Extended natural numbers

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

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

Applications

In graph theory, the extended natural numbers are used to define distances in graphs, with $\mathrm{\infty }$ 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 topos of right actions on the extended natural numbers is a category PRO of projection algebras.

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

References

• Folkman, Jon; Fulkerson, D.R. (1970). "Flows in Infinite Graphs". Journal of Combinatorial Theory 8 (1): 30–44. 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): 764–784. doi:10.2178/jsl.7803040. https://www.jstor.org/stable/43303679. 
• Koch, Sebastian (2020). "Extended Natural Numbers and Counters". Formalized Mathematics 28 (3): 239–249. doi:10.2478/forma-2020-0021. https://fm.mizar.org/fm28-3/counters.pdf. 
• Khanjanzadeh, Zeinab; Madanshekaf, Ali (2018). "Weak Ideal Topology in the Topos of Right Acts Over a Monoid". Communications in Algebra 46 (5): 1868–1888. doi:10.1080/00927872.2017.1360330. 
• Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. ISBN 978-0-521-84425-3. 

Further reading

• Robert, Leonel (3 September 2013). "The Cuntz semigroup of some spaces of dimension at most two". arXiv:0711.4396 [math.OA].
• Lightstone, A. H. (1972). "Infinitesimals". The American Mathematical Monthly 79 (3): 242–251. doi:10.1080/00029890.1972.11993024. 
• Khanjanzadeh, Zeinab; Madanshekaf, Ali (2019). "On Projection Algebras". Southeast Asian Bulletin of Mathematics 43 (2). 

External links

• Extended natural number in nLab