Results 1 to 3 of 3

Math Help - Are all defined bracketings equivalent for a partial associative operator?

  1. #1
    Newbie
    Joined
    Jun 2013
    From
    United Kingdom
    Posts
    2

    Are all defined bracketings equivalent for a partial associative operator?

    Hi, I cannot find an answer to this question, but it seems intuitively true. Say an operator is partial and associative in the following sense:

    a * (b * c) = (a * b) * c if both sides are defined

    That is, not the stronger constraint that "either both sides are defined and equal or both sides are undefined". In this case, is it true that all possible defined bracketings of a1 * a2 *.... * aN are equivalent? If so, what would be a proof argument? The standard induction-based proof used for total operators does not seem to work as some bracketings may not be defined:

    http://unafold.math.rpi.edu/Teaching...Binary_Ops.pdf
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,513
    Thanks
    769

    Re: Are all defined bracketings equivalent for a partial associative operator?

    Quote Originally Posted by mraza View Post
    In this case, is it true that all possible defined bracketings of a1 * a2 *.... * aN are equivalent?
    What do you mean by saying that the bracketings are equivalent? That expressions with all possible bracketings are defined simultaneously and are equal when defined? It could be that a * (b * c) is defined while (a * b) * c is not; then the fact that * is associative says nothing for these a, b and c. Or are you saying that they are equivalent in the sense that only those expressions with all possible bracketings that are defined are equal?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2013
    From
    United Kingdom
    Posts
    2

    Re: Are all defined bracketings equivalent for a partial associative operator?

    Yes I asked if "all DEFINED bracketings" are equivalent, that is, there may be many bracketings of a1*....*aN of which some are defined and some are not. I want to know whether all the defined bracketings will be equal under this definition of associativity. Or if not, a counterexample will be helpful.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: July 26th 2011, 07:38 PM
  2. Commutative and Associative 71.5
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 26th 2009, 09:12 AM
  3. commutative, associative
    Posted in the Algebra Forum
    Replies: 8
    Last Post: October 22nd 2009, 09:52 PM
  4. Replies: 2
    Last Post: December 20th 2008, 04:09 AM
  5. associative law
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: July 14th 2008, 03:08 PM

Search Tags


/mathhelpforum @mathhelpforum