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Math Help - Intersection of collection of algebras

  1. #1
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    Exclamation Intersection of collection of algebras

    I want proof of the following statement
    "Intersection of collection of algebras is also an algebra"...

    Please help
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  2. #2
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    Re: Intersection of collection of algebras

    Do you mean algebras over a field or algebraic structures? Are you sure these topics belong in an undergraduate subforum?
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  3. #3
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    Re: Intersection of collection of algebras

    This discussion includes topics from graduate course of algebra

    BTW I am talking about algebra of sets, which are closed under the operation of complementation and finite union
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  4. #4
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    Re: Intersection of collection of algebras

    Unless there are more restrictions on the definition of algebra, it is obvious that the intersection of any collection of algebras is again an algebra.
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  5. #5
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    Re: Intersection of collection of algebras

    This seems obvious. When trying to prove that the union of algebras is an algebra there is the following difficulty. Suppose A,B\in\mathcal{F}_1\cup\mathcal{F}_2. Then it may happen that A\in\mathcal{F}_1 and B\in\mathcal{F}_2. Even though \mathcal{F}_1 and \mathcal{F}_2 by themselves are closed under union, there is no reason to believe that A\cup B\in\mathcal{F}_1 or A\cup B\in\mathcal{F}_2. But with intersection of algebras, this can't happen.

    Edit: Next time, post similar questions to the Advanced Algebra subforum since it is intended for university-level questions. The current subforum is for pre-university level questions.
    Last edited by emakarov; September 22nd 2013 at 09:11 AM.
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  6. #6
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    Re: Intersection of collection of algebras

    ok thanks
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