indexing set

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

In such situations, the elements of $S$ could be referenced by using the indexing set $\mathrm{\Lambda }$, such as $f(\lambda )$ for some $\lambda \in \mathrm{\Lambda }$. 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 $\mathrm{\Lambda }$, such as $s_{\lambda }$ for some $\lambda \in \mathrm{\Lambda }$. If, however, the surjection from $\mathrm{\Lambda }$ to $S$ were called $s$, this notation would be quite 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 $f_i:B\to X$ defined by $f_i(j)=f(i,j)$ is a partial function;

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

Title	indexing set
Canonical name	IndexingSet
Date of creation	2013-03-22 16:07:51
Last modified on	2013-03-22 16:07:51
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	9
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 03E99
Synonym	index set
Defines	subscript
Defines	index
Defines	indices
Defines	indexed by
Defines	double indices
Defines	multiple indices