External (mathematics) - Generalizations

Generalizations

The external concept is a generalization rather than a specialization, and as such, it is different from many terms in mathematics. A similar but opposite concept is that of an internal binary function from R to S, defined as a function . Internal binary functions are like binary functions, but are a form of specialization, so they only accept a subset of the domains of binary functions. Here we list these terms with the function signatures they imply, along with some examples:

• (binary function)
  • Example: exponentiation ( as in ),
  • Example: set membership ( where is the category of sets)
  • Examples: matrix multiplication, the tensor product, and the Cartesian product
• (internal binary function)
  • Example: internal binary relations
  • Examples: the dot product, the inner product, and metrics.

• (external binary operation)
  • Examples: dynamical system flows, group actions, projection maps, and scalar multiplication.

• (binary operation).
  • Examples: addition, multiplication, permutations, and the cross product.