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
Famous quotes containing the word set:
“Where be your gibes now, your gambols, your songs, your
flashes of merriment, that were wont to set the table on a
roar?”
—William Shakespeare (1564–1616)