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 \({\mathbb{E}^{d}}\).
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Flor, P., Gruber, P.M. A note on semigroups, groups and geometric lattices. Arch. Math. 93, 253–258 (2009). https://doi.org/10.1007/s00013-009-0029-0
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DOI: https://doi.org/10.1007/s00013-009-0029-0