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Commutative monoids have binary products

  1. Mar 12, 2010 #1
    1. The problem statement, all variables and given/known data

    I'd like to prove that the category cmon of commutative monoids has binary products.

    3. The attempt at a solution

    actually i'm aware that i have to use cartesian products
    given monoids (M, [tex]\bullet[/tex]m, [tex]e^{}_{m}[/tex]) and (N, [tex]\bullet[/tex]n, [tex]e^{}_{n}[/tex])

    it follows that (M[tex]\times[/tex]N) [tex]\times[/tex] (M[tex]\times[/tex]N) [tex]\rightarrow[/tex] M[tex]\times[/tex] N

    and ((m,n), (m',n')) |---> (m [tex]\bullet[/tex]m m', n [tex]\bullet[/tex]n n') ....

    Thanks in advance for any help!
  2. jcsd
  3. Mar 12, 2010 #2


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    Staff Emeritus
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    Gold Member

    Well, what have you managed to do successfully, and where are you stuck? Can you, at least, write an outline of the individual steps you have to do even if you can't do them?

    P.S. I'm assuming you don't have any general structure theorems available -- e.g. that any variety of universal algebra has products.
  4. Mar 12, 2010 #3
    actually i've got the solution for the particular problem for category monoid of monoids but i can't figure out what is needed to add to the proof to satisfy the proposition that cmon of commutative monoids have binary products.
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