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Groups, monoids and nonempty subsets

  1. Feb 26, 2009 #1
    1. The problem statement, all variables and given/known data

    A) If M is any monois, let M' denote the set of all nonempty subsets of M and define an operation on M' by XY = {xy | x in X, y in Y}. show that M' is a monoid, commutative if M is, and find the units.

    B) If ab=ba in a monoid M, prove that (ab)^n = a^nb^n for all n >= 0.

    3. The attempt at a solution
    A) we need to show that M' is associative and has inverse.
    E = {e} in M' -- identity in M'
    X in M' XE = {xe | x in X} = {x | x inX} = X so that EX = X
    It seems right, but I have no idea how to show associative..
    Z in M' X(YZ) = {x(yz) | x in X, y in Y, and z in Z} = (XY)Z = { (xy)z |x in X, y in Y, and z in Z} is it right for associative??
    find all units in M'
    If a in M -- a unit in M
    {a} in M' -- a unit in M
    {a}{a^-1} = {aa^-1} = {e} = E
    {a}{a^-1} = {e} = E ----------I think it's correct..


    B)By induction. If n = 1, (ab)^1 = a^1b^1 which is true. If k >= 1, we assume that 1,2,...,k are all true, (ab)^k = a^kb^k. we must show that k+1 is also true.
    (ab)^k+1 = (ab)^k * ab
    = a^kb^k * ab
    = a^k (b^k * a)b
    = a^k(a*b^k)b = (a^k*a)(b^k*b) = a^k+1 b^k+1 complete induction.
    I think it's correct, but it was wrong somewhere..
    I don't know where..

    Thank you.
     
  2. jcsd
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