Detecting divisibility in uniquely divisible $\mathfrak{N}$-semigroups via homomorphisms to ${\mathbb{R}}_{>0}$

Abstract

An $\mathfrak{N}$-semigroup is an archimedean, idempotent-free, cancellative and commutative semigroup. For example, the additive semigroup ${\mathbb{R}}_{>0}$ of strictly positive real numbers is a $\mathfrak{N}$-semigroup. Hewitt-Zuckerman showed that every $\mathfrak{N}$-semigroup admits a semigroup homomorphism to ${\mathbb{R}}_{>0}$. We show that every uniquely divisible $\mathfrak{N}$-semigroup $S$ admits enough semigroup homomorphisms to ${\mathbb{R}}_{>0}$ to detect divisibility in $S$. %They were studied by Tamura and Kobayashi. The main ingredient is a version of the Eidelheit and Kakutani hyperplane separation theorem for vector spaces over $\mathbb{Q}$. A counterexample due to Ravsky explains why this result does not generalise to the larger class of $\mathfrak{N}$-semigroups.

arXiv comment7 pages