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Groups, rings and fields are equipped with binary operations. These can be applied repeatedly: $a_1+a_2+a_3+\dots$ to produce a sum of many elements, perhaps countably many. Can this be done also for uncountable sums? That is, is it possible to define the sum of an uncountable set of elements, using the binary operations?

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What do you mean with a sum of a countable set of element in a group? I think you need a topology, at least. –  Andrea Jan 9 '12 at 14:05
Related: math.stackexchange.com/q/106102 –  Jonas Meyer Aug 22 at 4:20

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Given a binary associative operation on a set $S$, you can, inductively, define a derived $n$-ary operation for any positive integer $n$.

Less pedantically: if you have an operation $\cdot\colon S\times S\to S$, written, as usual, in infix notation (so we write $ab$ instead of $\cdot(a,b)$), which is associative, then you can define an operation with $n$ terms for any positive integer $n$: the unary operation is just the identity function, $s\mapsto s$; the binary operation is $\cdot$, $(a,b)\mapsto ab$; and if you have already defined the $n$ary operation, we get the $(n+1)$-ary operation by $$(a_1,\ldots,a_n,a_{n+1}) \mapsto (a_1\cdots a_n)\cdot a_{n+1}.$$

In general, however, there is no way to go to even infinite countable operations. They are not well-defined in terms of a finitary operation.

Instead, you need some auxiliary notions to even try to define them. In the presence of a topology, you can try to define an infinite countable sum as the limit of a sequence; this is what we do in calculus. We never actually "add up" an infinite series $$\sum_{i=1}^{\infty} a_i;$$ instead, we consider the sequence of partial sums: $$\begin{align*} s_1 &= a_1\\ s_2 &= a_1+a_2\\ s_3 &= a_1+a_2+a_3\\ &\vdots\\ s_n &= a_1+\cdots+a_n\\ &\vdots\\ \end{align*}$$ and then consider whether this sequence converges or not; if it does, then we refer to the limit of this sequence as "the value of the sum" (even though we never actually add up more than finitely many terms). If it does not converge, we say the series "diverges" and has no value.

There are a handful of cases where all such terms actually converge so that you do have all countable "sums"/"products". For example, if $S$ is the power set of a set $X$ and the operation is binary union (or intersection) (this is not a group, though). Even so, for any infinite cardinal $\kappa$, you can construct examples where all sums/products of cardinality less than $\kappa$ "make sense", but there are some of cardinality $\kappa$ that do not "make sense."

But in general, if all you have is a group/ring/field, you can only do sums/products with finitely many terms; you can't even get to countably infinite.

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Thanks!! Arturo Magidin –  kssss Jan 10 '12 at 5:42

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