06/15/2013, 08:02 PM

I'm a bit confused

You said that the only assumptions were the commutativity of two operations, but you did not use the assumption of the the commutativity of the operation in the proof. (in fact it is not important imo):

you asumptions are:

1-commutativity and associativity of the operator .

2-the existence of a unique right identity element (that is the left id. element too if it is commutative) of the operation .

And these assuptions make you proof valid to show that the only operation with these properties is the addition.

You said that the only assumptions were the commutativity of two operations, but you did not use the assumption of the the commutativity of the operation in the proof. (in fact it is not important imo):

you asumptions are:

1-commutativity and associativity of the operator .

2-the existence of a unique right identity element (that is the left id. element too if it is commutative) of the operation .

And these assuptions make you proof valid to show that the only operation with these properties is the addition.