# Combined Algebraic Properties of IP* and Central* Sets Near 0

Authors: Dibyendu De; Ram Krishna Paul
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

It is known that for an IP* setA in ℕ and a sequence 〈xn〉n=1∞ there exists a sum subsystem 〈yn〉n=1∞ of 〈xn〉n=1∞ such that FS(〈yn〉n=1∞)∪FP(〈yn〉n=1∞)⊆A. Similar types of results also have been proved for central* sets. In this present work we will extend the results for dense subsemigroups of ((0,∞),+).


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## Body

## 1. Introduction

One of the famous Ramsey theoretic results is Hindman's Theorem.Theorem 1.1.
Given a finite coloringℕ=⋃i=1r‍Ai there exists a sequence 〈xn〉n=1∞ in ℕ and i∈{1,2,…,r} such that
(1.1)FS(〈xn〉n=1∞)={∑n∈F‍xn:F∈Pf(N)}⊆Ai,
where for any set X,𝒫f(X) is the set of finite nonempty subsets of X.The original proof of this theorem was combinatorial in nature. But later using algebraic structure ofβℕ a very elegant proof of this theorem was established in [1, Corollary 5.10]. First we give a brief description of algebraic structure of βSd for a discrete semigroup (S,·).We take the points ofβSd to be the ultrafilters on S, identifying the principal ultrafilters with the points of S and thus pretending that S⊆βSd. Given A⊆S,
(1.2)clA=A¯={p∈βSd:A∈p}
is a basis for the closed sets of βSd. The operation · on S can be extended to the Stone-Čech compactification βSd of S so that (βSd,·) is a compact right topological semigroup (meaning that for any p∈βSd, the function ρp:βSd→βSd defined by ρp(q)=q·p is continuous) with S contained in its topological center (meaning that for any x∈S, the function λx:βSd→βSd defined by λx(q)=x·q is continuous). A nonempty subset I of a semigroup T is called a left ideal ofS if TI⊂I, a right ideal if IT⊂I, and a two-sided ideal (or simply an ideal) if it is both a left and right ideal. A minimal left ideal is the left ideal that does not contain any proper left ideal. Similarly, we can define minimal right ideal and smallest ideal.Any compact Hausdorff right topological semigroupT has a smallest two-sided ideal:
(1.3)K(T)=⋃‍{L:LisaminimalleftidealofT}=⋃‍{R:RisaminimalrightidealofT}.Given a minimal left idealL and a minimal right ideal R, L∩R is a group, and in particular contains an idempotent. An idempotent in K(T) is a minimal idempotent. If p and q are idempotents in T we write p≤q if and only if pq=qp=p. An idempotent is minimal with respect to this relation if and only if it is a member of the smallest ideal.Givenp,q∈βS, and A⊆S, A∈p·q if and only if {x∈S:x-1A∈q}∈p, where x-1A={y∈S:x·y∈A}. See [1] for an elementary introduction to the algebra of βS and for any unfamiliar details.A
⊆
ℕ is called an IP* set if it belongs to every idempotent in βℕ. Given a sequence 〈xn〉n=1∞ in ℕ, we let FP(〈xn〉n=1∞) be the product analogue of Finite Sum. Given a sequence 〈xn〉n=1∞ in ℕ, we say that 〈yn〉n=1∞ is a sum subsystem of 〈xn〉n=1∞ provided there is a sequence 〈Hn〉n=1∞ of nonempty finite subsets of ℕ such that maxHn<minHn+1 and yn=∑t∈Hn‍xt for each n∈ℕ.Theorem 1.2.
Let〈xn〉n=1∞ be a sequence in ℕ and let A be an IP* set in (ℕ,+). Then there exists a sum subsystem 〈yn〉n=1∞ of 〈xn〉n=1∞ such that
(1.4)FS(〈yn〉n=1∞)∪FP(〈yn〉n=1∞)⊆A.Proof.
See [2, Theorem 2.6] or see [1, Corollary 16.21].Definition 1.3.
A subsetC⊆S is called central if and only if there is an idempotent p∈K(βS) such that C∈p.The algebraic structure of the smallest ideal ofβS has played a significant role in Ramsey Theory. It is known that any central subset of (ℕ,+) is guaranteed to have substantial additive structure. But Theorem 16.27 of [1] shows that central sets in (ℕ,+) need not have any multiplicative structure at all. On the other hand, in [2] we see that sets which belong to every minimal idempotent of N, called central* sets, must have significant multiplicative structure. In fact central* sets in any semigroup (S,·) are defined to be those sets which meet every central set.Theorem 1.4.
IfA is a central* set in (ℕ,+) then it is central in (ℕ,·).Proof.
See [2, Theorem 2.4].In case ofcentral* sets a similar result has been proved in  [3] for a restricted class of sequences called minimal sequences, where a sequence 〈xn〉n=1∞ in ℕ is said to be a minimal sequence if
(1.5)⋂m=1∞‍FS(〈xn〉n=m∞)¯∩K(βN)≠∅.Theorem 1.5.
Let〈yn〉n=1∞ be a minimal sequence and let A be a central* set in (ℕ,+). Then there exists a sum subsystem 〈xn〉n=1∞ of 〈yn〉n=1∞ such that
(1.6)FS(〈xn〉n=1∞)∪FP(〈xn〉n=1∞)⊆A.Proof.
See [3, Theorem 2.4].A strongly negative answer to the partition analogue of Hindman's theorem was presented in [4]. Given a sequence 〈xn〉n=1∞ in ℕ, let us denote PS(〈xn〉n=1∞)= {xm+xn:m,n∈ℕ and m≠n} and PP(〈xn〉n=1∞)={xm·xn:m,n∈ℕ and m≠n}.Theorem 1.6.
There exists a finite partitionℛ of ℕ with no one-to-one sequence 〈xn〉n=1∞ in ℕ such that PS(〈xn〉n=1∞)∪PP(〈xn〉n=1∞) is contained in one cell of the partition ℛ.Proof.
See [4, Theorem 2.11].A similar result in this direction in the case of dyadic rational numbers has been proved by V. Bergelson et al..Theorem 1.7.
There exists a finite partition𝔻∖{0}=⋃i=1r‍Ai such that there do not exist i∈{1,2,…,r} and a sequence 〈xn〉n=1∞ with
(1.7)FS(〈xn〉n=1∞)∪PP(〈xn〉n=1∞)⊆Ai.Proof.
See [5, Theorem 5.9].In [5], the authors also presented the following conjecture and question.Conjecture 1.8.
There exists a finite partitionℚ∖{0}=⋃i=1r‍Ai such that there do not exists i∈{1,2,…,r} and a sequence 〈xn〉n=1∞ with
(1.8)FS(〈xn〉n=1∞)∪FP(〈xn〉n=1∞)⊆Ai.Question 1.
Does there exist a finite partitionℝ∖{0}=⋃i=1r‍Ai such that there do not exist i∈{1,2,…,r} and a sequence 〈xn〉n=1∞ with
(1.9)FS(〈xn〉n=1∞)∪FP(〈xn〉n=1∞)⊆Ai?In the present paper our aim is to extend Theorems1.2 and 1.5 for dense subsemigroups of ((0,∞),+).Definition 1.9.
IfS is a dense subsemigroup of ((0,∞),+) one defines 0+(S)={p∈βSd:(forallϵ>0)((0,ϵ)∈p)}.It is proved in [6], that 0+(S) is a compact right topological subsemigroup of (βSd,+) which is disjoint from K(βSd) and hence gives some new information which are not available from K(βSd). Being compact right topological semigroup 0+(S) contains minimal idempotents of 0+(S). A subset A of S is said to be IP*-set near 0 if it belongs to every idempotent of 0+(S) and a subset C of S is said to be central* set near 0 if it belongs to every minimal idempotent of 0+(S). In [7] the authors applied the algebraic structure of 0+(S) on their investigation of image partition regularity near 0 of finite and infinite matrices. Article [8] used algebraic structure of 0+(ℝ) to investigate image partition regularity of matrices with real entries from ℝ.


## 2. IP* and Central* Set Near 0

In the following discussion, we will extend Theorem1.2 for a dense subsemigroup of ((0,∞),+) in the appropriate context.Definition 2.1.
LetS be a dense subsemigroup of ((0,∞),+). A subset A of S is said to be an IP set near 0 if there exists a sequence 〈xn〉n=1∞ such that ∑n=1∞‍xn converges and such that FS(〈xn〉n=1∞)⊆A. One calls a subset D of S an IP*set near 0 if for every subset C of S which is IP set near 0, C∩D is IP set near 0.From [6, Theorem 3.1] it follows that for a dense subsemigroup S of ((0,∞),+) a subset A of S is an IP set near 0 if and only if there exists some idempotent p∈0+(S) with A∈p. Further it can be easily observed that a subset D of S is an IP* set near 0 if and only if it belongs to every idempotent of 0+(S).Givenc∈ℝ∖{0} and p∈βℝd∖{0}, the product c·p is defined in (βℝd,·). One has A⊆ℝ is a member of c·p if and only if c-1A={x∈ℝ:c·x∈A} is a member of p.Lemma 2.2.
LetS be a dense subsemigroup of ((0,∞),+) such that S∩(0,1) is a subsemigroup of ((0,1),·). If A is an IP set near 0 in S then sA is also an IP set near 0 for every s∈S∩(0,1). Further if A is a an IP* set near 0 in (S,+) then s-1A is also an IP* set near 0 for every s∈S∩(0,1).Proof.
SinceA is an IP set near 0 then by [6, Theorem 3.1] there exists a sequence 〈xn〉n=1∞ in S with the property that ∑n=1∞‍xn converges and FS(〈xn〉n=1∞)⊆A. This implies that ∑n=1∞‍(s·xn) is also convergent and FS(〈sxn〉n=1∞)⊆sA. This proves that sA is also an IP* set near 0.
For the second letA be a an IP* set near 0 and s∈S∩(0,1). To prove that s-1A is a an IP* set near 0 it is sufficient to show that if B is any IP set near 0 then B∩s-1A≠∅. Since B is an IP set near 0, sB is also an IP set near 0 by the first part of the proof, so that A∩sB≠∅. Choose t∈sB∩A and k∈B such that t=sk. Therefore k∈s-1A so that B∩s-1A≠∅.GivenA⊆S and s∈S, s-1A={t∈S: st∈A}, and -s+A={t∈S: s+t∈A}.Theorem 2.3.
LetS be a dense subsemigroup of ((0,∞),+) such that S∩(0,1) is a subsemigroup of ((0,1),·). Also let 〈xn〉n=1∞ be a sequence in S such that ∑n=1∞‍xn converges and let A be a IP* set near 0 in S. Then there exists a sum subsystem 〈yn〉n=1∞ of 〈xn〉n=1∞ such that
(2.1)FS(〈yn〉n=1∞)∪FP(〈yn〉n=1∞)⊆A.Proof.
Since∑n=1∞‍xn converges, from [6, Theorem 3.1] it follows that we can find some idempotent p∈0+(S) for which FS(〈xn〉n=1∞)∈p. In fact T=⋂m=1∞‍cℓβSdFS(〈yn〉n=m∞)⊆0+(S) and p∈T. Again, since A is a IP* set near 0 in S, by Lemma 2.2 for every s∈S∩(0,1), s-1A∈p. Let A⋆={s∈A: -s+A∈p}. Then by  [1, Lemma 4.14] A⋆∈p. We can choose y1∈A⋆∩FS(〈xn〉n=1∞). Inductively let m∈ℕ and 〈yi〉i=1m, 〈Hi〉i=1m in 𝒫f(ℕ) be chosen with the following properties:(1)
i∈{1,2,…,m-1}maxHi<minHi+1;(2)
ifyi=∑t∈Hi‍xt then ∑t∈Hm‍xt∈A⋆ and FP(〈yi〉i=1m)⊆A.
We observe that{∑t∈H‍xt : H∈𝒫f(ℕ),minH>maxHm}∈p. Let B={∑t∈H‍xt : H∈𝒫f(ℕ),minH>maxHm}, let E1=FS(〈yi〉i=1m) and E2=FP(〈yi〉i=1m). Now consider
(2.2)D=B∩A⋆∩⋂s∈E1‍(-s+A⋆)∩⋂s∈E2‍(s-1A⋆).
Then D∈p. Now choose ym+1∈D and Hm+1∈𝒫f(ℕ) such that minHm+1>maxHm. Putting ym+1=∑t∈Hm+1‍xt shows that the induction can be continued and proves the theorem.If we turn our attention tocentral* sets then the above result holds for a restricted class of sequences which we call minimal sequence near 0.Definition 2.4.
LetS be a dense subsemigroup of ((0,∞),+). A sequence 〈xn〉n=1∞ in S is said to be a minimal sequence near 0 if
(2.3)⋂m=1∞‍FS(〈xn〉n=m∞)¯∩K(0+(S))≠∅.The notion of piecewise syndetic set near 0 was first introduced in [6].Definition 2.5.
For a dense subsemigroupS of ((0,∞),+), a subset A of S is piecewise syndetic near 0 if and only if cℓβSdA∩K(0+(S))≠∅.The following theorem characterizes minimal sequences near 0 in terms of piecewise syndetic set near 0.Theorem 2.6.
LetS be a dense subsemigroup of ((0,∞),+). Then the following conditions are equivalent:  (a)
〈xn〉n=1∞ is a minimal sequence near 0.(b)
FS(〈xn〉n=1∞) is piecewise syndetic near 0.(c)
There is an idempotent in⋂m=1∞‍FS(〈xn〉n=m∞)¯∩K(0+(S))≠∅.Proof.
(
a
)
⇒
(
b
) follows from  (see [6, Theorem 3.5]).
To prove that (b) implies (a) let us consider thatFS(〈xn〉n=1∞) be a piecewise syndetic near 0. Then there exists a minimal left ideal L of 0+(S) such that L∩FS(〈xn〉n=1∞)¯≠∅. We choose q∈L∩FS(〈xn〉n=1∞)¯. By [6, Theorem 3.1], ⋂m=1∞‍cℓβSdFS(〈xn〉n=m∞) is a subsemigroup of 0+(S), so it suffices to show that for each m∈ℕ, L∩FS(〈xn〉n=m∞)¯≠∅. In fact minimal left ideals being closed, we can conclude that L∩⋂n=m∞‍FS(〈xn〉n=m∞)¯≠∅ and so L∩⋂n=m∞‍FS(〈xn〉n=m∞)¯ is a compact right topological semigroup so that it contains idempotents. To this end, let m∈ℕ with m>1. Then FS(〈xn〉n=1∞)=FS(〈xn〉n=m∞)∪FS(〈xn〉n=1m-1)∪⋃‍{t+FS(〈xn〉n=m∞):t∈FS(〈xn〉n=1m-1)}. So we must have one of the following: (i)
FS(〈xn〉n=m∞)∈q,(ii)
FS(〈xn〉n=1m-1)∈q,(iii)
t+FS(〈xn〉n=m∞)∈q for some t∈FS(〈xn〉n=1m-1).
Clearly (ii) does not hold, because in that caseq becomes a member of S while it is a member of minimal left ideal. If (iii) holds then we have t+FS(〈xn〉n=m∞)∈q for some t∈FS(〈xn〉n=1m-1). Since q∈0+(S), we have (0,t)∩S∈q. But (0,t)∩(t+FS(〈xn〉n=m∞))=∅, a contradiction. Hence (i) must hold so that q∈L∩FS(〈xn〉n=m∞)¯.
(
a
)
⇔
(
c
) is obvious.Let us recall following lemma for our purpose.Lemma 2.7.
LetS be a dense subsemigroup of ((0,∞),+) such that S∩(0,1) is a subsemigroup of ((0,1),·) and assume that for each y∈S∩(0,1) and each x∈S, x/y∈S and yx∈S. If A⊆S and y-1A is a central set near 0, then A is also a central set near 0.Proof.
See [6, Lemma 4.8].Lemma 2.8.
LetS be a dense subsemigroup of ((0,∞),+) such that S∩(0,1) is a subsemigroup of ((0,1),·) and assume that for each s∈S∩(0,1) and each t∈S, t/s∈S and st∈S. If A is central set near 0 in S then sA is also central set near 0.Proof.
Sinces-1(sA)=A and A is central set near 0 then by Lemma 2.7, sA is central set near 0.Lemma 2.9.
LetS be a dense subsemigroup of ((0,∞),+) such that S∩(0,1) is a subsemigroup of ((0,1),·) and assume that for each s∈S∩(0,1) and each t∈S, t/s∈S and st∈S. If A is a central* set near 0 in (S,+) then s-1A is also central* set near 0.Proof.
LetA be a central* set near 0 and s∈S∩(0,1). To prove that s-1A is a central* set near 0 it is sufficient to show that for any central set near 0 C, C∩s-1A≠∅. Since C is central set near 0, sC is also central set near 0 so that A∩sC≠∅. Choose t∈sC∩A and k∈C such that t=sk. Therefore k∈s-1A so that C∩s-1A≠∅.We end this paper by following generalization of Theorem2.3, whose proof is also straight forward generalization of Theorem 2.3 and hence omitted.Theorem 2.10.
LetS be a dense subsemigroup of ((0,∞),+) such that S∩(0,1) is a subsemigroup of ((0,1),·) and assume that for each s∈S∩(0,1) and each t∈S, t/s∈S and st∈S. Also let 〈xn〉n=1∞ be a minimal sequence near 0 and let A be a central* set near 0 in S. Then there exists a sum subsystem 〈yn〉n=1∞ of 〈xn〉n=1∞ such that
(2.4)FS(〈yn〉n=1∞)∪FP(〈yn〉n=1∞)⊆A.



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