ktuple

A ktuple, sometimes written as a k-tuple, is an ordered list of k elements drawn from a given set A. It is typically denoted (a1, a2, ..., ak), where each ai is an element of A and k is a nonnegative integer. The length of the tuple is k. The special case k = 1 yields a single element, and k = 0 yields the empty tuple, which is often used as the identity element in product constructions.

This concept is formalized by the Cartesian product A × A × ... × A (k factors), commonly written A^k. The set A^k consists of all k-tuples with entries from A. In this context, order matters, and repetition of elements is allowed unless explicitly restricted.

ktuples are used across mathematics and computer science. In geometry, a real k-tuple corresponds to a point in R^k. In databases, a k-tuple can describe a record with k fields. In programming languages that support tuples, a fixed-size heterogenous container may be called a k-tuple, with components possibly of different types.

Related notions include the n-tuple for any fixed n, and the concept of an empty tuple for k = 0. In logic and category theory, finite tuples model n-ary relations and finite products.