Abstract
Given any cancellative continuous semigroup operation \(\star \) on the positive real numbers \(\mathbf {R}_+\) with the ordinary topology, we completely characterize the set \(\mathcal {D}_\star (\mathbf {R}_+)\) of all cancellative continuous semigroup operations on \(\mathbf {R}_+\) which are distributed by \(\star \) in terms of homeomorphism. As a consequence, we show that an arbitrary semigroup operation in \(\mathcal {D}_\star (\mathbf {R}_+)\) is homeomorphically isomorphic to the ordinary addition \(+\) on \(\mathbf {R}_+\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aczél, J.: Sur les opérations défines pour nombres réels. Bull. Soc. Math. Fr. 76, 59–64 (1948)
Aczél, J.: The state of the second part of Hilbert’s fifth problem. Bull. Am. Math. Soc. 20, 153–163 (1989)
Anbe, S., Takahasi, S.-E., Tsukada, M., Miura, T.: Commutative semigroup operations on \(\mathbf{R}^2\) compatible with the ordinary additive operation. Linear Nonlinear Anal. 1(1), 89–93 (2015)
Craigen, R., Páles, Z.: The associativity equation revisited. Aeqationes Math. 37, 306–312 (1989)
Kobayashi, Y., Nakasuji, Y., Takahasi, S.-E., Tsukada, M.: Continuous semigroup structures on \(\mathbb{R}\), cancellative semigroups and bands. Semigroup Forum 90, 518–531 (2015). https://doi.org/10.1007/s00233-014-9624-x
Nakasuji, Y., Takahasi, S.-E.: A reconsideration of Jensen’s inequality and its applications. J. Inequal. Appl. 2013, 408 (2013). https://doi.org/10.1186/1029-242X-2013-408
Takahasi, S.-E., Takagi, H., Miura, T., Oka, H.: Semigroup operations distributed by the ordinary multiplication or addition on the real numbers. Publ. Math. Debr. 91(3–4), 297–307 (2017)
Acknowledgements
We would like to express our sincere gratitude to the referee for his/her helpful comments to improve the original manuscript. The second author is partially supported by JSPS KAKENHI Grant Numbers (C)-15K04921 and 16K05172. This work was supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jimmie D. Lawson.
Dedicated to the memory of Professor Hiroyuki Takagi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Takahasi, SE., Miura, T. & Oka, H. Characterization of distributive semigroup operations on the positive real numbers. Semigroup Forum 100, 585–604 (2020). https://doi.org/10.1007/s00233-019-10006-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-019-10006-3