0tuple

A 0tuple, also known as the empty tuple or arity-zero tuple, is a tuple containing no elements. It is the unique element of the set of k-tuples when k equals zero, and is commonly represented by the symbol () in many languages. As a conceptual limit case, the 0tuple serves as the neutral or identity element for concatenation of tuples: appending a 0tuple to any k-tuple yields the original k-tuple, and prepending a 0tuple to a k-tuple also yields the same k-tuple.

Mathematically, the 0tuple corresponds to the empty product over an empty index set. In set theory this product is a singleton set, reflecting the idea that there is exactly one way to have zero components. In category theory terms, the product of zero objects is a terminal object, which aligns with the notion that the 0tuple contains no data.

In programming and type systems, a 0tuple often aligns with the unit type, a type that has exactly one value. The unit type functions as a harmless, always-present value for functions with no meaningful input or output. Some languages use an empty tuple literal to denote this idea; for example, the literal () can represent the 0tuple, while other languages may use a dedicated unit type rather than an actual empty sequence.

See also: empty tuple, unit type, arity, nullary.