- 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 f : R imes S ightarrow S and a right external binary operation on "S" over "R" is a function f : S imes R ightarrow S where "S" is the set the operation is defined on, and "R" is 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 f : R imes R ightarrow 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:
* f : Q imes R ightarrow S (
binary function )
** Example:exponentiation (z^q : Bbb{Z} imes Bbb{Q} ightarrow Bbb{C} as in 1)}^{1/2} = i),
** Example: set membership (in) : S imes mathbf{Set} ightarrow Bbb{B} where mathbf{Set} is thecategory of sets )
** Examples:matrix multiplication , thetensor product , and theCartesian product
* f : R imes R ightarrow S (internal binary function)
** Example: internalbinary relations (le) : R imes R ightarrow Bbb{B})
** Examples: thedot product , the inner product, and metrics.* f : R imes S ightarrow S (external binary operation)
** Examples:dynamical system flows, group actions, projection maps, andscalar multiplication .* f : S imes S ightarrow S (
binary operation ).
** Examples:addition ,multiplication , permutations, and thecross product .External monoids
Since
monoid s are defined in terms ofbinary 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 S, imes) over "R" is a set "S" with an external binary operation. This satisfies r imes s in S for all s in S, r in R (external closure).
* An externalsemigroup S, imes) over R, cdot) is an external magma that satisfies r_1 cdot r_2) imes s = r_1 imes (r_2 imes s) for all s in S, r_1, r_2 in R (externallyassociative ).
* An externalmonoid S, imes) over R, cdot) is an external semigroup in which there exists 1 in R such that 1 imes s = s for all s in S (has externalidentity element ).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 otimes is externallydistributive in S, oplus, otimes) over the ring R, +, cdot)iff :
* r otimes (s_1 oplus s_2) = (r otimes s_1) oplus (r otimes s_2) for all s_1,s_2 in S, r in R and:
* r_1 + r_2) otimes s = (r_1 otimes s) oplus (r_2 otimes s) for all s in S, r_1,r_2 in RUsing these terminology we can make the following local generalizations:
* An external semiring S, oplus, otimes) over thesemiring R, +, cdot) is acommutative monoid S, oplus) and an external monoid S, otimes) where otimes is externallydistributive in S, oplus, otimes) over thesemiring R, +, cdot).
* An external ring S, oplus, otimes) over the ring R, +, cdot) is anAbelian group S, oplus) and an external monoid S, otimes) where otimes is externallydistributive in S, oplus, otimes) over the ring R, +, cdot).Other examples
Now that we have all the terminology we need, we can make simple connections between various structures:
* Complex exponentiation forms an externalmonoid Bbb{C}, uparrow) over theAbelian group Bbb{C}, cdot).
* Prime factorization forests form an externalsemiring Bbb{N}, cdot, uparrow) over thesemiring Bbb{N}, +, cdot).
* A dynamical system T, S, Phi) is an external monoid S, Phi) over themonoid T, {+}).
* Asemimodule is an external semiring over asemiring .
* A module is an external ring over a ring.
* Avector space is an external ring over a field.Usefulness
It could be argued that we already have terms for the concepts described here, like
dynamical systems ,group actions , modules, andvector 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.
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