Example of Commutative but Not Associative Binary Operation

[Maths Lover]

hi ,

I met lot's of binary operation which is associative and commtative and I also met lot's of binary operation which is associative and not abelian

but
is there an example for a binary operation which is commtative and not associative ?
I don't remmber that I've met one likes this .

and what about a binary operation which is not commutative and not abelian ?

I know that there is no relation between associative and commutative laws

but , all books don't mentions operations like this ?!

 

[Hurkyl]

Let's make a random one! With 3 values to keep it small.

Commutativity means the multiplication table should be symmetric.

  abc
 +---
a|aca
b|cac
c|acb

(ab)c = cc = b
a(bc) = ac = a

Ah good, my first guess worked out. Would probably have been better to construct the multiplication table systematically to ensure that it wouldn't be associative, but my intuition says that "most" randomly chosen operations should be non-associative.

 

[Hurkyl]

Ah, but I can guess you're about to ask for a "natural" example. (be careful that you're not asking simply because you find distasteful the idea that examples exist!)

The first examples of symmetric binary operations that arise "naturally" spring to mind are symmetric polynomials. Here's a quadratic polynomial as an example:

$$f(x,y) = x^2 + y^2$$

If we use this function to define a binary operation on real numbers, we have

$$a \star(b \star c) = a^2 + (b^2 + c^2)^2 = a^2 + b^4 + c^4 + 2b^2 c^2$$
$$(a \star b) \star c = (a^2 + b^2)^2 + c^2 = a^4 + b^4 + c^2 + 2a^2 b^2$$

 

[Maths Lover]

Ah, but I can guess you're about to ask for a "natural" example. (be careful that you're not asking simply because you find distasteful the idea that examples exist!)

The first examples of symmetric binary operations that arise "naturally" spring to mind are symmetric polynomials. Here's a quadratic polynomial as an example:

$$f(x,y) = x^2 + y^2$$

If we use this function to define a binary operation on real numbers, we have

$$a \star(b \star c) = a^2 + (b^2 + c^2)^2 = a^2 + b^4 + c^4 + 2b^2 c^2$$
$$(a \star b) \star c = (a^2 + b^2)^2 + c^2 = a^4 + b^4 + c^2 + 2a^2 b^2$$

I think that this example is great !

thank you very much :)

 

[D H]

Here's a "natural" example:

·|rps
-+---
r|rpr
p|pps
s|rss

"Natural" because kids play this everywhere. It's rock paper scissors.

 

[Dead Boss]

and what about a binary operation which is not commutative and not abelian ?

Commutative is abelian. I take it you mean not commutative and not associative? If so then there's subtraction for example. Or exponentiation ($a^b$).