The generalisation of ordered pair to something having more positions is usually called a tuple (or ordered tuple). More particularly one gets the term -tuple, which refers to a list, , with entries from some set, ; here is a natural number. It thus corresponds to an element in the -fold product set, . The various elements of the -tuple are usually called its components and sometimes it is useful to call the set of components the support or range of the tuple.
The term ‘tuple’ is usually used for an -tuple for a specific number . If we wish to speak of an -tuple for an arbitrary (particularly without specifying that ), then we may speak of a list (which has other terminology, described on that page). Then the set of lists is the disjoint union over of the sets of -tuples.
The term ‘tuple’ is usually used for an -tuple for a finite number . If we wish to speak of an -tuple for an infinite (or possibly infinite) , then we may speak of a sequence.
See ordered pair for methods of formalising ordered pairs (which are -tuples) in various foundations of mathematics. Some of these generalise immediately to -tuples for arbitrary ; otherwise, we may define -tuples recursively: a triple is an ordered pair whose (say) first component is an ordered pair; a quadruple is an ordered pair whose first component is a triple, etc.