A Special Noncommutative Semigroup Operation on the Real Numbers and Left or Right Distribution

Let R be the space of real numbers with the ordinary topology. Define x ⋆ 1 y = | x | y ( x , y ∈ R ) and x ⋆ 2 y = x | y | ( x , y ∈ R ) . We show that any cancellative continuous semigroup operation on R which is left-distributed by ⋆ 1 is equal to a semigroup operation induced by the ordinary add...

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Bibliographic Details
Published inResultate der Mathematik Vol. 77; no. 2
Main Authors Miura, Takeshi, Niwa, Norio, Oka, Hirokazu, Takahasi, Sin-Ei
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.04.2022
Subjects
Mathematics
Mathematics and Statistics
right-distribution
left-distribution
Secondary 06F05
Primary 22A15
Cancellative continuous semigroup operation
homeomorphic isomorphism
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Summary:Let R be the space of real numbers with the ordinary topology. Define x ⋆ 1 y = | x | y ( x , y ∈ R ) and x ⋆ 2 y = x | y | ( x , y ∈ R ) . We show that any cancellative continuous semigroup operation on R which is left-distributed by ⋆ 1 is equal to a semigroup operation induced by the ordinary addition on R and a special homeomorphism from R onto itself.Also we show that there is no cancellative continuous semigroup operation on R which is right-distributed by ⋆ 1 and that there is no cancellative continuous semigroup operation on R which is distributive over ⋆ 1 . Similar results hold for the operation ⋆ 2 .
ISSN:1422-6383
1420-9012
DOI:10.1007/s00025-021-01588-y