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.