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Owner confidence rating: Very high Entry average rating: No information on entry rating
indexing set (Definition)

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.



"indexing set" is owned by Wkbj79.
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Other names:  index set
Also defines:  subscript, index, indices, indexed by, double indices, multiple indices
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Cross-references: partial function, map, ordered pairs, intersections, unions, products, summations, nets, sequences, function, surjective, surjection
There are 186 references to this entry.

This is version 6 of indexing set, born on 2006-07-31, modified 2007-08-08.
Object id is 8202, canonical name is IndexingSet.
Accessed 6469 times total.

Classification:
AMS MSC: 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)
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please read before editing indexing set by Wkbj79 on 2006-07-31 07:55:27
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
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