Extended natural numbers

In mathematics, the extended natural numbers is a set which contains the values [math]\displaystyle{ 0, 1, 2, \dots }[/math] and [math]\displaystyle{ \infty }[/math] (infinity). That is, it is the result of adding a maximum element [math]\displaystyle{ \infty }[/math] to the natural numbers. Addition and multiplication work as normal for finite values, and are extended by the rules [math]\displaystyle{ n+\infty=\infty+n=\infty }[/math] ([math]\displaystyle{ n\in\mathbb{N}\cup \{\infty\} }[/math]), [math]\displaystyle{ 0\times \infty=\infty \times 0=0 }[/math] and [math]\displaystyle{ m\times \infty=\infty\times m=\infty }[/math] for [math]\displaystyle{ m\neq 0 }[/math]. With addition and multiplication, [math]\displaystyle{ \mathbb{N}\cup \{\infty\} }[/math] is a semiring but not a ring, as [math]\displaystyle{ \infty }[/math] lacks an additive inverse. The set can be denoted by [math]\displaystyle{ \overline{\mathbb{N}} }[/math], [math]\displaystyle{ \mathbb{N}_\infty }[/math] or [math]\displaystyle{ \mathbb{N}^\infty }[/math]. It is a subset of the extended real number line, which extends the real numbers by adding [math]\displaystyle{ -\infty }[/math] and [math]\displaystyle{ +\infty }[/math].

Applications

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

References

• 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). https://www.jstor.org/stable/43303679. 
• Koch, Sebastian (2020). "Extended Natural Numbers and Counters". Formalized Mathematics 28 (3). 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). 
• 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.
• 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). 

External links

• Extended natural number in nLab