External (mathematics)

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 ${\displaystyle f:R\times R\rightarrow S}$. 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:

• ${\displaystyle f:Q\times R\rightarrow S}$ (binary function)
  • Example: exponentiation (${\displaystyle z^{q}:\mathbb {Z} \times \mathbb {Q} \rightarrow \mathbb {C} }$ as in ${\displaystyle {(-1)}^{1/2}=i}$)
  • Examples: matrix multiplication, the tensor product, and the Cartesian product
• ${\displaystyle f:R\times R\rightarrow S}$ (internal binary function)
  • Example: internal binary relations (${\displaystyle (\leq ):R\times R\rightarrow \mathbb {B} }$)
  • Example: set membership (${\displaystyle (\in ):\mathbf {Set} \times \mathbf {Set} \rightarrow \mathbb {B} }$ where ${\displaystyle \mathbf {Set} }$ is the category of sets)
  • Examples: the dot product, the inner product, and metrics
• ${\displaystyle f:R\times S\rightarrow S}$ (external binary operation)
  • Examples: dynamical system flows, group actions, projection maps, and scalar multiplication
• ${\displaystyle f:S\times S\rightarrow S}$ (binary operation)
  • Examples: addition, multiplication, permutations, and the cross product

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:

• An external magma ${\displaystyle (S,\times )}$ over R is a set S with an external binary operation. This satisfies ${\displaystyle r\times s\in S}$ for all ${\displaystyle s\in S,r\in R}$ (external closure).
• An external semigroup ${\displaystyle (S,\times )}$ over ${\displaystyle (R,\cdot )}$ is an external magma that satisfies ${\displaystyle (r_{1}\cdot r_{2})\times s=r_{1}\times (r_{2}\times s)}$ for all ${\displaystyle s\in S,r_{1},r_{2}\in R}$ (externally associative).
• An external monoid ${\displaystyle (S,\times )}$ over ${\displaystyle (R,\cdot )}$ is an external semigroup in which there exists ${\displaystyle 1\in R}$ such that ${\displaystyle 1\times s=s}$ for all ${\displaystyle s\in S}$ (has external identity element).

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 ${\displaystyle \otimes }$ is externally distributive in ${\displaystyle (S,\oplus ,\otimes )}$ over the ring ${\displaystyle (R,+,\cdot )}$ iff:

• ${\displaystyle r\otimes (s_{1}\oplus s_{2})=(r\otimes s_{1})\oplus (r\otimes s_{2})}$ for all ${\displaystyle s_{1},s_{2}\in S,r\in R}$ and:
• ${\displaystyle (r_{1}+r_{2})\otimes s=(r_{1}\otimes s)\oplus (r_{2}\otimes s)}$ for all ${\displaystyle s\in S,r_{1},r_{2}\in R}$

Using these terminology we can make the following local generalizations:

• An external semiring ${\displaystyle (S,\oplus ,\otimes )}$ over the semiring ${\displaystyle (R,+,\cdot )}$ is a commutative monoid ${\displaystyle (S,\oplus )}$ and an external monoid ${\displaystyle (S,\otimes )}$ where ${\displaystyle \otimes }$ is externally distributive in ${\displaystyle (S,\oplus ,\otimes )}$ over the semiring ${\displaystyle (R,+,\cdot )}$.
• An external ring ${\displaystyle (S,\oplus ,\otimes )}$ over the ring ${\displaystyle (R,+,\cdot )}$ is an abelian group ${\displaystyle (S,\oplus )}$ and an external monoid ${\displaystyle (S,\otimes )}$ where ${\displaystyle \otimes }$ is externally distributive in ${\displaystyle (S,\oplus ,\otimes )}$ over the ring ${\displaystyle (R,+,\cdot )}$.

Now that we have all the terminology we need, we can make simple connections between various structures:

• Complex exponentiation forms an external monoid ${\displaystyle (\mathbb {C} ,\uparrow )}$ over the abelian group ${\displaystyle (\mathbb {C} ,\cdot )}$.
• Prime factorization forests form an external semiring ${\displaystyle (\mathbb {N} ,\cdot ,\uparrow )}$ over the semiring ${\displaystyle (\mathbb {N} ,+,\cdot )}$.
• A dynamical system ${\displaystyle (T,S,\Phi )}$ is an external monoid ${\displaystyle (S,\Phi )}$ over the monoid ${\displaystyle (T,{+})}$.
• A semimodule is an external semiring over a semiring.
• A module is an external ring over a ring.
• A vector space is an external ring over a field.