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]
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]
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]
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]