Semigroup operations distributed by natural noncancellative semigroup operations on the positive real numbers

  • Hirokazu OkaEmail author
  • Takeshi Miura
  • Sin-Ei Takahasi
Research Article


Let \({\mathbf {R}}_+\) be the space of positive real numbers with the ordinary topology. Let \(\star \) be the natural noncancellative continuous semigroup operation \(\times \) or \(\diamond \) on \({\mathbf {R}}_+\) as defined in first section. We 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 multiplication on \({\mathbf {R}}_+\).


Cancellative continuous semigroup operation Distributive law Noncancellative semigroup operation Homeomorphic isomorphism 

Mathematics Subject Classification

Primary 22A15 Secondary 06F05 



We would like to express our sincere gratitude to the referee for his/her helpful comments to improve the original manuscript. This work was supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of EngineeringIbaraki UniversityHitachiJapan
  2. 2.Department of Mathematics, Faculty of ScienceNiigata UniversityNiigataJapan
  3. 3.Laboratory of Mathematics and GamesFunabashiJapan

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