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 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 .