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词条 IP set
释义

  1. Hindman's theorem

  2. Semigroups

  3. See also

  4. References

{{Citation style|date=September 2009}}

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 Furstenberg and Weiss[1] to abbreviate "infinite-dimensional parallelepiped". Serendipitously, the abbreviation IP can also be expanded to "idempotent"[2] (a set is IP if and only if it is a member of an idempotent ultrafilter).

Hindman's theorem

If is an IP set and , then at least one contains an IP set.

This is known as Hindman's theorem or the finite sums theorem.[3][4] In different terms, Hindman's theorem states that the class of IP sets is partition regular.

Since the set of natural numbers itself is an IP set and partitions can also be seen as colorings, one can reformulate a special case of Hindman's theorem in more familiar terms: Suppose the natural numbers are "colored" with n different colors; each natural number gets one and only one of the n colors. Then there exists a color c and an infinite set D of natural numbers, all colored with c, such that every finite sum over D also has color c.

The Milliken–Taylor theorem is a common generalisation of Hindman's theorem and Ramsey's theorem.

Semigroups

The definition of being IP has been extended from subsets of the special semigroup of natural numbers with addition to subsets of semigroups and partial semigroups in general. A variant of Hindman's theorem is true for arbitrary semigroups.[5][6]

See also

  • Ergodic Ramsey theory
  • Piecewise syndetic set
  • Syndetic set
  • Thick set

References

1. ^{{Cite book|url=https://www.worldcat.org/oclc/889248822|title=Recurrence in ergodic theory and combinatorial number theory|last=Harry,|first=Furstenberg,|isbn=9780691615363|location=Princeton, New Jersey|oclc=889248822}}
2. ^{{Cite journal|last=Bergelson|first=V.|last2=Leibman|first2=A.|date=2016|title=Sets of large values of correlation functions for polynomial cubic configurations|url=https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/sets-of-large-values-of-correlation-functions-for-polynomial-cubic-configurations/65D3C9FC85AFFBECE602FC7BC41A5414|journal=Ergodic Theory and Dynamical Systems|pages=1–24|doi=10.1017/etds.2016.49|issn=0143-3857}}
3. ^{{Cite journal|last=Hindman|first=Neil|title=Finite sums from sequences within cells of a partition of N|url=https://doi.org/10.1016/0097-3165(74)90023-5|journal=Journal of Combinatorial Theory, Series A|volume=17|issue=1|pages=1–11|doi=10.1016/0097-3165(74)90023-5}}
4. ^{{Cite journal|last=Baumgartner|first=James E|title=A short proof of Hindman's theorem|url=https://doi.org/10.1016/0097-3165(74)90103-4|journal=Journal of Combinatorial Theory, Series A|volume=17|issue=3|pages=384–386|doi=10.1016/0097-3165(74)90103-4}}
5. ^{{Cite journal|last=Golan|first=Gili|last2=Tsaban|first2=Boaz|title=Hindmanʼs coloring theorem in arbitrary semigroups|url=http://linkinghub.elsevier.com/retrieve/pii/S0021869313004419|journal=Journal of Algebra|volume=395|pages=111–120|doi=10.1016/j.jalgebra.2013.08.007|arxiv=1303.3600}}
6. ^{{Cite book|url=https://www.worldcat.org/oclc/39368501|title=Algebra in the Stone-Čech compactification : theory and applications|last=Hindman|first=Neil|last2=Strauss|first2=Dona|date=1998|publisher=Walter de Gruyter|others=|year=|isbn=311015420X|location=New York|pages=|oclc=39368501}}
  • Vitaly Bergelson, I. J. H. Knutson, R. McCutcheon "Simultaneous diophantine approximation and VIP Systems" Acta Arith. 116, Academia Scientiarum Polona, (2005), 13-23
  • Vitaly Bergelson, "Minimal Idempotents and Ergodic Ramsey Theory" Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310, Cambridge Univ. Press, Cambridge, (2003)
  • {{cite journal |quotes= |last= |first= |author= Bergelson, V. |authorlink= |author2=Hindman, N. |year= 2001|title= Partition regular structures contained in large sets are abundant|journal= J. Comb. Theory A|volume=93 |issue= |page= |pages=18–36 |publisher= |location= |pmid= |pmc= |doi=10.1006/jcta.2000.3061 |bibcode= |oclc= |id= |url= |language= |format= |accessdate= |laysummary= |laysource= |laydate= |quote=}} A publicly available copy is hosted by one of the authors.
  • H. Furstenberg, B. Weiss, "Topological Dynamics and Combinatorial Number Theory", J. Anal. Math. 34 (1978), pp. 61–85
  • J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (2000), pp. 317–332
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3 : Semigroup theory|Ergodic theory|Ramsey theory
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