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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=a+b+ab

a*b=a+b-ab

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

https://mathoverflow.net/questions/139331/generalizing-detropicalization

https://mathoverflow.net/questions/139215/commutative-associative-rational-binary-operations

So another possibility is ab/(a+b), (a+b)/(1-ab). But they are not continuous at a+b=0 and 1-ab=0.

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KH Kim
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    There are an extreme amount of such if you don't require anything but commutativity and associativity. – Tobias Kildetoft Sep 12 '13 at 13:47
  • What if we assumed continuity? – KH Kim Sep 12 '13 at 13:48
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    If, for any $a \in \mathbb{R}$, we define $_a:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ by $x_ay=a$ for all $(x,y) \in \mathbb{R} \times \mathbb{R}$ then $*_a$ is a commutative and associative binary operation. – Rebecca J. Stones Sep 12 '13 at 13:49
  • @Rebecca Good example! But a little trivial. I thought ab=0 can be included by rab by r=0. but ab=c can also satisfy the both. I updated the question. – KH Kim Sep 12 '13 at 13:57
  • @Kildetoft I think when it's over rational numbers, there's not so much space that an operations can differ. Or maybe we can assume continuity. I am kind of curious whether addition and multiplication is so basic that all associative and commutative operations are made from addition and multiplication. – KH Kim Sep 12 '13 at 14:01

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The question doesn't make much sense because when you ask for associative commutative binary operations on the set of real numbers, you forget the whole structure of $\mathbb{R}$. It is just a set, and for the question only the cardinality matters, which is known as the continuum $c$. If $X$ is any set with cardinality $c$ and some associative commutative binary operation, we get a corresponding one on the set of real numbers. For example the set of integer sequences has cardinality $c$, and we can take its addition, or its multiplication.

Martin Brandenburg
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  • I am not a mathematician, so I am not sure if I understood you. But I got the impression that for every associative commutative binary operation over integers, we can construct associative commutative binary operation over real numbers. And then, what will be those operations? Can they all be expressed by multiplication and addition? – KH Kim Sep 12 '13 at 14:10
  • I can think of + with mod. Then we can expand it over reals but it is not continuous. So I think the continuity have to be assumed... – KH Kim Sep 12 '13 at 14:14