Mathematical indices are defined as functions where the domain is a finite set. This approach allows for the representation of data through tables or formulas, similar to standard functions on real numbers.
Functions can be defined on Cartesian products of finite sets, resulting in multi-indexed functions. These structures are analogous to multidimensional arrays in programming, where values are indexed by multiple arguments.
Operations such as addition and multiplication are performed on these functions by applying the operations to their values at corresponding arguments. New functions can also be constructed by combining existing ones.
Summation is a key operation for index functions, analogous to integration for continuous functions. Indices used in summation are termed 'dummy' indices, while those remaining in the result are 'free' indices.
Multiple summations are commutative, meaning the order of summation does not affect the final result. This property holds for both simple and multi-indexed functions, allowing for flexible notation.
