Abstract

It is known that for an IP* set 𝐴 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,∞),+).

1. Introduction

One of the famous Ramsey theoretic results is Hindman's Theorem.

Theorem 1.1. Given a finite coloring ⋃ℕ=𝑟𝑖=1𝐴𝑖 there exists a sequence ⟨𝑥𝑛⟩∞𝑛=1 in ℕ and 𝑖∈{1,2,…,𝑟} such that 𝐹𝑆⟨𝑥𝑛⟩∞𝑛=1=𝑛∈𝐹𝑥𝑛∶𝐹∈𝒫𝑓(ℕ)⊆𝐴𝑖,(1.1) 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 of 𝛽ℕ 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 of 𝛽𝑆𝑑 to be the ultrafilters on 𝑆, identifying the principal ultrafilters with the points of 𝑆 and thus pretending that 𝑆⊆𝛽𝑆𝑑. Given 𝐴⊆𝑆, 𝑐ℓ𝐴=𝐴=𝑝∈𝛽𝑆𝑑∶𝐴∈𝑝(1.2) 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 of 𝑆 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 semigroup 𝑇 has a smallest two-sided ideal: =𝐾(𝑇)={𝐿∶𝐿isaminimalleftidealof𝑇}{𝑅∶𝑅isaminimalrightidealof𝑇}.(1.3)

Given a minimal left ideal 𝐿 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.

Given 𝑝,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 𝐹𝑆⟨𝑦𝑛⟩∞𝑛=1∪𝐹𝑃⟨𝑦𝑛⟩∞𝑛=1⊆𝐴.(1.4)

Proof. See [2, Theorem 2.6] or see [1, Corollary 16.21].

Definition 1.3. A subset 𝐶⊆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 of 𝛽𝑆 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. If 𝐴 is a central∗ set in (ℕ,+) then it is central in (ℕ,⋅).

Proof. See [2, Theorem 2.4].

In case of central∗ 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 ∞𝑚=1𝐹𝑆⟨𝑥𝑛⟩∞𝑛=𝑚∩𝐾(𝛽ℕ)≠∅.(1.5)

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 𝐹𝑆⟨𝑥𝑛⟩∞𝑛=1∪𝐹𝑃⟨𝑥𝑛⟩∞𝑛=1⊆𝐴.(1.6)

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 partition ℛ 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 partition ⋃𝔻⧵{0}=𝑟𝑖=1𝐴𝑖 such that there do not exist 𝑖∈{1,2,…,𝑟} and a sequence ⟨𝑥𝑛⟩∞𝑛=1 with 𝐹𝑆⟨𝑥𝑛⟩∞𝑛=1∪𝑃𝑃⟨𝑥𝑛⟩∞𝑛=1⊆𝐴𝑖.(1.7)

Proof. See [5, Theorem 5.9].

In [5], the authors also presented the following conjecture and question.

Conjecture 1.8. There exists a finite partition ⋃ℚ⧵{0}=𝑟𝑖=1𝐴𝑖 such that there do not exists 𝑖∈{1,2,…,𝑟} and a sequence ⟨𝑥𝑛⟩∞𝑛=1 with 𝐹𝑆⟨𝑥𝑛⟩∞𝑛=1∪𝐹𝑃⟨𝑥𝑛⟩∞𝑛=1⊆𝐴𝑖.(1.8)

Question 1. Does there exist a finite partition ⋃ℝ⧵{0}=𝑟𝑖=1𝐴𝑖 such that there do not exist 𝑖∈{1,2,…,𝑟} and a sequence ⟨𝑥𝑛⟩∞𝑛=1 with 𝐹𝑆⟨𝑥𝑛⟩∞𝑛=1∪𝐹𝑃⟨𝑥𝑛⟩∞𝑛=1⊆𝐴𝑖?(1.9)

In the present paper our aim is to extend Theorems 1.2 and 1.5 for dense subsemigroups of ((0,∞),+).

Definition 1.9. If 𝑆 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 Theorem 1.2 for a dense subsemigroup of ((0,∞),+) in the appropriate context.

Definition 2.1. Let 𝑆 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=1𝑥𝑛 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).

Given 𝑐∈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. Let 𝑆 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. Since 𝐴 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=1𝑥𝑛 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 let 𝐴 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≠∅.

Given 𝐴⊆𝑆 and 𝑠∈𝑆, 𝑠−1𝐴={𝑡∈𝑆: 𝑠𝑡∈𝐴}, and −𝑠+𝐴={𝑡∈𝑆: 𝑠+𝑡∈𝐴}.

Theorem 2.3. Let 𝑆 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=1𝑥𝑛 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 𝐹𝑆⟨𝑦𝑛⟩∞𝑛=1∪𝐹𝑃⟨𝑦𝑛⟩∞𝑛=1⊆𝐴.(2.1)

Proof. Since ∑∞𝑛=1𝑥𝑛 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=1𝑐ℓ𝛽𝑆𝑑𝐹𝑆(⟨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⟩𝑚𝑖=1, ⟨𝐻𝑖⟩𝑚𝑖=1 in 𝒫𝑓(ℕ) be chosen with the following properties:(1)𝑖∈{1,2,…,𝑚−1}max𝐻𝑖<min𝐻𝑖+1; (2)if 𝑦𝑖=∑𝑡∈𝐻𝑖𝑥𝑡 then ∑𝑡∈𝐻𝑚𝑥𝑡∈𝐴⋆ and 𝐹𝑃(⟨𝑦𝑖⟩𝑚𝑖=1)⊆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⟩𝑚𝑖=1) and 𝐸2=𝐹𝑃(⟨𝑦𝑖⟩𝑚𝑖=1). Now consider 𝐷=𝐵∩𝐴⋆∩𝑠∈𝐸1−𝑠+𝐴⋆∩𝑠∈𝐸2𝑠−1𝐴⋆.(2.2) 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 to central∗ sets then the above result holds for a restricted class of sequences which we call minimal sequence near 0.

Definition 2.4. Let 𝑆 be a dense subsemigroup of ((0,∞),+). A sequence ⟨𝑥𝑛⟩∞𝑛=1 in 𝑆 is said to be a minimal sequence near 0 if ∞𝑚=1𝐹𝑆⟨𝑥𝑛⟩∞𝑛=𝑚0∩𝐾+(𝑆)≠∅.(2.3)

The notion of piecewise syndetic set near 0 was first introduced in [6].

Definition 2.5. For a dense subsemigroup 𝑆 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. Let 𝑆 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=1𝐹𝑆(⟨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 that 𝐹𝑆(⟨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=1𝑐ℓ𝛽𝑆𝑑𝐹𝑆(⟨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=𝑚𝐹𝑆(⟨𝑥𝑛⟩∞𝑛=𝑚)≠∅ and so ⋂𝐿∩∞𝑛=𝑚𝐹𝑆(⟨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⟩𝑚−1𝑛=1⋃)∪{𝑡+𝐹𝑆(⟨𝑥𝑛⟩∞𝑛=𝑚)∶𝑡∈𝐹𝑆(⟨𝑥𝑛⟩𝑚−1𝑛=1)}. So we must have one of the following: (i)𝐹𝑆(⟨𝑥𝑛⟩∞𝑛=𝑚)∈𝑞, (ii)𝐹𝑆(⟨𝑥𝑛⟩𝑚−1𝑛=1)∈𝑞, (iii)𝑡+𝐹𝑆(⟨𝑥𝑛⟩∞𝑛=𝑚)∈𝑞 for some 𝑡∈𝐹𝑆(⟨𝑥𝑛⟩𝑚−1𝑛=1).
Clearly (ii) does not hold, because in that case 𝑞 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⟩𝑚−1𝑛=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. Let 𝑆 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. Let 𝑆 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. Since 𝑠−1(𝑠𝐴)=𝐴 and 𝐴 is central set near 0 then by Lemma 2.7, 𝑠𝐴 is central set near 0.

Lemma 2.9. Let 𝑆 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. Let 𝐴 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 Theorem 2.3, whose proof is also straight forward generalization of Theorem 2.3 and hence omitted.

Theorem 2.10. Let 𝑆 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 𝐹𝑆⟨𝑦𝑛⟩∞𝑛=1∪𝐹𝑃⟨𝑦𝑛⟩∞𝑛=1⊆𝐴.(2.4)

Acknowledgments

The authors would like to acknowledge the referee for her/his constructive report. The first named author is partially supported by DST-PURSE programme. The work of this paper was a part of second named authors Ph.D. dissertation which was supported by a CSIR Research Fellowship.