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Math Help - Set closed under under multiplication

  1. #1
    MHF Contributor Bruno J.'s Avatar
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    Set closed under under multiplication

    Hello, this is a nice problem.

    Suppose S is a set of real numbers, closed under multiplication.
    Suppose S is partitioned into A, B, such that both A,B are closed under multiplication of three elements (so that a_1,a_2,a_3 \in A implies a_1a_2a_3 \in A, and similarily for B).

    Show that either A or B is closed under multiplication.
    Last edited by Bruno J.; June 20th 2009 at 07:30 PM.
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    Quote Originally Posted by Bruno J. View Post
    Hello, this is a nice problem.

    Suppose S is a set of real numbers, closed under multiplication.
    Suppose S is partitioned into A, B, such that both A,B are closed under multiplication of three elements (so that a_1,a_2,a_3 \in A implies a_1a_2a_3 \in A, and similarily for B).

    Show that either A or B is closed under multiplication.
    nice but pretty easy! S doesn't have to be a subset of real numbers. the result is true for any semigroup S:

    suppose neither A nor B is closed under multiplication. then, since S is multiplicatively closed, there must exist a_1,a_2 \in A, \ b_1,b_2 \in B such that a_1a_2 \in B, \ b_1b_2 \in A. but then we would get

    something impossible: a_1a_2b_1b_2=(a_1a_2)b_1b_2=a_1a_2(b_1b_2) \in A \cap B = \emptyset. \ \ \Box
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Yup that is perfect!
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