After I got the answer here, Is multiplication the only operation that satisfies the associative, commutative and distributive law?
I got to wonder how many and different operations can satisfy both associative and commutative law over rational numbers and real numbers.
Here's some examples.
a*b=rab (r: constant)
a*b=c (c: constant)
Are there any other operations that can satisfy both associative and commutative law over rational numbers and real numbers?
Can we imagine all the binary operations that can satisfy both laws?
And are there any reference I can look up?
I googled a bit and found these
So another possibility is ab/(a+b), (a+b)/(1-ab). But they are not continuous at a+b=0 and 1-ab=0.