In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set.
The finite sums of a set D of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of D. The set of all finite sums over D is often denoted as FS(D).
A set A of natural numbers is an IP set if there exists an infinite set D such that FS(D) is a subset of A.
Some authors give a slightly different definition of IP sets. They require that FS(D) equal A instead of just being a subset.
Sources disagree on the origin of the name IP set. Some claim it was coined by Furstenberg and Weiss to abbreviate "Infinite-dimensional Parallelepiped", others that it abbreviates "idempotent" (since a set is IP if and only if it is a member of an idempotent ultrafilter).
Read more about IP Set: Hindman's Theorem, Semigroups
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