Question about the Distributive Property

Question

I couldn't find an answer to this question in other posts or in my textbooks.

I know that everyone knows the following, but bear with me. Suppose I define a scalar field as follows:

1. There is a set of elements (called scalars), of either finite or infinite cardinality.
2. There are two operators. Let's call them "Bill" and "Ted". You will see why later.
3. Both operators are binary-to-unary; both operators accept only arguments that come from the set of scalars; both operators return only a result that is also in that set.
4. Both operators are commutative.
5. Both operators are associative.
6. Each operator has one and only one identity element in the set of scalars, and they cannot share the same identity element. Call the identity elements "Bill's identity element" and "Ted's identity element."
7. Every element in the set of scalars must be invertible under "Bill". I.e., for every x in the set of scalars, there is another element -x in that set such that Bill(x, -x) = Bill's identity element.
8. Every element in the set of scalars, \*except for Bill's identity element\*, is invertible under "Ted". I.e., for every x in the set of scalars, except for Bill's identity element, there is another element \~x such that Ted(x, \~x) = Ted's identity element.

And last comes the distributive property. One of the operators must be distributively superior to the other one, if only so that we can write unambiguous expressions containing both operators. Now, finally, for my question:

Do rules 7 and 8 force us to say that it must be "Ted" that distributes over "Bill," or are we still free to say that "Bill" distributes over "Ted"?

I think the answer is that yes, rules 7 and 8 force us to say that "Ted" is distributive over "Bill", but I don't know how to show this, or if I'm wrong, I don't understand why.

Can anyone help? Thank you so much.

Answer 1

I need to show what I've tried so far, to meet Rule 1:  


Suppose "Bill" distributes over "Ted." This means:  


Bill(a, Ted(b,c)) = Ted(Bill(a, b), Bill(a, c))  


If a is Bill's identity element, this becomes:  
Bill(a, Ted(b,c)) = Ted(b,c) on the LHS and Ted(b,c) on the RHS, which seems okay.  


Now suppose b is Bill's identity element, so that Bill(a, b) = a. We get:  
Bill(a, Ted(b,c)) = Ted(a, Bill(a,c))

I'm suspicious of this result because the "b" has just vanished---but "b" is Bill's identity element, not Ted's. So its presence on the LHS in the "Ted" operator makes me want to say that it must still be on the RHS somewhere.