Special noncancellative commutative semigroup operations on the real numbers and 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 = max { 1 , x } + max { 1 , y } ( x , y ∈ R ) . We show that there is no cancellative continuous semigroup operation which is distributed by ⋆ i ( i = 1 , 2 ) . Conversely we show that there is no cancellative continuous semigroup operation which is distributive over ⋆ i ( i = 1 , 2 ) . Moreover we discuss the above arguments for a null semigroup operation on R .