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Extended natural numbers

From Wikipedia, the free encyclopedia

In mathematics, the extended natural numbers is a set which contains the values and (infinity). That is, it is the result of adding a maximum element to the natural numbers. Addition and multiplication work as normal for finite values, and are extended by the rules (), and for .

With addition and multiplication, is a semiring but not a ring, as lacks an additive inverse.[1] The set can be denoted by , or .[2][3][4] It is a subset of the extended real number line, which extends the real numbers by adding and .[2]

Applications

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In graph theory, the extended natural numbers are used to define distances in graphs, with 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 are a one-point compactification of the natural numbers, yielding the set of non-increasing binary sequences i.e. such that . The sequence represents , while the sequence represents . It is a retract of and the claim that implies the limited principle of omniscience.[3]

Notes

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References

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  • 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.

Further reading

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  • 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).
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