be expressed as an application of the monoidal category to some particular setting, acting on some particular objects. Quotient algebras A number of important subspaces of the tensor algebra can be constructed as quotients: these include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the universal enveloping algebra in general. The exterior algebra is constructed from the exterior product. Given a vector space , the exterior product is defined as Note that when the underlying field of does not have characteristic 2, then this definition is equivalent to The image of in the exterior product is usually denoted and satisfies, by construction, Similar constructions are possible for ( factors), giving rise to the th exterior power of . The latter notion is the basis of differential -forms. The symmetric algebra is constructed in a similar manner, from the symmetric product More generally That is, in the symmetric algebra two adjacent vectors (and therefore all of them) can be interchanged. The resulting objects are called symmetric tensors. Tensor product in programming Array programming languages Array programming languages may have this pattern built in. For example, in APL the tensor product is expressed as ○.× (for example A ○.× B or A ○.× B ○.× C). In J the tensor product is the dyadic form of */ (for example a */ b or a */ b */ c). Note that J's treatment also allows the representation of some tensor fields, as a and b may be functions instead of constants. This product of two functions is a derived function, and if a and b are differentiable, then a */ b is differentiable. However, these kinds of notation are not universally present in array languages. Other array languages may require explicit treatment of indices (for example, MATLAB), and/or may not support higher-order functions such as the Jacobian derivative (for example, Fortran/APL). See also Notes References Operations on vectors Operations on structures Bilinear maps