In mathematics, the extended natural numbers is a set which contains the values
(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
With addition and multiplication,
is a semiring but not a ring, as
lacks an additive inverse.
[1] The set can be denoted by
[2][3][4] It is a subset of the extended real number line, which extends the real numbers by adding
[2] 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.
represents
represents
It is a retract of
and the claim that
implies the limited principle of omniscience.