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The set of sets of elements of $R$, also known as powerset of $R$ can be typeset $2^R$. I am now interested in the set of multisets of elements in $R$. How is it called? Is there a standard notation?

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    The set of finite subsets of $R$ is *not* the power set. [Here's the wikipedia page](http://en.wikipedia.org/wiki/Power_set) about power sets.2012-05-23
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    The notation $2^R$ denotes the set of *all* subsets of $R$, not just the finite ones.2012-05-23
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    Multisets are just functions $R\to\mathbb{N}$, so $\mathbb{N}^R$ might be it. However, I am not sure that there is established notation, it varies among the books and articles and I think it ts best to _clearly_ state what you mean, just to avoid confusion.2012-05-23
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    @dtldarek: Just to be clear, you're including $0$ in $\mathbb{N}$, right? (not everyone does this) Without $0$, then $\mathbb{N}^R$ would just be the multisets with at least one copy of every element.2012-05-23
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    I would be a little surprised if $\{1,2\}$ was deemed to "contain" the multiset $\{\{1,1,1,1,1,1,1,1\}\}$, as the comments in this thread suggest. Certainly one does not usually say that the multiset $\{\{1,2\}\}$ contains $\{\{1,1,1,1,1,1,1,1\}\}$. I hope @Halladba will clarify what he or she is looking for.2012-05-23
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    @ZevChonoles Yeah, for me $0 \in \mathbb{N}$. Thanks for pointing that out (and shame on all those who think differently ^^ (of course, this is a joke))!2012-05-23
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    @Mark: I'm not sure which comment suggests that. I don't think the terminology "multiset of elements of $R$" implies that $R$ is to be interpreted as (always) containing it. Of course, even with vanilla subsets, containment is partial order; for multisets I think a reasonable definition would be that $f\subseteq g$ for $f,g\in \mathbb{N}^R$ when $f(r)\leq g(r)$ for all $r\in R$.2012-05-23
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    Ok, sorry for the finite aspect. I edited that out. @Mark, $R$ is not assumed to be a multiset itself, just a set. So if $R$ is $\{1,2\}$, $\{1,1,1\}$ is indeed a multiset of elements of $R$ and would be in the set of multisets of elements of $R$.2012-05-23

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As requested, the comment rewritten as an answer.

Multisets are just functions $R \to \mathbb{N}$ (with $0 \in \mathbb{N}$, thanks to Zev Chonoles for emphasizing that), so $\mathbb{N}^R$ might be what you are looking for. However, I am not sure that there is established notation, it varies among the books and articles and I think it ts best to clearly state what you mean, just to avoid confusion.