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GCD on a union magma

  1. Feb 22, 2014 #1
    1. The problem statement, all variables and given/known data

    Consider the following magma, S is not empty; P(S) is the power set.

    (P(S), U)

    Now, let A and B be in P(S).

    What is the GCD of A and B?

    2. Relevant equations



    3. The attempt at a solution

    If I choose a common divisor of A and B under unions, call it X, I get that X is a subset of A and B as the only requirement.

    The only way I can interpret "GCD" is that X is the largest such set. Then X = (A intersect B.)

    Agree?
     
  2. jcsd
  3. Feb 22, 2014 #2

    Dick

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    You should give a few more definitions here. But it sounds plausible if 'divisor' means set containment and 'greater' is also defined in terms of set containment.
     
  4. Feb 22, 2014 #3

    HallsofIvy

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    I thought "magma" had something to do with volcanos!
     
  5. Feb 22, 2014 #4
    The operation on the magma is union, so X is a divisor of A if there exists some set L in P(S) so that (X U L) = A.

    I think this is equivalent to set containment (X is a subset of A).

    I took "greater" to mean larger cardinality in this case. I reckon that the largest set that is a subset of A and B is (A intersect B), and that's why I think it is the GCD of A and B.
     
  6. Feb 22, 2014 #5

    Dick

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    Ah ok, I see. So yes, L 'divides' A is equivalent to L is a subset of A. Defining "greater" in terms of cardinality can get you into some trouble with infinite sets. If the intersection of A and B is infinite, there may be many subsets with the same cardinality contained in it.
     
  7. Feb 22, 2014 #6

    Dick

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    I had to look it up too. A "magma" is a set with a binary operation and that's it. No other properties necessary. Bourbaki invented the terminology.
     
  8. Feb 22, 2014 #7
    I didn't think of that.. perhaps instead of cardinality, I could say that the greatest common divisor is a superset of all other common divisors?

    EDIT: Sorry about the terminology confusion, it is what my professors calls them, yes, a magma (M,*) is a set M with a binary operation *: M -> M.

    When we talk about divisibility in a magma (M,*), * becomes analogous to multiplication and M becomes analagous to Z, regardless of what they actually are.
     
    Last edited: Feb 22, 2014
  9. Feb 22, 2014 #8

    Dick

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    Or just define A is 'greater' than B to mean B is a subset of A. Which amounts to the same thing.
     
    Last edited: Feb 22, 2014
  10. Feb 23, 2014 #9
    Thank you for the assistance.
     
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