![]() Products over other kinds of algebraic structures include: Products over other algebraic structures ![]() However, the concept of the empty product is more general, and requires special treatment in logic, set theory, computer programming and category theory. The empty product on numbers and most algebraic structures has the value of 1 (the identity element of multiplication), just like the empty sum has the value of 0 (the identity element of addition). Many of these are Cartesian closed categories. ![]() The class of all things (of a given type) that have Cartesian products is called a Cartesian category. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b)-where a ∈ A and b ∈ B. In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. Other kinds of products in linear algebra include: More precisely, a monoidal category is the class of all things (of a given type) that have a tensor product. That is, the monoidal category captures precisely the meaning of a tensor product it captures exactly the notion of why it is that tensor products behave the way they do. In general, whenever one has two mathematical objects that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the internal product of a monoidal category. The class of all objects with a tensor product The outer product is simply the Kronecker product, limited to vectors (instead of matrices). The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its intrinsic definition. The tensor product, outer product and Kronecker product all convey the same general idea. įor infinite-dimensional vector spaces, one also has the: Where V * and W * denote the dual spaces of V and W. Term + term summand + summand addend + addend augend + addend } =
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