PreprintCC BY-NC-SA 4.0arXiv
Detecting divisibility in uniquely divisible N-semigroups via homomorphisms to ℝ>0
Abstract
An N-semigroup is an archimedean, idempotent-free, cancellative and commutative semigroup. For example, the additive semigroup ℝ>0 of strictly positive real numbers is a N-semigroup. Hewitt-Zuckerman showed that every N-semigroup admits a semigroup homomorphism to ℝ>0. We show that every uniquely divisible N-semigroup S admits enough semigroup homomorphisms to 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 ℚ. A counterexample due to Ravsky explains why this result does not generalise to the larger class of N-semigroups.
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BibTeX
@misc{tan2026detecting,
title = {{Detecting divisibility in uniquely divisible $\mathfrakN$-semigroups via homomorphisms to $\mathbbR_>0$}},
author = {Colin Tan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32133},
eprint = {2609.32133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32133},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Detecting divisibility in uniquely divisible N-semigroups via homomorphisms to ℝ_>0
AU - Colin Tan
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32133
UR - https://arxiv.org/abs/2609.32133
AB - An N-semigroup is an archimedean, idempotent-free, cancellative and commutative semigroup. For example, the additive semigroup ℝ_>0 of strictly positive real numbers is a N-semigroup. Hewitt-Zuckerman showed that every N-semigroup admits a semigroup homomorphism to ℝ_>0. We show that every uniquely divisible N-semigroup S admits enough semigroup homomorphisms to 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 ℚ. A counterexample due to Ravsky explains why this result does not generalise to the larger class of N-semigroups.
ER -