Supernatural number

In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz[1]^:249–251 in 1910 as a part of his work on field theory.

Hasse diagram of the lattice of supernatural numbers; primes other than 2 and 3 are omitted for simplicity.

A supernatural number ${\displaystyle \omega }$ is a formal product:

${\displaystyle \omega =\prod _{p}p^{n_{p}},}$

where ${\displaystyle p}$ runs over all prime numbers, and each ${\displaystyle n_{p}}$ is zero, a natural number or infinity. Sometimes ${\displaystyle v_{p}(\omega )}$ is used instead of ${\displaystyle n_{p}}$. If no ${\displaystyle n_{p}=\infty }$ and there are only a finite number of non-zero ${\displaystyle n_{p}}$ then we recover the positive integers. Slightly less intuitively, if all ${\displaystyle n_{p}}$ are ${\displaystyle \infty }$, we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide ${\displaystyle \omega }$ "infinitely often," by taking that prime's corresponding exponent to be the symbol ${\displaystyle \infty }$.

There is no natural way to add supernatural numbers, but they can be multiplied, with ${\displaystyle \prod _{p}p^{n_{p}}\cdot \prod _{p}p^{m_{p}}=\prod _{p}p^{n_{p}+m_{p}}}$. Similarly, the notion of divisibility extends to the supernaturals with ${\displaystyle \omega _{1}\mid \omega _{2}}$ if ${\displaystyle v_{p}(\omega _{1})\leq v_{p}(\omega _{2})}$ for all ${\displaystyle p}$. The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining

${\displaystyle \displaystyle \operatorname {lcm} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\sup(v_{p}(\omega _{i}))}}$

${\displaystyle \displaystyle \operatorname {gcd} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\inf(v_{p}(\omega _{i}))}}$

With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number. We can also extend the usual ${\displaystyle p}$-adic order functions to supernatural numbers by defining ${\displaystyle v_{p}(\omega )=n_{p}}$ for each ${\displaystyle p}$

Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of a finite field.[2] They are also used implicitly in many number-theoretical proofs, such as the density of the square-free integers and bounds for odd perfect numbers.

Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.