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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 (a n-ary product type), and in this case, it characterizes the expressions which behave as tuples of expressions of the corresponding types. A degenerated 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, ..., typen is written type1
*typen in ML and
) in Haskell. In both these languages, tuples are written
) 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.
In mathematics, a binary operation or dyadic operation is a calculation that combines two elements to produce another element. More formally, a binary operation is an operation of arity two.
In mathematics and computer science, currying is the technique of converting a function that takes multiple arguments into a sequence of functions that each takes a single argument. For example, currying a function that takes three arguments creates three functions:
In logic, mathematics and linguistics, And is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is typically written as or ⋅ .
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Arity is the number of arguments or operands taken by a function or operation in logic, mathematics, and computer science. In mathematics, arity may also be named rank, but this word can have many other meanings in mathematics. In logic and philosophy, it is also called adicity and degree. In linguistics, it is usually named valency.
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In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence of n elements, where n is a non-negative integer. There is only one 0-tuple, referred to as the empty tuple. An n-tuple is defined inductively using the construction of an ordered pair.
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In functional programming, a monad is an abstraction that allows structuring programs generically. Supporting languages may use monads to abstract away boilerplate code needed by the program logic. Monads achieve this by providing their own data type, which represents a specific form of computation, along with two procedures:
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