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(sub)semigroup with an endomorphism

  1. Jul 2, 2009 #1
    Let's have a semigroup S and a proper sub-semigroup B of S.
    If we have also an endomorphism [tex]f:S\rightarrow S[/tex], is it possible that the subset [tex]B'=\{f(b)|b\in B \}[/tex] is not a sub-semigroup anymore?
     
  2. jcsd
  3. Jul 2, 2009 #2

    HallsofIvy

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    The only requirement in order that a set, A, be a "sub-semigroup" of a semigroup S is that it be closed under the group operation. If x and y are in B', then x= f(a) and y= f(b) for some a, b in B'. But xy= f(a)f(b)= f(ab) is in B' because ab is in B.
     
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