produktoperasjonen

Produktoperasjonen, in mathematics often referred to simply as the product operation, is a binary operation that combines two elements to produce a third. It is usually denoted by symbols such as juxtaposition (ab), a dot (a · b), or a times sign (a × b). Formally, it is a function from a set S × S to S: (a, b) ↦ a · b.

In algebraic structures, the product operation is defined on a set with a rule that specifies how elements multiply. The operation is commonly associative, meaning (a · b) · c = a · (b · c). If there exists an identity element e with e · a = a · e = a for all a in S, the structure is a monoid; if every element also has an inverse, it forms a group. When multiplication distributes over addition, the structure becomes a ring or a field, depending on the presence of multiplicative inverses.

Different contexts use different flavors of product. Internal product refers to multiplying elements within a single set, while external products (such as Cartesian product) construct new objects from sets. In algebra and linear algebra, important variants include the matrix product, the tensor product of vector spaces, and the direct product of algebraic structures. The cross product is a specialized binary operation in three dimensions that yields a vector orthogonal to the operands.

Examples range from the familiar integers under ordinary multiplication to matrices, functions under composition, and Cartesian products that form pairs. Properties of the product operation, such as commutativity (which holds in many familiar settings like the real numbers) and distributivity over addition, depend on the specific mathematical context.

See also: multiplication, Cartesian product, tensor product, direct product, monoid.