Extended natural numbers

In mathematics, the extended natural numbers is a set which contains the values ${\displaystyle 0,1,2,\dots }$ and ${\displaystyle \infty }$ (infinity). That is, it is the result of adding a maximum element ${\displaystyle \infty }$ to the natural numbers. Addition and multiplication work as normal for finite values, and are extended by the rules ${\displaystyle n+\infty =\infty +n=\infty }$ (${\displaystyle n\in \mathbb {N} \cup \{\infty \}}$), ${\displaystyle 0\times \infty =\infty \times 0=0}$ and ${\displaystyle m\times \infty =\infty \times m=\infty }$ for ${\displaystyle m\neq 0}$.

With addition and multiplication, ${\displaystyle \mathbb {N} \cup \{\infty \}}$ is a semiring but not a ring, as ${\displaystyle \infty }$ lacks an additive inverse.^[1] The set can be denoted by ${\displaystyle {\overline {\mathbb {N} }}}$, ${\displaystyle \mathbb {N} _{\infty }}$ or ${\displaystyle \mathbb {N} ^{\infty }}$.^[2][3][4] It is a subset of the extended real number line, which extends the real numbers by adding ${\displaystyle -\infty }$ and ${\displaystyle +\infty }$.^[2]

In graph theory, the extended natural numbers are used to define distances in graphs, with ${\displaystyle \infty }$ being the distance between two unconnected vertices.^[2] They can be used to show the extension of some results, such as the max-flow min-cut theorem, to infinite graphs.^[5]

In topology, the topos of right actions on the extended natural numbers is a category PRO of projection algebras.^[4]

In constructive mathematics, the extended natural numbers ${\displaystyle \mathbb {N} _{\infty }}$ 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 ${\displaystyle \forall i\in \mathbb {N} :x_{i}\geq x_{i+1}}$. The sequence ${\displaystyle 1^{n}0^{\omega }}$ represents ${\displaystyle n}$, while the sequence ${\displaystyle 1^{\omega }}$ represents ${\displaystyle \infty }$. It is a retract of ${\displaystyle 2^{\mathbb {N} }}$ and the claim that ${\displaystyle \mathbb {N} \cup \{\infty \}\subseteq \mathbb {N} _{\infty }}$ implies the limited principle of omniscience.^[3]

• Folkman, Jon; Fulkerson, D.R. (1970). "Flows in Infinite Graphs". Journal of Combinatorial Theory. 8 (1). doi:10.1016/S0021-9800(70)80006-0.
• Escardó, Martín H (2013). "Infinite Sets That Satisfy The Principle of Omniscience in Any Variety of Constructive Mathematics". Journal of Symbolic Logic. 78 (3).
• Koch, Sebastian (2020). "Extended Natural Numbers and Counters" (PDF). Formalized Mathematics. 28 (3).
• Khanjanzadeh, Zeinab; Madanshekaf, Ali (2018). "Weak Ideal Topology in the Topos of Right Acts Over a Monoid". Communications in Algebra. 46 (5).
• Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. ISBN 978-0-521-84425-3. Zbl 1188.68177.

• Robert, Leonel (3 September 2013). "The Cuntz semigroup of some spaces of dimension at most two". arXiv:0711.4396.
• Lightstone, A. H. (1972). "Infinitesimals". The American Mathematical Monthly. 79 (3).
• Khanjanzadeh, Zeinab; Madanshekaf, Ali (2019). "On Projection Algebras". Southeast Asian Bulletin of Mathematics. 43 (2).

• Extended natural number at the nLab