| dbo:abstract |
In der Mathematik bezeichnet der Begriff IP-Menge eine Menge natürlicher Zahlen, die alle endlichen Summen einer unendlichen Menge von natürlichen Zahlen enthält. Die Bezeichnung IP-Menge (IP-set) geht auf Hillel Fürstenberg und zurück; IP steht dabei für „Infinite-dimensional Parallelepiped“. (de) 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). Slightly more generally, for a sequence of natural numbers (ni), one can consider the set of finite sums FS((ni)), consisting of the sums of all finite length subsequences of (ni). 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. Equivalently, one may require that A contains all finite sums FS((ni)) of a sequence (ni). Some authors give a slightly different definition of IP sets: They require that FS(D) equal A instead of just being a subset. The term IP set was coined by Hillel Furstenberg and Benjamin Weiss to abbreviate "infinite-dimensional parallelepiped". Serendipitously, the abbreviation IP can also be expanded to "idempotent" (a set is IP if and only if it is a member of an idempotent ultrafilter). (en) |
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http://home.hia.no/~ingerjh/forskning/bhm2jun03.pdf http://nhindman.us/research/large.pdf https://web.archive.org/web/20061117184558/http:/www.mtholyoke.edu/~jmcleod/somenotionsofsize.pdf http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf |
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| rdfs:comment |
In der Mathematik bezeichnet der Begriff IP-Menge eine Menge natürlicher Zahlen, die alle endlichen Summen einer unendlichen Menge von natürlichen Zahlen enthält. Die Bezeichnung IP-Menge (IP-set) geht auf Hillel Fürstenberg und zurück; IP steht dabei für „Infinite-dimensional Parallelepiped“. (de) 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). Slightly more generally, for a sequence of natural numbers (ni), one can consider the set of finite sums FS((ni)), consisting of the sums of all finite length subsequences of (ni). (en) |
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IP-Menge (de) IP set (en) |
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