A note on semigroups, groups and geometric lattices

Abstract

Let G be a closed, additive semigroup in a Hausdorff topological vector space. Then G is a group if and only if it satisfies natural convexity conditions of algebraic or geometric-topological type. This yields a characterization of the geometric lattices among the discrete, additive semigroups of Euclidean d-space \({\mathbb{E}^{d}}\) and, more generally, of direct sums of subspaces and lattices in \({\mathbb{E}^{d}}\).