IndexingSet 2006-07-31 07:47:44 2007-08-08 22:39:04 Definition subscript index indices indexed by double indices multiple indices \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{psfrag} \usepackage{graphicx} \usepackage{amsthm} \usepackage{xypic} Let $\Lambda$ and $S$ be sets such that there exists a surjection $f \colon \Lambda \to S$. Then $\Lambda$ is an \emph{indexing set} for $S$. Also, $S$ is \emph{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 \emph{subscripts} (also called \emph{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 \PMlinkescapetext{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: \begin{enumerate} \item For every $x\in X$, there exist $i\in A$ and $j\in B$ such that $f(i,j)=x$; \item 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; \item 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. \end{enumerate}