IP set

In mathematics, the term IP - quantity denotes a set of natural numbers which contains all finite sums of an infinite set.

The finite sums of a set D of natural numbers are the numbers that can be represented as a sum of elements of a non-empty finite subset of D. The set of all finite sums of D is also referred to as FS (D); while FS stands for Finite Sums.

Sometimes a slightly different definition is used: one then that even A = FS ( D) for a suitable D demands ..

The term IP quantity ( IP set) goes back to Hillel Furstenberg and Barak Weiss; IP stands for " Infinite - dimensional parallelepiped ."

The set of Hindman

The set of Hindman, or the Finite Sums Theorem is as follows:

Since the set of natural numbers is also an IP quantity yourself and you can interpret partitions as colorings, can be the following special case of the theorem of Hindman formulate:

Semigroups

The IP property can not only for the natural numbers, which with the addition of a semigroup, define, but also more generally for semigroups and partial semigroups.

Swell

  • V. Bergelson, IJH Knutson, R. McCutcheon: Simultaneous diophantine approximation and VIP system (PDF, 127 kB) Acta Arith. 116, Academia Scientiarum Polona, (2005), 13-23
  • V. Bergelson: Minimal Idempotents and Ergodic Ramsey Theory ( PDF, 349 kB) Topics in Dynamics and Ergodic Theory 8-39, London Math Soc. Lecture Note Series 310, Cambridge Univ. Press, Cambridge, (2003)
  • H. Furstenberg, B. Weiss: Topological Dynamics and Combinatorial Number Theory, J. d'Analyse Math 34 (1978), 61-85
  • Group Theory
  • Combinatorics
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