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Math Help - Monoid proof

  1. #1
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    Smile Monoid proof

    I'm a little bit stuck on an assignment question.

    The question is:
    Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula

    x * y  = xy + c(x+y) + c^2 - c

    for all integers x, y, and z.

    Is (Z, *) a monoid? If so, what is it's identity element?
    I already proved that (Z, *) is a semigroup. I also showed that x*e = e*x . I just don't know how to find e. (That is, if e exists at all)

    This is what I have so far:

    x*e = xe + c(x+e) + c^2 - c

    <br />
e*x = ex + c(e+x) + c^2 - c<br /> <br />

    Any help would be much appreciated.

    EDIT: I'm pretty sure it's NOT a monoid. Do you think I need to prove this though? Can I just say something along the lines of "clearly, there is no identity element" :P

    x * e = xe + c(x+e) + c^2 - c \neq x
    Last edited by pikminman; January 25th 2010 at 09:20 AM. Reason: Error in Latex code
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  2. #2
    Super Member girdav's Avatar
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    I guess you proved associativity.
    The law * is commutative, hence you only have to find e such as x*e= x.
    You can isolate e in this equation.
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  3. #3
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    Quote Originally Posted by girdav View Post
    I guess you proved associativity.
    The law * is commutative, hence you only have to find e such as x*e= x.
    You can isolate e in this equation.
    Well how could I do that? I'm really confused.
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  4. #4
    Super Member girdav's Avatar
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    We have x*e =x so xe+c\left(x+e\right) +c^2-c =x  \Rightarrow e\left(x+c\right)+cx+c^2-c = x so we obtain
    e\left(x+c\right)= \left(1-c\right)x+c\left(1-c\right)
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by girdav View Post
    We have x*e =x so xe+c\left(x+e\right) +c^2-c =x  \Rightarrow e\left(x+c\right)+cx+c^2-c = x so we obtain
    e\left(x+c\right)= \left(1-c\right)x+c\left(1-c\right)
    You of course need to verify that when solved the above yields an identity element in the underlying set of the supposed monoid.
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