Extended natural numbers

With addition and multiplication, ${\displaystyle \mathbb{N}\cup \{\mathrm{\infty }\}}$ is a semiring but not a ring, as ${\displaystyle \mathrm{\infty }}$ lacks an additive inverse. The set can be denoted by ${\displaystyle \overline{\mathbb{N}}}$, ${\displaystyle {\mathbb{N}}_{\mathrm{\infty }}}$ or ${\displaystyle {\mathbb{N}}^{\mathrm{\infty }}}$. It is a subset of the extended real number line, which extends the real numbers by adding ${\displaystyle -\mathrm{\infty }}$ and ${\displaystyle +\mathrm{\infty }}$.

Applications

In graph theory, the extended natural numbers are used to define distances in graphs, with ${\displaystyle \mathrm{\infty }}$ being the distance between two unconnected vertices. They can be used to show the extension of some results, such as the max-flow min-cut theorem, to infinite graphs.

In topology, the Grothendieck topos of right actions on the extended natural numbers is a category PRO of projection algebras.

In constructive mathematics, the extended natural numbers ${\displaystyle {\mathbb{N}}_{\mathrm{\infty }}}$ are a one-point compactification of the natural numbers, yielding the set of non-increasing binary sequences i.e. ${\displaystyle (x_0,x_1,\dots )\in 2^{\mathbb{N}}}$ such that ${\displaystyle \mathrm{\forall }i\in \mathbb{N}:x_i\ge x_{i+1}}$. The sequence ${\displaystyle 1^n{0}^{\omega }}$ represents ${\displaystyle n}$, while the sequence ${\displaystyle 1^{\omega }}$ represents ${\displaystyle \mathrm{\infty }}$. It is a retract of ${\displaystyle 2^{\mathbb{N}}}$ and the claim that ${\displaystyle \mathbb{N}\cup \{\mathrm{\infty }\}\subseteq {\mathbb{N}}_{\mathrm{\infty }}}$ implies the limited principle of omniscience.