View source for Fundamental unit (number theory)

In [[mathematics]], an '''IP set''' is a set of [[natural number]]s 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 [[non-empty set|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]]). 

== Hindman's Theorem ==
If <math>S\,</math> is an IP set and <math>S = C_1 \cup C_2 \cup ... \cup C_n</math>, then at least one <math>C_i\,</math> is an IP set. 
This is known as Hindman's Theorem, or the Finite Sums Theorem.

Since the set of natural numbers itself is an IP-set and partitions can also be seen as colorings, 
we 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''.

Hindman's Theorem states that the class of IP sets is [[partition regular]].

== Semigroups ==

The definition of being IP has been extended from subsets of the special semigroup of natural numbers with addition to subsets of [[semigroup]]s and partial semigroups in general.

== See also ==
* [[Ergodic Ramsey theory]]
* [[Piecewise syndetic set]]
* [[Syndetic set]]
* [[Thick set]]

== References ==
{{Citation style|date=September 2009}}
* [[Vitaly Bergelson]], I. J. H. Knutson, R. McCutcheon "[http://home.hia.no/~ingerjh/forskning/bhm2jun03.pdf Simultaneous diophantine approximation and VIP Systems]" ''Acta Arith.'' '''116''', Academia Scientiarum Polona, (2005), 13-23
* [[Vitaly Bergelson]], "[http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf 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= |coauthors=  Hindman, N. |date= |year= 2001|month= |title= Partition regular structures contained in large sets are abundant|journal= J. Comb. Theory (Series 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 [http://mysite.verizon.net/nhindman/research/large.pdf publicly available copy] is hosted by one of the authors.
* [[H. Furstenberg]], B. Weiss, "Topological Dynamics and Combinatorial Number Theory", ''J. d'Analyse Math.'' '''34''' (1978), pp.&nbsp;61-85
* J. McLeod, "[http://www.mtholyoke.edu/%7Ejmcleod/somenotionsofsize.pdf Some Notions of Size in Partial Semigroups]", ''Topology Proceedings'', Vol. '''25''' (2000), pp.&nbsp;317-332

{{DEFAULTSORT:Ip Set}}
[[Category:Semigroup theory]]
[[Category:Ergodic theory]]
[[Category:Ramsey theory]]