Proving associativity is a structural property

1. Feb 9, 2009

kathrynag

1. The problem statement, all variables and given/known data

Give a proof for the operation * is commutative being a structural property.

2. Relevant equations

3. The attempt at a solution
* is commutative
I know this means that I have to show (a*b)*c=a*(b*c)
I'm not sure where to go now

2. Feb 9, 2009

Citan Uzuki

What is the definition of *, and what do you mean by "structural property"?

3. Feb 9, 2009

Tom Mattson

Staff Emeritus
You're confusing "commutative" and "associative".

4. Feb 9, 2009

kathrynag

oops, I menat to say associative. So, is that the right first step?

5. Feb 9, 2009

Tom Mattson

Staff Emeritus
There's no way to tell if you're going in the right direction until you answer Citran's question. What is the definition of "structural property"?

6. Feb 9, 2009

kathrynag

A structural property of a binary structure is one that must be shared by any isomorphic structure.

7. Feb 9, 2009

kathrynag

* is just any arbitrary operation

8. Feb 9, 2009

Citan Uzuki

Okay, so you want to show that associativity of a binary operation is preserved under isomorphisms. That is, if $(S,\ \ast)$ and $(T,\ \star)$ are isomorphic binary structures, and $\ast$ is associative, then $\star$ is also associative. Is that right?

If so, then what you will want to do is start by letting $(S,\ \ast)$ and $(T,\ \star)$ be arbitrary isomorphic binary structures with an isomorphism $\varphi: S \rightarrow T$ between them. Then assume that $\ast$ is associative, and use that and the isomorphism to show that $\star$ is associative.

9. Feb 9, 2009

kathrynag

That makes a lot more sense to me.

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