Extended_natural_numbers Knowpia

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

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

Applications edit

In graph theory, the extended natural numbers are used to define distances in graphs, with $\infty$ 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 $\mathbb {N} _{\infty }$ are a one-point compactification of the natural numbers, yielding the set of non-increasing binary sequences i.e. $(x_{0},x_{1},\dots )\in 2^{\mathbb {N} }$ such that $\forall i\in \mathbb {N} :x_{i}\geq x_{i+1}$ . The sequence $1^{n}0^{\omega }$ represents $n$ , while the sequence $1^{\omega }$ represents $\infty$ . It is a retract of $2^{\mathbb {N} }$ and the claim that $\mathbb {N} \cup \{\infty \}\subseteq \mathbb {N} _{\infty }$ implies the limited principle of omniscience.^[3]

Notes edit

^ Sakarovitch (2009), p. 28.
^ ^a ^b ^c Koch (2020).
^ ^a ^b Escardó (2013).
^ ^a ^b Khanjanzadeh & Madanshekaf (2018).
^ Folkman & Fulkerson (1970).

References edit

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.

External links edit

Extended natural number at the nLab

[FOOTNOTESakarovitch200928-1] Sakarovitch (2009), p. 28.

[FOOTNOTEKoch2020-2] Koch (2020).

[FOOTNOTEEscardó2013-3] Escardó (2013).

[FOOTNOTEKhanjanzadehMadanshekaf2018-4] Khanjanzadeh & Madanshekaf (2018).

[FOOTNOTEFolkmanFulkerson1970-5] Folkman & Fulkerson (1970).

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