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Quotient space explanation

  1. Dec 4, 2011 #1
    I'm having some troubles understanding the concepts of quotient algebra.
    May someone explain me what exactly they are, giving some concrete examples?

    I know that a quotient set is the set of all equivalence classes, but it sounds very vague for me and i can't make the analogy with quotient spaces, or quotient groups.
     
    Last edited: Dec 4, 2011
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  3. Dec 6, 2011 #2

    Deveno

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    what do you mean by "quotient algebra"? there are different kinds of algebras:

    1. universal algebras (defined by a set and a signature of arity, and identities involving n-ary operations, etc.)
    2. associative algebras (vector spaces with an associative ring structure).
     
  4. Dec 6, 2011 #3

    Fredrik

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    If A is an algebra and I is an ideal in A, the quotient algebra A/I is defined in the following way. Define a relation ~ on A by saying that x~y if x-y is a member of I. It's easy to show that ~ is an equivalence relation. The equivalence class that contains x is denoted by x+I. The set of all equivalence classes is given the structure of an algebra by the definitions

    (x+I)+(y+I)=(x+y)+I
    a(x+I)=(ax)+I
    (x+I)(y+I)=xy+I
     
  5. Dec 7, 2011 #4

    Deveno

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    yes, that follows by definition 2. in the algebras of type 1, you generally don't have ideals, but rather congruences. i was asking "which" definition of algebra he meant.
     
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