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Proving associativity is a structural property

  1. Feb 9, 2009 #1
    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. jcsd
  3. Feb 9, 2009 #2
    What is the definition of *, and what do you mean by "structural property"?
     
  4. Feb 9, 2009 #3

    Tom Mattson

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    You're confusing "commutative" and "associative".
     
  5. Feb 9, 2009 #4
    oops, I menat to say associative. So, is that the right first step?
     
  6. Feb 9, 2009 #5

    Tom Mattson

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    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"?
     
  7. Feb 9, 2009 #6
    A structural property of a binary structure is one that must be shared by any isomorphic structure.
     
  8. Feb 9, 2009 #7
    * is just any arbitrary operation
     
  9. Feb 9, 2009 #8
    Okay, so you want to show that associativity of a binary operation is preserved under isomorphisms. That is, if [itex](S,\ \ast)[/itex] and [itex](T,\ \star)[/itex] are isomorphic binary structures, and [itex]\ast[/itex] is associative, then [itex]\star[/itex] is also associative. Is that right?

    If so, then what you will want to do is start by letting [itex](S,\ \ast)[/itex] and [itex](T,\ \star)[/itex] be arbitrary isomorphic binary structures with an isomorphism [itex]\varphi: S \rightarrow T[/itex] between them. Then assume that [itex]\ast[/itex] is associative, and use that and the isomorphism to show that [itex]\star[/itex] is associative.
     
  10. Feb 9, 2009 #9
    That makes a lot more sense to me.
     
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