Extended natural numbers
In mathematics, the extended natural numbers is a set which contains the values and . 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. The set can be denoted by, or. It is a subset of the extended real number line, which extends the real numbers by adding and.
Applications
In graph theory, the extended natural numbers are used to define distances in graphs, with 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 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.