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# indexing set

Let $\Lambda$ and $S$ be sets such that there exists a surjection $f\colon\Lambda\to S$. Then $\Lambda$ is an *indexing set* for $S$. Also, $S$ is *indexed by* $\Lambda$.

In such situations, the elements of $S$ could be referenced by using the indexing set $\Lambda$, such as $f(\lambda)$ for some $\lambda\in\Lambda$. On the other hand, quite often, indexing sets are used without explicitly defining a surjective function. When this occurs, the elements of $S$ are referenced by using *subscripts* (also called *indices*) which are elements of $\Lambda$, such as $s_{{\lambda}}$ for some $\lambda\in\Lambda$. If, however, the surjection from $\Lambda$ to $S$ were called $s$, this notation would be quite similar to the function notation: $s(\lambda)=s_{{\lambda}}$.

Indexing sets are quite useful for describing sequences, nets, summations, products, unions, and intersections.

Multiple indices are possible. For example, consider the set $X=\{x_{{aa}},x_{{ab}},x_{{ac}},x_{{bb}},x_{{bc}},x_{{cc}}\}$. Some people would consider the indexing set for $X$ to be $\{aa,ab,ac,bb,bc,cc\}$. Others would consider the indexing set to be $\{a,b,c\}\times\{a,b,c\}$. (The double indices can be considered as ordered pairs.) Thus, in the case of multiple indices, it need not be the case that the underlying function $f$ be a surjection. On the other hand, $f$ must be a partial surjection. For example, if a set $X$ is indexed by $A\times B$, the following must hold:

1. For every $x\in X$, there exist $i\in A$ and $j\in B$ such that $f(i,j)=x$;

2. For every $i\in A$, the map $f_{i}\colon B\to X$ defined by $f_{i}(j)=f(i,j)$ is a partial function;

3. For every $j\in B$, the map $f_{j}\colon A\to X$ defined by $f_{j}(i)=f(i,j)$ is a partial function.

## Mathematics Subject Classification

03E99*no label found*

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## Comments

## please read before editing indexing set

My main concerns about the entry ``indexing set'' are the following:

MSC: I would imaging that this entry could fall under many MSC's. If you know of any, please add them.

uses for indexing sets: If you can think of more ways that indexing sets are used, please add them.

Changes elsewhere in the entry are not as essential to me but of course are welcome. I would not have made the entry world-editable otherwise. On the other hand, unlike my other world-editable objects, I will probably only have this object as world-editable for a limited time. So, if you want to put your two cents in, now is the time!

Thanks,

Warren

## Re: please read before editing indexing set

The expression "s_\lambda" is usually understood to be a stylistic variant of "s(\lambda)". How are you interpreting it so as to make sense of "This notation means that, if a surjection $f$ were explicitly defined, then $f(\lambda)=s_\lambda$ for every $\lambda \in \Lambda$"?

## Re: please read before editing indexing set

My interpretation probably comes from the fact that I have never had subscripts explained to me that way before. Your explanation makes a lot of sense! I will edit the object accordingly. Thanks.