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Directed sets

  1. Sep 12, 2007 #1
    1. The problem statement, all variables and given/known data
    http://en.wikipedia.org/wiki/Directed_set

    The definition of a directed set at the site above makes no sense to me. The part that does not make sense is: "for any two elements a and b in A, there exists an element c in A (not necessarily distinct from a,b) with"

    If c does not need to be distinct from a or b, why does this add any restrictions on the binary relation because a possible c is always just max(a,b), where max is defined in the natural way?




    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 12, 2007 #2

    Hurkyl

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    We have no guarantee that a and b are comparable (i.e. we may have [itex]a \not\leq b[/itex] and [itex]b \not\leq a[/itex]), and thus cannot define a maximum operator.
     
  4. Sep 12, 2007 #3
    I see. Thanks.
     
  5. Sep 12, 2007 #4

    HallsofIvy

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    For example consider a colection of SETS with [itex]\le[/itex] defined by [itex]A\e B[/itex] if and only if [itex]A\subset B[/itex]. It is quite possible to have A and B, [itex]A\me B[/itex] such that A is not a subset of B and B is not a subset of A.-
     
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