space $ \mathbb{R}^{n} $ has a basis consisting non-negative vectors, (i.e., vectors which contain only non-negative coordinates). Applying this theorem, he also showed that an arbitrary (not necessarily of the maximal dimension) lattice has such a basis if and only if it is generated by all its non-negative vectors. Next, these results are generalized for arbitrary convex cones (note that the set of all non-negative vectors is a convex cone). Finally, he showed that each lattice of dimension $ n \geq 2 $ in $ \mathbb{R}^{n} $ has a basis in any translation of every convex cone of dimension $ n $.