Product type

In programming languages and type theory, a product of types is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the product. An instance of a product type retains the fixed order, but otherwise may contain all possible instances of its primitive data types. The expression of an instance of a product type will be a tuple, and is called a "tuple type" of expression. A product of types is a direct product of two or more types.

If there are only two component types, it can be called a "pair type". For example, if two component types A and B are the set of all possible values of that type, the product type written A × B contains elements that are pairs (a,b), where "a" and "b" are instances of A and B respectively. The pair type is a special case of the dependent pair type, where the type B may depend on the instance picked from A.

In many languages, product types take the form of a record type, for which the components of a tuple can be accessed by label. In languages that have algebraic data types, as in most functional programming languages, algebraic data types with one constructor are isomorphic to a product type.

In the Curry–Howard correspondence, product types are associated with logical conjunction (AND) in logic.

The notion directly extends to the product of an arbitrary finite number of types (an n-ary product type), and in this case, it characterizes the expressions that behave as tuples of expressions of the corresponding types. A degenerate form of product type is the unit type: it is the product of no types.

In call-by-value programming languages, a product type can be interpreted as a set of pairs whose first component is a value in the first type and whose second component is a value in the second type. In short, it is a cartesian product and it corresponds to a product in the category of types.

Most functional programming languages have a primitive notion of product type. For instance, the product of type₁, ..., type_n is written type₁*...*type_n in ML and (type₁,...,type_n) in Haskell. In both these languages, tuples are written (v₁,...,v_n) and the components of a tuple are extracted by pattern-matching. Additionally, many functional programming languages provide more general algebraic data types, which extend both product and sum types. Product types are the dual of sum types.

References

product type at the nLab
Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study. See section 1.5.

v t e Data types
Uninterpreted	Bit Byte Trit Tryte Word Bit array
Numeric	Arbitrary-precision or bignum Complex Decimal Fixed point Floating point Reduced precision Minifloat Half precision bfloat16 Single precision Double precision Quadruple precision Octuple precision Extended precision Long double Integer signedness Interval Rational
Pointer	Address physical virtual Reference
Text	Character String null-terminated
Composite	Algebraic data type generalized Array Associative array Class Dependent Equality Inductive Intersection List Object metaobject Option type Product Record or Struct Refinement Set Union tagged
Other	Boolean Bottom type Collection Enumerated type Exception Function type Opaque data type Recursive data type Semaphore Stream Strongly typed identifier Top type Type class Empty type Unit type Void
Related topics	Abstract data type Boxing Data structure Generic Kind metaclass Parametric polymorphism Primitive data type Interface Subtyping Type constructor Type conversion Type system Type theory Variable

See also

References