Contents
Context
Arithmetic
number theory
number
- natural number, integer number, rational number, real number, irrational number, complex number, quaternion, octonion, adic number, cardinal number, ordinal number, surreal number
arithmetic
arithmetic geometry, function field analogy
Arakelov geometry
Algebra
- algebra, higher algebra
- universal algebra
- monoid, semigroup, quasigroup
- nonassociative algebra
- associative unital algebra
- commutative algebra
- Lie algebra, Jordan algebra
- Leibniz algebra, pre-Lie algebra
- Poisson algebra, Frobenius algebra
- lattice, frame, quantale
- Boolean ring, Heyting algebra
- commutator, center
- monad, comonad
- distributive law
Group theory
Ring theory
Module theory
Contents
Definition
In Heyting fields
Given a Heyting field , let us define the type of all terms in apart from 0:
The reciprocal or reciprocal function is a partial function such that for all we have and
In a discrete field, the reciprocal is a function defined as
for and
for , where . This is because for a discrete field , the set is a decidable subset of .
In dense sequentially Cauchy complete ordered integral domains
Let be a ordered integral domain, and for all elements and let be the open subinterval containing all elements greater than and less than . Then the sequences
indexed by natural number are Cauchy sequences for all elements , and if is sequentially Cauchy complete, it has a limit for elements as
If is also dense with given element , then there are sequences of sequences
indexed by natural numbers and , the first which is Cauchy for elements and the second which is Cauchy for . Since is sequentially Cauchy complete, both have limits as
which themselves are Cauchy, and thus have limits
Since both and go to infinity as goes to infinity, the domain of is and the domain of is .
The reciprocal is a piecewise defined partial function defined as
Thus, every dense sequentially Cauchy complete ordered integral domain is an ordered field.
In sequentially Cauchy complete ordered integral rational algebras
Let be a ordered integral domain which is a -algebra, and for all elements and let be the open subinterval containing all elements greater than and less than . Then the sequence
indexed by natural number is a Cauchy sequence for all elements , and if is sequentially Cauchy complete, it has a limit for elements as
There is a sequence of sequences
indexed by natural numbers and , which is Cauchy for .
Since is sequentially Cauchy complete, the function has a limit as
which itself is Cauchy, and thus has a limit
called the natural logarithm. Since goes to infinity as goes to infinity, the domain of is .
The exponential function is defined as as
and the reciprocal function is defined as
See also