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Operation exists that is valid for a group limited to the natural

  1. Jul 16, 2009 #1
    What kind of operation exists that is valid for a group limited to the natural numbers?
     
  2. jcsd
  3. Jul 16, 2009 #2
    Re: Groups/Operations

    I can't think of any really natural or appealing ones, but you can easily cook up some fairly contrived group operations on [tex] \mathbb{N} [/tex] by bijecting it with [tex] \mathbb{Z} [/tex]. For example, for [tex] a,b \in \mathbb{N} [/tex], define the function [tex] f(a,b) [/tex] as follows:

    [tex]
    \displaystyle f(a,b) \equiv (-1)^{2 \left\{ \frac{a}{2} \right\} } \left\lfloor \frac{a}{2} \right\rfloor + (-1)^{2 \left\{ \frac{b}{2} \right\} } \left\lfloor \frac{b}{2} \right\rfloor \textrm{.}
    [/tex]

    Then let

    [tex]
    \displaystyle a \star b \equiv
    \left\{
    \begin{aligned}
    2f(a,b) \quad \textrm{if} \; f(a,b) \geq 0\\
    |2f(a,b)| + 1 \quad \textrm{if} \; f(a,b) < 0 \textrm{.}
    \end{aligned}
    \right.
    [/tex]

    Then [tex] \star [/tex] is a group operation, with identity element [tex] 1 [/tex], such that [tex] n^{-1} = n + (-1)^n [/tex].

    The natural structure of [tex] \mathbb{N} [/tex] is that of a monoid, i.e., a "group without inverses." By the way, just out of curiosity, why do you ask? Is there some problem you're working on that requires you to treat [tex] \mathbb{N} [/tex] as a group in some way?
     
  4. Jul 17, 2009 #3
    Re: Groups/Operations

    Thanks for the help.
    The reason I asked was simply out of curiosity (while thinking, I couldn't think of a naturally occurring one).
     
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