Indexstel

Indexstel is a term used primarily in mathematics and computer science to describe a set of indices that label elements within a larger structure such as a sequence, array, matrix, or tensor. The concept is fundamental to the formal description of indexed families, allowing precise reference to each component of a structure.

In set theory, an indexstel I is a nonempty collection of objects that serve as labels. An indexed family is then a mapping from I to another set, typically written as {a_i | i ∈ I}. For instance, the sequence of natural numbers can be viewed as an indexed family {n_i | i ∈ {1,2,3,…}} where each i specifies the position of n in the sequence. Similar notation applies to finite-dimensional vector spaces, where a basis {e_i | i ∈ I} can be described using an indexstel for coordinates.

In computer science, indexstels underpin data structures such as arrays and matrices. When programming languages refer to an array Index, the underlying indexstel is typically the set {0, 1, …, n−1} for an n‑element array, providing a convenient numeric labeling system for memory addresses.

The usage of indexstels is common in proofs that involve permutation of elements, summation over multiple indices, and in the definition of Kronecker products. When multiple indexstels are involved, notation such as {i, j, … ∈ I} is used to indicate product sets, enabling concise representation of multidimensional data.

Historically, the adoption of explicit indexstels dates back to the early 20th century, with the formalization of set theory by Cantor and the development of linear algebra by Hamilton and Cayley. They have become standard in modern mathematical literature and are often introduced in introductory courses on discrete mathematics and linear algebra.