Proving associativity is a structural property

[kathrynag]

Homework Statement

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

Homework Equations

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

 

[Citan Uzuki]

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

 

[quantumdude]

The Attempt at a Solution

* is commutative
I know this means that I have to show (a*b)*c=a*(b*c)

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

 

[kathrynag]

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

 

[quantumdude]

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"?

 

[kathrynag]

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

 

[kathrynag]

* is just any arbitrary operation

 

[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.

 

[kathrynag]

That makes a lot more sense to me.