External (mathematics)

The term external is useful for describing certain algebraic structures. The term comes from the concept of an external binary operation which is a binary operation that draws from some "external set". To be more specific, a left external binary operation on "S" over "R" is a function formula_1 and a right external binary operation on "S" over "R" is a function formula_2 where "S" is the set the operation is defined on, and "R" is the external set (the set the operation is defined "over").
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 formula_3. 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:
External monoids.
Since monoids are defined in terms of binary operations, we can define an "external monoid" in terms of "external binary operations". For the sake of simplicity, unless otherwise specified, a "left" external binary operation is implied. Using the term "external", we can make the generalizations:
Modules as external rings.
Much of the machinery of modules and vector spaces are fairly straightforward, or discussed above. The only thing not covered yet is their distribution axioms. The external ring multiplication formula_25 is externally distributive in formula_26 over the ring formula_27 iff:
Using these terminology we can make the following local generalizations:
Other examples.
Now that we have all the terminology we need, we can make simple connections between various structures:
Usefulness.
It could be argued that we already have terms for the concepts described here, like dynamical systems, group actions, modules, and vector spaces. However, there is still no other terminology available for an external monoid for which this terminology gives us a concise expression. Above all else, this is a reason this term should be of use in the mathematical community.