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Central Sets Theorem near zero
Topology and its Applications 210 (2016) 70–80
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Topology and its Applications www.elsevier.com/locate/topol
Central Sets Theorem near zero E. Bayatmanesh, M. Akbari Tootkaboni ∗ Department of Mathematics, Faculty of Basic Science, Shahed University, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 29 November 2015 Received in revised form 21 June 2016 Accepted 23 June 2016 Available online 21 July 2016
a b s t r a c t In this paper, we introduce notions of J-set near zero and C-set near zero for a dense subsemigroup of ((0, +∞), +) and state the Central Sets Theorem near zero. Among the other results for a dense subsemigroup S ⊆ ((0, +∞), +), we give some sufficient and equivalent algebraic conditions on a subset A ⊂ S to be J-set near zero and to be C-set near zero. © 2016 Elsevier B.V. All rights reserved.
MSC: primary 54D35, 22A15 secondary 05D10, 54D80 Keywords: Central Sets Theorem The Stone–Čech compactification C-set J-set Piecewise syndetic set near zero
1. Introduction Let (S, +) be a discrete semigroup. The collection of all ultrafilters on S is called the Stone–Čech compactification of S and denoted by βS. For A ⊆ S, define A = {p ∈ βS : A ∈ p}, then {A : A ⊆ S} is a basis for the open sets (also for the closed sets) of βS. We identify the principal ultrafilters with the points of S and thus pretend that S ⊆ βS. There is a unique extension of the operation to βS, making (βS, +) a right topological semigroup (i.e. for each p ∈ βS, the right translation ρp is continuous, where ρp (q) = q + p) and also for each x ∈ S, the left translation λx is continuous, where λx (q) = x + q. The principal ultrafilters identified by the points of S and S is a dense subset of βS. For p, q ∈ βS and A ⊆ S, we have A ∈ p + q if and only if {x ∈ S : −x + A ∈ q} ∈ p, where −x + A = {y ∈ S : x + y ∈ A}. A nonempty subset L of a semigroup (S, +) is called a left ideal of S if S + L ⊆ L, a right ideal if L + S ⊆ L, and a two sided ideal (or simply an ideal) if it is both a left and a right ideal. A minimal left ideal is a left ideal that does not contain any proper left ideal. In the same way, we can define minimal right ideal and smallest ideal. * Corresponding author. E-mail addresses: [email protected] (E. Bayatmanesh), [email protected] (M. Akbari Tootkaboni). http://dx.doi.org/10.1016/j.topol.2016.06.014 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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Any compact Hausdorff right topological semigroup (S, +) has a smallest two sided ideal, denoted by K(S), which is the union of all minimal left ideals, and also the union of all minimal right ideals, as well. 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(S) is called a minimal idempotent. For more details see [8]. For A ⊆ S and p ∈ βS, we define A∗ (p) = {s ∈ A : −s + A ∈ p}. Lemma 1.1. Let (S, +) be a semigroup, p + p = p ∈ βS, and let A ∈ p. Then for each s ∈ A∗ (p), −s + A∗ (p) ∈ p. Proof. See Lemma 4.14 in [8].
2
Now we review the definition of partition regularity. In this paper, the collection of all nonempty finite subsets of S is denoted by Pf (S) and P(S) is the set of all subsets of S. Definition 1.1. Let R be a nonempty set of subsets of S. R is partition regular if and only if whenever F is a finite subset of P(S) and F ∈ R, there exist A ∈ F and B ∈ R, such that B ⊆ A. / R. Let Theorem 1.2. Let R ⊆ P(S) be a nonempty set and assume ∅ ∈ R↑ = {B ∈ P(S) : A ⊆ B for some A ∈ R}. Then (a), (b) and (c) are equivalent. (a) R is partition regular. (b) Whenever A ⊆ P(S) has the property that every finite nonempty subfamily of A has an intersection which is in R↑ , there is U ∈ βS, such that A ⊆ U ⊆ R↑ . (c) Whenever A ∈ R, there is U ∈ βS such that A ∈ U ⊆ R↑ . Proof. [8, Theorem 3.11]. 2 Definition 1.3. Let (S, .) be a discrete semigroup and A ⊆ S. Then A is a central set if and only if there exists an idempotent p ∈ K(βS) with A ∈ p. We have been considering semigroups which are dense in ((0, ∞), +) with the natural topology. When discussing the Stone–Čech compactification of such a semigroup S, we will deal with Sd , which is the set S with the discrete topology. Definition 1.4. Let S be a dense subset of ((0, ∞), +). Then 0+ (S) = {p ∈ βSd : (∀ > 0) (0, ) ∩ S ∈ p}. By Lemma 2.5 in [7], 0+ (S) is a compact right topological subsemigroup of (βSd , +), and 0+ (S) ∩ K(βSd ) = ∅. Since 0+ (S) is a compact right topological semigroup, so 0+ (S) contains idempotents. The set 0+ (S) of all non-principal ultrafilters on S = ((0, ∞), +) that are convergent to 0 is a semigroup under the restriction of the usual ‘+’ on βSd , the Stone–Čech compactification of the discrete semigroup S = ((0, ∞), +), see [7]. In [2], the authors used the algebraic structure of 0+ (S) in their investigation of image partition regularity near 0 of finite and infinite matrices. In [5], the algebraic structure of 0+ (R) was used to investigate image partition regularity of matrices with real entries from R. Central sets near zero were introduced by N. Hindman and I. Leader in [7] as
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central sets. Central sets near zero, too, enjoy rich combinatorial structure. Central sets are ideal objects for Ramsey theory. The central sets theorem was first introduced by H. Furstenberg (see [6]) for the semigroups N and Z. The most general version of Central Sets Theorem is available in [1], [3] and [9]. In section 3, we will introduce the Central Sets Theorem near zero and define C-set near zero, J-set near zero and derive some results for dense subsemigroups of ((0, ∞), +). 2. Preliminary Central subsets of a discrete semigroup S have very strong combinatorial properties, which are consequences of the Central Sets Theorem. There is an elementary description of central sets, which was given in [8]. In this paper, the collection of all functions f : N → S is denoted by T =N S. Also, A ⊂ B means that A ⊆ B and A = B. Theorem 2.1 (Central Sets Theorem [1]). Let (S, +) be a commutative semigroup. Let A be a central subset of S. There exist functions α : Pf (T ) → S and H : Pf (T ) → Pf (N), such that (1) if F, G ∈ Pf (T ) and F ⊂ G, then maxH(F ) < minH(G), and (2) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m α(Gi ) + i=1
fi (t) ∈ A.
t∈H(Gi )
We define a set to be a C-set if and only if it satisfies the conclusion of the Central Sets Theorem. Definition 2.2. Let S be a commutative semigroup, A ⊆ S, and T =N S. The set A is a C-set if and only if there exist functions α : Pf (T ) → S and H : Pf (T ) → Pf (N), such that (1) if F, G ∈ Pf (T ) and F ⊂ G, then maxH(F ) < minH(G), and (2) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m i=1
α(Gi ) +
fi (t) ∈ A.
t∈H(Gi )
C-sets are the important objects in combinatorial theory. In many semigroups, they are the objects that behavior like central sets, see [10]. In [1], the authors obtained a simple characterization of C-sets in an arbitrary discrete semigroup. Definition 2.3. Let (S, +) be a commutative semigroup, T =N S, and A ⊆ S is a J-set if and only if whenever F ∈ Pf (T ), there exist a ∈ S and H ∈ Pf (N), such that for each f ∈ F , a + t∈H f (t) ∈ A. Now we recall the notions of syndetic near zero and piecewise syndetic near zero in [8]. Definition 2.4. A subset B of (0, 1) is syndetic near zero if and only if for every > 0 there exist some F ∈ Pf (0, ) and some δ > 0 such that (0, δ) ⊆ t∈F (−t + B). Definition 2.5. a) A subset A of (0, 1) is piecewise syndetic near zero if and only if there exist sequences ∞ {Fn }∞ n=1 and {δn }n=1 such that
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1 1 (1) for each n ∈ N, Fn ∈ Pf ((0, )) and δn ∈ (0, ), and n n 1 (2) for all G ∈ Pf ((0, )) and all μ > 0 there is some x ∈ (0, μ) such that for all n ∈ N, (G ∩ (0, δn )) + x ⊆ n t∈Fn (−t + A). A is a central set near zero if and only if there exists an idempotent p in the smallest ideal of 0+ (S) with A ∈ p. In this paper, the minimal ideal in 0+ (S) is denoted by K. Theorem 2.6. Let S be a dense subsemigroup of ((0, +∞), +) and p ∈ 0+ (S). The following statements are equivalent. (a) p ∈ K. (b) For every A ∈ p, {x ∈ S : −x + A ∈ p} is syndetic near 0. (c) For every r ∈ 0+ (S), p ∈ 0+ (S) + r + p. Proof. See [7] or see Theorem 3.4 in [11]. 2 Theorem 2.7. Let A ⊆ S, then K ∩ A = ∅ if and only if A is piecewise syndetic near 0. Proof. See Theorem 13.35 in [8]. 2 3. Concepts near zero The results and theorems of this section correspond to the results of [8] in the same order they appear in [8]. In [8], concepts of central set and their relation with the algebraic structure of βS is thoroughly stated. So, we use this to investigate concepts near zero, frequently. Some proofs are similar to the discrete case. However, we try to keep the process of stating theorems on track. In this section we define J-sets near zero and C-sets near zero. Definition 3.1. Let S be a dense subsemigroup of (0, ∞), + . The set of sequences in S converging to 0 is denoted by T0 . Definition 3.2. Let S be a dense subsemigroup of ((0, ∞), +) and let A ⊆ S. Then A is a J-set near zero if and only if whenever F ∈ Pf (T0 ) and δ > 0, there exist a ∈ S ∩ (0, δ) and H ∈ Pf (N) such that for each f ∈ F , a + t∈H f (t) ∈ A. Of course, we can say that A ⊆ S is a J-set near zero if and only if for each F ∈ Pf (T0 ) and for each δ > 0, there exist a ∈ S and H ∈ Pf (N) such that a + t∈H f (t) ∈ A ∩ (0, δ) for each f ∈ F , i.e. for each δ > 0, A ∩ (0, δ) is a J-set. It is obvious that every J-set near zero with respect to this definition is a J-set near zero by Definition 3.2. So we focus on Definition 3.2. Lemma 3.3. Let S be a dense subsemigroup of (0, ∞), + and A ⊆ S be a J-set near zero. Whenever m ∈ N, F ∈ Pf (T0 ) and δ > 0, there exist a ∈ S ∩ (0, δ) and H ∈ Pf (N) such that minH > m and for each f ∈ F , a + t∈H f (t) ∈ A. Proof. See Lemma 14.8.2 in [8].
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Theorem 3.4. Let S be a dense subsemigroup of (0, ∞), + and A be a subset of S. If A is piecewise syndetic near zero, then A is a J-set near zero. l Proof. Let F ∈ Pf (T0 ), and l = |F |, and enumerate F as {f1 , · · · , fl }. Let Y = t=1 0+ (S). Then by l → − s ∈ t=1 S, then λ→ Theorem 2.22 in [8], Y is a compact right topological semigroup and if − s is continuous. For i ∈ N and δ > 0, let Ii,δ =
a+
f1 (t), · · · , a +
t∈H
fl (t) : a ∈ S ∩ (0, δ), H ∈ Pf (N), and minH > i
t∈H
and let Ei,δ = Ii,δ ∪ {(a, · · · , a) : a ∈ S ∩ (0, δ)}.
Let E = i∈N,δ>0 Ei,δ and I = i∈N,δ>0 Ii,δ . It is obvious that E ⊆ Y and I ⊆ Y . We claim that E is a subsemigroup of Y and I is an ideal of E. To this end, let p, q ∈ E. We will show that p + q ∈ E and if either p ∈ I or q ∈ I, then p + q ∈ I. Pick δ > 0, then U = clβSd (0, δ) ∩ S is an open neighborhood of p + q and let i ∈ N. Since ρq is continuous, pick a neighborhood V of p such that V + q ⊆ U . Pick → − → → → x ∈ Ei, δ ∩ V with − x ∈ Ii, δ if p ∈ I. If − x ∈ Ii, δ so that − x = (a + t∈H f1 (t), · · · , a + t∈H fl (t)) for 3 3 3 − some a ∈ S ∩ (0, 3δ ) and some H ∈ Pf (N) with minH > i, let j = maxH. Otherwise, let j = i. Since λ→ x is → → → x + W ⊆ U . Pick − y ∈ Ej, δ ∩ W with − y ∈ Ij, δ if q ∈ I. continuous, pick a neighborhood W of q such that − 3 3 → → → → x +− y ∈ Ei,δ ∩ U and if either p ∈ I or q ∈ I, then − x +− y ∈ Ii,δ ∩ U . Then − l By Theorem 2.23 in [8], K(Y ) = t=1 K(0+ (S)). Pick by Theorem 2.7 some p ∈ K(0+ (S)) ∩ A. Then p = (p, · · · , p) ∈ K(Y ). We claim that p ∈ E. To see this, let U be a neighborhood of p, i ∈ N, and
l l pick C1 , · · · , Cl ∈ p such that t=1 Ct ⊆ U . Pick a ∈ t=1 Ct . Then a = (a, · · · , a) ∈ U ∩ Ei,δ . Thus p ∈ K(Y ) ∩ E and consequently K(Y ) ∩ E = ∅. Then by Theorem 1.65 in [8], we have K(E) = K(Y ) ∩ E l l → z ∈ I1,δ ∩ t=1 A, a ∈ S ∩ (0, δ), and and so p ∈ K(E) ⊆ I. Then I1,δ ∩ t=1 A = ∅ for each δ > 0, so let − H ∈ Pf (N) such that − → z = a+
f1 (t), · · · , a +
t∈H
fl (t) .
2
t∈H
Theorem 3.5 (Central Sets Theorem near zero). Let S be a dense subsemigroup of (0, ∞), + . Let A be a central subset of S near zero. Then for each δ ∈ (0, 1), there exist functions αδ : Pf (T0 ) → S and Hδ : Pf (T0 ) → Pf (N) such that (1) αδ (F ) < δ for each F ∈ Pf (T0 ), (2) if F, G ∈ Pf (T0 ) and F ⊂ G, then maxHδ (F ) < minHδ (G) and (3) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m i=1
αδ (Gi ) +
fi (t) ∈ A.
t∈Hδ (Gi )
Proof. Pick a minimal idempotent p of 0+ (S) such that A ∈ p. Let A∗ = {x ∈ A : −x + A ∈ p}, so A∗ ∈ p. Also by Lemma 4.14 in [8], if x ∈ A∗ , then −x + A∗ ∈ p. We define αδ (F ) ∈ S and Hδ (F ) ∈ Pf (N) for F ∈ Pf (T0 ) by induction on |F | satisfying the following inductive hypotheses: (1) αδ (G) < δ for each G ∈ T0 , (2) if F, G ∈ Pf (T0 ) and F ⊂ G, then maxHδ (F ) < minHδ (G) and
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(3) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m αδ (Gi ) + i=1
fi (t) ∈ A∗ .
t∈Hδ (Gi )
Assume that F = {f }. Since A∗ is piecewise syndetic near zero, let by Theorem 3.4, for δ > 0, a ∈ S∩(0, δ) and L ∈ Pf (N), a + t∈L f (t) ∈ A∗ . Let αδ ({f }) = a and Hδ ({f }) = L. Let |F | > 1, αδ (G) and Hδ (G) be defined for all proper subsets G of F and for each δ > 0. Pick δ > 0, let Kδ =
{Hδ (G) : G is a non-empty proper subset of F }
and let m = maxKδ . Let Mδ =
n i=1
αδ (Gi ) +
n fi (t) : n ∈ N, ∅ = G1 ⊂ · · · ⊂ Gn ⊂ F, and {fi }ni=1 ∈ Gi . i=1
t∈Hδ (Gi )
Then Mδ is finite and by hypothesis (3), Mδ ⊆ A∗ . Let B = A∗ ∩ x∈Mδ (−x + A∗ ). Then B ∈ p so pick by Theorem 3.4 and Lemma 3.3, a ∈ S ∩ (0, δ) and L ∈ Pf (N), a + t∈L f (t) ∈ B for each f ∈ F . Let αδ (F ) = a and Hδ (F ) = L. The hypothesis (1) is obvious. Since minL > m, we have the hypothesis (2) is satisfied. To verify n hypothesis (3), pick δ > 0 and n ∈ N, let ∅ ⊂ G1 ⊂ · · · ⊂ Gn = F , and let {fi }ni=1 ∈ i=1 Gi . If n = 1, then α(G1 ) + t∈H(G1 ) f1 (t) = a + t∈L f1 (t) ∈ B ⊆ A∗ . So assume that n > 1 and let y = n n−1 αδ (Gi ) + t∈Hδ (Gi ) fi (t) . Then y ∈ Mδ so a + t∈L f1 (t) ∈ B ⊆ (−y+A∗ ) and thus i=1 αδ (Gi ) + i=1 ∗ t∈Hδ (Gi ) fi (t) = y + a + t∈L f1 (t) ∈ A as required. 2 Definition 3.6. Let S be a dense subsemigroup of (0, ∞), + and let A ⊆ S. We say A is a C-set near zero if and only if for each δ ∈ (0, 1), (a) there exist functions αδ : Pf (T0 ) → S and Hδ : Pf (T0 ) → Pf (N), such that (1) αδ (F ) < δ for each F ∈ Pf (T0 ), (2) if F, G ∈ Pf (T0 ) and F ⊂ G, then maxHδ (F ) < minHδ (G) and (3) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m
αδ (Gi ) +
i=1
fi (t) ∈ A.
t∈Hδ (Gi )
b) J0 (S) = {p ∈ 0+ (S) : for all A ∈ p, A is a J − set near zero}. Let Φ be the set of all functions f : N → N for which f (n) ≤ n for each n ∈ N. Theorem 3.7. Let S be a dense subsemigroup of (0, ∞), + , and let A be a C-set near zero in S, and for each l ∈ N, let {yl,n }n∈N ∈ T0 . There exist a sequence {an }n∈N in S and a sequence {Hn }n∈N in Pf (N) such that an → 0, maxHn < minHn+1 for each n ∈ N and for each f ∈ Φ F S {an + yf (n),t }n∈N ⊆ A. t∈Hn
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In particular, the above conclusion applies if A is a central set near zero in S. Proof. Give α and H as guaranteed by Definition 3.6. We may assume that the sequences {yl,n }n∈N are distinct. For n ∈ N, let Fn = {{yl,t }t∈N , {y2,t }t∈N , · · · , {yn,t }t∈N } and let an = α n1 (Fn ) < n1 and Hn = H n1 (Fn ). Let f ∈ Φ be given. To see that F S {an + yf (n),t }n∈N ⊆ A, t∈Hn
let K ∈ Pf (N). Let K = {n1 , · · · , nm } where n1 < n2 , · · · < nm . Then Fn1 ⊂ Fn2 ⊂ · · · ⊂ Fnm and for each i ∈ {1, · · · , m}, {yf (ni ),t }t∈N ∈ Fni so
(an +
n∈K
yf (n),t ) =
m i=1
t∈Hn
(α n1 (Fni ) + i
t∈H
1 ni
yf (ni ),t ) ∈ A.
(Fni )
The “in particular” part is obvious. 2 Lemma 3.8. Let S be a dense subsemigroup of (0, ∞), + , and A1 and A2 be subsets of S. If A1 ∪ A2 is a J-set near zero, then either A1 or A2 is a J-set near zero. Proof. Suppose not and pick F1 and F2 in Pf (T◦ ) and δ > 0 such that for each i ∈ {1, 2}, each H ∈ Pf (N), each a ∈ S ∩ (0, δ), there is some f ∈ Fi such that a + t∈H f (t) ∈ / Ai . Let F = F1 ∪ F2 , k =| F |, and write F = {f1 , f2 , ..., fk }. Pick by Exercise 14.2.1., in [8], some n ∈ N such that whenever length n words over the alphabet {1, 2, ..., k} are 2-colored, there is a variable word w(v) such that {w(l) : l ∈ {1, 2, ..., k}} is monochromatic. 1 Let h ∈ T0 . For each t ∈ N, we can choose st ∈ N such that h(st ) < 2t+1 δ. Since we could replace each ∞ fi by the sequence fi , defined by fi (t) = fi (st ), we may suppose that t=1 h(t) < 12 δ. Let W be the set of length n words over {1, 2, ..., k}. It is obvious that W is a finite set. If w = b1 b2 · · · bn ∈ n W (where each bi ∈ {1, 2, · · · , k}), define gw (t) = i=1 fbi (nt + i). Since {gw : w ∈ W } is finite, there exists c ∈ (0, 12 δ) and K ∈ Pf (N) such that c+ t∈K gw (t) ∈ A for each w ∈ W . Define ϕ : W → {1, 2} by ϕ(w) = 1 if c + t∈K gw (t) ∈ A1 and ϕ(w) = 2 otherwise. We can choose a variable word w(v) of length n, such that ϕ is constant on {w(l) : ł ∈ {1, 2, ..., k}}. We may suppose that ϕ(w(j)) = 1 for every j ∈ {1, 2, ..., k}. Suppose that w(v) = b1 b2 · · · bn . Let C = {i ∈ {1, · · · , n} : bi ∈ {1, · · · , k}} and let I = {i ∈ {1, · · · , n} : bi = v}. Then, for each j ∈ {1, 2, · · · , k}, we have c+
t∈K
gw(j) (t) = c +
fbi (nt + i) +
t∈K i∈C
fj (nt + i).
t∈K i∈I
Let a = c + t∈K i∈C fbi (nt + i). Then a ∈ (0, δ) and a + t∈K i∈I fj (nt + i) ∈ A1 for every j ∈ {1, 2, · · · , k}. Putting H = nK + I, we have t∈K i∈I
fj (nt + i) =
fj (t)
t∈H
because the sets of the form nK + i, with i ∈ I, are pairwise disjoint. It follows that a + for every j ∈ {1, 2, · · · , k} – a contradiction. 2
t∈H
fj (t) ∈ A1
Theorem 3.9. Let S be a dense subsemigroup of (0, ∞), + . Then J0 (S) is a compact two sided ideal of 0+ .
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Proof. Trivially J0 (S) is closed subset of βS. Let p ∈ J0 (S) and let q ∈ βS. We show q + p ∈ J0 (S) and p + q ∈ J0 (S). To see q + p ∈ J0 (S), let A ∈ q + p, F ∈ Pf (T0 ) and pick δ > 0. Then {b ∈ S : −b + A ∈ p} ∈ q so pick b ∈ S ∩ (0, 2δ ) such that −b + A ∈ p. Pick H ∈ Pf (N) and a ∈ S ∩ (0, 2δ ) such that for each f ∈ F , a + t∈H f (t) ∈ −b + A. Therefore for each F ∈ P( T0 ) and each δ > 0, there exists H ∈ Pf (N) and c = a + b ∈ S ∩ (0, δ) such that a + b + t∈H f (t) ∈ A. To see p + q ∈ J0 (S), let A ∈ p + q and let B = {x ∈ S : −x + A ∈ q}. Then B ∈ p so for F ∈ Pf (T0 ) and δ > 0, pick H ∈ Pf (N) and a ∈ S ∩ (0, 2δ ) such that for each f ∈ F , a + t∈H f (t) ∈ B. Then
δ
(0, 2 ). Therefore for each f ∈F (−(a + t∈H f (t)) + A) ∈ q so pick b ∈ f ∈F − (a + t∈H f (t)) + A F ∈ Pf (T0 ) and each δ > 0, there exists H ∈ Pf (N) and c = a +b ∈ S ∩(0, δ) such that a +b + t∈H f (t) ∈ A. This implies that p + q ∈ J0 (S). 2 Theorem 3.10. Let S be a dense subsemigroup of (0, ∞), + , let A ⊆ S. Then A ∩ J0 (S) = ∅ if and only if A is a J-set near zero. Proof. The necessity is trivial. By Lemma 3.8, J-sets near zero are partition regular. So, if A is a J-set near zero, by Theorem 1.2, there is some p ∈ βS such that A ∈ p and for every B ∈ p, B is a J-set near zero. 2 As a consequence, we have the following corollary similar to Theorem 3.4. Corollary 3.11. Let S be a dense subsemigroup of (0, ∞), + , and let A be a piecewise syndetic near zero subset of S. Then A is a J-set near zero. Proof. By Theorem 2.7, A∩K(0+ (S)) = ∅. Since K(0+ (S)) ⊆ J0 (S), thus A∩J0 (S) = ∅ so by Theorem 3.10, A is a J-set near zero. 2 Corollary 3.12. Let S be a dense subsemigroup of ((0, ∞), +) and A be a central set near zero in S. Then A is a C-set near zero. Proof. It is obvious. 2 Lemma 3.13. Let J be a set, (D, ≤) be a directed set, and S be a dense subsemigroup of ((0, ∞), +). Let {Ti }i∈D be a decreasing family of nonempty subsets of S, such that 1) 0 ∈ clR Ti ,
2) i∈D Ti = ∅, and 3) for each i ∈ D and each x ∈ Ti , there is some j ∈ D such that x + Tj ⊆ Ti .
Let Q = i∈D clβSd Ti . Then Q is a compact subsemigroup of 0+ (S). Let {Ei }i∈D and {Ii }i∈D be decreasing families of nonempty subsets of Πt∈J S with the following properties: (a) for each i ∈ D, Ii ⊆ Ei ⊆ Πt∈J Ti , → − x ∈ Ii , there exists j ∈ D such that → x + Ej ⊆ Ii , and (b) for each i ∈ D and each − → − − x ∈ Ei \ Ii , there exists j ∈ D such that → x + Ej ⊆ Ei and → x + Ij ⊆ Ii . (c) for each i ∈ D and each −
Let Y = Πt∈J 0+ (S), E = i∈D clY Ei , and I = i∈D clY Ii . Then E is a subsemigroup of Πt∈J Q and I is an ideal of E. If, in addition, either (d) for each i ∈ D, Ti = S and {a ∈ S : a ∈ / Ei } is not piecewise syndetic near zero,
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or (e) for each i ∈ D and each a ∈ Ti , a ∈ Ei , then given any p ∈ K(Q), one has p ∈ E ∩ K(Πt∈J Q) = K(E) ⊆ I. Proof. By Theorem 4.20 in [8], Q is a subsemigroup of 0+ (S). For the proof that E is a subsemigroup of Πt∈J Q and I is an ideal of E, see the proof of Lemma 14.9 in [8]. To complete the proof, assume that (d) or (e) holds. It suffices to establish if p ∈ K(Q), then p ∈ E.
(∗)
Indeed, assume we have established (∗). Then p ∈ E ∩ Πt∈J K(Q) and Πt∈J K(Q) = K(Πt∈J Q) by Theorem 2.23 in [8]. Then by Theorem 1.65 in [8], K(E) = E ∩ K(Πt∈J Q) and, since I is an ideal of E, K(E) ⊆ I. To establish (∗), let p ∈ K(Q) be given. To see that p ∈ E, let i ∈ D be given and let U be a neighborhood
of p. Pick F ∈ Pf (J) and for each t ∈ F , pick some At ∈ p such that t∈F πt−1 [clβSd At ] ⊆ U , where πt is a projection for t ∈ J. Assume now that (d) holds. Since p ∈ K(0+ (S)) and {a ∈ S : a ∈ / Ei } are not piecewise syndetic near zero, so by Theorem 2.7, {a ∈ S : a ∈ / Ei } ∈ / p and hence {a ∈ S : a ∈ Ei } ∈ p. Now pick
a∈ t∈F At ∩ {a ∈ S : a ∈ Ei }. Then a ∈ U ∩ Ei . If (e) holds, see the proof of Lemma 14.9 in [8]. 2 The proof of the following theorem is similar to the proof of Theorem 2.7 in [4]. Theorem 3.14. Let S be a dense subsemigroup of ((0, ∞), +) and A ⊆ S. Then A is a C-set near zero if and only if there is an idempotent in A ∩ J0 (S). Proof. Sufficiency. Let A∗ = {x ∈ A : −x + A ∈ p}. By Lemma 4.14 in [8] if x ∈ A∗ then −x + A∗ ∈ p. For every δ ∈ (0, 1) we define αδ (F ) and Hδ (F ) for F ∈ Pf (T0 ) by induction on |F | such that (1) if F, G ∈ Pf (T0 ) and F ⊂ G, then maxHδ (F ) < minHδ (G) and (2) if m ∈ N, G1 , ..., Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gn = F , and for all i ∈ {1, 2, ..., m}, fi ∈ Gi , then m i=1
(αδ (Gi ) +
fi (t)) ∈ A∗ .
t∈Hδ (Gi )
If F = {f }, since A∗ is a J-set near zero, pick αδ (F ) ∈ S ∩ (0, δ) and Hδ (F ) ∈ Pf (N) such that αδ (F ) + t∈Hδ (Gi ) fi (t) ∈ A∗ . Now assume that |F | > 1 and αδ (K) and Hδ (K) have been chosen for all K with ∅ = K ⊂ F . Let
m r = max {Hδ (K) : ∅ = K ⊂ F }. Let B = A∗ ∩ {−( i=1 (αδ (Gi ) + t∈Hδ (Gi ) fi (t))) + A∗ : m ∈ N, G1 , G2 , ..., Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gn = F, and (∀i ∈ {1, 2, ..., m}) (fi ∈ Gi )}. Then B ∈ p so pick by Lemma 3.3, αδ (F ) ∈ S ∩ (0, δ) and Hδ (F ) ∈ Pf (N) such minHδ (F ) > r and for each f ∈ F , αδ (F ) + t∈Hδ (F ) f (t) ∈ B. Necessity. For each k ∈ N, pick α k1 and H k1 as guaranteed by the definition of C-set near zero. For F ∈ Pf (T0 ) and k ∈ N, let
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m (α k1 (Gi ) +
TF,k = {
i=1
79
fi (t)) : m ∈ N, G1 , G2 , ..., Gm ∈ Pf (T0 ),
t∈H 1 (Gi ) k
F ⊂ G1 ⊂ G2 ⊂ · · · ⊂ Gm , and (∀i ∈ {1, 2, ..., m}) (fi ∈ Gi )}.
Then each TF,k = ∅. Let Qk = F ∈Pf (T0 ) TF,k . We show first that Qk is semigroup. Given F, G ∈ Pf (T0 ), TF ∪G,k ⊆ TF,k ∩ TG,k and so Qk = ∅. By Theorem 4.20 in [8], (∀F ∈ Pf (T0 ))(∀x ∈ TF,k )(∃X ∈ Pf (T0 ))(TX,k ⊆ −x + TF,k ) so let F ∈ Pf (T0 ) and let x ∈ TF,k . Pick m, G1 , G2 , ..., Gm , and f1 , f2 , ..., fm as in the definition of TF,k so m that x = i=1 (α k1 (Gi ) + t∈H 1 (Gi ) fi (t)). Then TGm ,k ⊆ −x + TF,k . k
We show that K(Qk ) ⊆ A ∩ J0 (S), so that any idempotent in K(Qk ) establishes the result. We have that each TF,k ⊆ A so Qk ⊆ A. Now let p ∈ K(Qk ) and let B ∈ p. We need to show that B is J-set near zero, so let F ∈ Pf (T0 ) be given. Let k = |F | and write F = {f1 , f2 , ..., fk }. Let D = {G ∈ Pf (T0 ) : F ⊆ G} and
note that Qk = G∈D TG,k . → Let Y = Πki=1 0+ (S). By Theorem 2.22 in [8], Y is a right topological semigroup and if − x ∈ Πki=1 S, then − λ→ x is continuous. For G ∈ D and for k ∈ N, let 1 → IG,k = {− x ∈ Πki=1 TG,k : (∃d ∈ S ∩ (0, ))(∃L ∈ Pf (N)) k → (− x = (d + f (t), ..., d + f (t)))} 1
t∈L
k
t∈L
and let EG,k = IG,k ∪ {(d, d, ..., d) : d ∈ TG,k }. Let Ik = G∈D clY IG,k and let Ek = G∈D clY EG,k . We claim that Ek is a subsemigroup of Πli=1 Qk and that Ik is an ideal of Ek . Trivially Ek ⊆ Πli=1 Qk . Given G1 , G2 ∈ D we have that IG1 ∪G2 ,k ⊆ IG1 ,k ∩ IG2 ,k so to see that Ik = ∅ it suffices to let G ∈ D and show that IG,k = ∅. Pick G1 , G2 ∈ D such that G ⊂ G1 ⊂ G2 . Let L = H k1 (G2 ). Let d = α k1 (G1 ) + t∈H 1 (G1 ) f1 (t) + α k1 (G2 ). Then (d + t∈L f1 (t), ..., d + t∈L fk (t)) ∈ IG,k . k → → → → → → → → q ,− r ∈ Ek . We show that − q +− r ∈ Ek and, if either − q ∈ Ik or − r ∈ Ik , then − q +− r ∈ Ik . To Now let − → − → − → − this end, let G ∈ D and let U be an open neighborhood of q + r . Pick a neighborhood V of q such that − → → → x ∈ V ∩ EG,k , with − x ∈ IG,k if − q ∈ Ik . For each i ∈ {1, 2, ..., k}, we have that xi ∈ TG,k V +− r ⊆ U . Pick → k → so pick Xi ∈ D such that TXi ,k ⊆ −xi + TG,k and let X = i=1 Xi . Pick a neighborhood W of − r such that → − → − → − → − → − → − → − → − x + W ⊆ U . Pick y ∈ W ∩ EX,k with y ∈ IX,k if r ∈ Ik . Then x + y ∈ EG,k and, if x ∈ IG,k or → − → → y ∈ I , then − x +− y ∈I . X,k
G,k
Recall that we have chosen p ∈ K(Qk ) and B ∈ p. We claim that p = (p, p, ..., p) ∈ Ek . To see this let G ∈ D be given and let U be a neighborhood of p in Y. Pick C ∈ p such that Πki=1 C ⊆ U and pick d ∈ C ∩ TG,k . Then (d, d, ..., d) ∈ U ∩ EG,k . By Theorem 2.23 in [8], we have that K(Πki=1 Qk ) = Πki=1 K(Qk ) so p ∈ Ek ∩ K(Πki=1 Qk ). Therefore by Theorem 1.65 in [8] we have that p ∈ K(Ek ) and, since Ik is an ideal → x ∈ Πki=1 B ∩ Πki=1 TF,k . of Ek , we have that p ∈ Ik . Since Πki=1 B is a neighborhood of p, we have some − Thus B is a J-set near zero as required. 2 Acknowledgements The authors are very grateful to the anonymous referee for his or her comments and suggestions, which have been very helpful in improving the presentation of this paper. We would like to express our thanks to the referee for bringing the Proof of Lemma 3.8 to our attention.
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References [1] D. De, N. Hindman, D. Strauss, A new and stronger central sets theorem, Fundam. Math. 199 (2008) 155–175. [2] D. De, N. Hindman, Image partition regularity near zero, Discrete Math. 309 (2009) 3219–3232. [3] D. De, N. Hindman, A simple characterization of sets satisfying the Central Sets Theorem, N.Y. J. Math. 15 (2009) 203–223. [4] N. Hindman, D. Strauss, Sets satisfying the Central Sets Theorem, Semigroup Forum 79 (2009) 480–506. [5] D. De, R.K. Paul, Image partition regularity near zero with real entries, N.Y. J. Math. 17 (2011) 149–161. [6] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, in: M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. [7] N. Hindman, I. Leader, The semigroup of ultrafilters near 0, Semigroup Forum 59 (1999) 33–55. ˘ [8] N. Hindman, D. Strauss, Algebra in the Stone–Cech Compactification, Theory and Applications, de Gruyter, Berlin, 2011. [9] N. Hindman, D. Strauss, Sets central with respect to certain subsemigroups of βSd , Topol. Proc. 33 (2009) 55–79. [10] N. Hindman, D. Strauss, Sets satisfying the Central Set Theorem, Semigroup Forum 79 (2009) 480–506. [11] M.A. Tootkaboni, T. Vahed, The semigroup of ultrafilters near an idempotent of a semitopological semigroup, Topol. Appl. 159 (16) (2012) 3494–3503.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Central Sets Theorem near zero E. Bayatmanesh, M. Akbari Tootkaboni ∗ Department of Mathematics, Faculty of Basic Science, Shahed University, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 29 November 2015 Received in revised form 21 June 2016 Accepted 23 June 2016 Available online 21 July 2016
a b s t r a c t In this paper, we introduce notions of J-set near zero and C-set near zero for a dense subsemigroup of ((0, +∞), +) and state the Central Sets Theorem near zero. Among the other results for a dense subsemigroup S ⊆ ((0, +∞), +), we give some sufficient and equivalent algebraic conditions on a subset A ⊂ S to be J-set near zero and to be C-set near zero. © 2016 Elsevier B.V. All rights reserved.
MSC: primary 54D35, 22A15 secondary 05D10, 54D80 Keywords: Central Sets Theorem The Stone–Čech compactification C-set J-set Piecewise syndetic set near zero
1. Introduction Let (S, +) be a discrete semigroup. The collection of all ultrafilters on S is called the Stone–Čech compactification of S and denoted by βS. For A ⊆ S, define A = {p ∈ βS : A ∈ p}, then {A : A ⊆ S} is a basis for the open sets (also for the closed sets) of βS. We identify the principal ultrafilters with the points of S and thus pretend that S ⊆ βS. There is a unique extension of the operation to βS, making (βS, +) a right topological semigroup (i.e. for each p ∈ βS, the right translation ρp is continuous, where ρp (q) = q + p) and also for each x ∈ S, the left translation λx is continuous, where λx (q) = x + q. The principal ultrafilters identified by the points of S and S is a dense subset of βS. For p, q ∈ βS and A ⊆ S, we have A ∈ p + q if and only if {x ∈ S : −x + A ∈ q} ∈ p, where −x + A = {y ∈ S : x + y ∈ A}. A nonempty subset L of a semigroup (S, +) is called a left ideal of S if S + L ⊆ L, a right ideal if L + S ⊆ L, and a two sided ideal (or simply an ideal) if it is both a left and a right ideal. A minimal left ideal is a left ideal that does not contain any proper left ideal. In the same way, we can define minimal right ideal and smallest ideal. * Corresponding author. E-mail addresses: [email protected] (E. Bayatmanesh), [email protected] (M. Akbari Tootkaboni). http://dx.doi.org/10.1016/j.topol.2016.06.014 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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Any compact Hausdorff right topological semigroup (S, +) has a smallest two sided ideal, denoted by K(S), which is the union of all minimal left ideals, and also the union of all minimal right ideals, as well. 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(S) is called a minimal idempotent. For more details see [8]. For A ⊆ S and p ∈ βS, we define A∗ (p) = {s ∈ A : −s + A ∈ p}. Lemma 1.1. Let (S, +) be a semigroup, p + p = p ∈ βS, and let A ∈ p. Then for each s ∈ A∗ (p), −s + A∗ (p) ∈ p. Proof. See Lemma 4.14 in [8].
2
Now we review the definition of partition regularity. In this paper, the collection of all nonempty finite subsets of S is denoted by Pf (S) and P(S) is the set of all subsets of S. Definition 1.1. Let R be a nonempty set of subsets of S. R is partition regular if and only if whenever F is a finite subset of P(S) and F ∈ R, there exist A ∈ F and B ∈ R, such that B ⊆ A. / R. Let Theorem 1.2. Let R ⊆ P(S) be a nonempty set and assume ∅ ∈ R↑ = {B ∈ P(S) : A ⊆ B for some A ∈ R}. Then (a), (b) and (c) are equivalent. (a) R is partition regular. (b) Whenever A ⊆ P(S) has the property that every finite nonempty subfamily of A has an intersection which is in R↑ , there is U ∈ βS, such that A ⊆ U ⊆ R↑ . (c) Whenever A ∈ R, there is U ∈ βS such that A ∈ U ⊆ R↑ . Proof. [8, Theorem 3.11]. 2 Definition 1.3. Let (S, .) be a discrete semigroup and A ⊆ S. Then A is a central set if and only if there exists an idempotent p ∈ K(βS) with A ∈ p. We have been considering semigroups which are dense in ((0, ∞), +) with the natural topology. When discussing the Stone–Čech compactification of such a semigroup S, we will deal with Sd , which is the set S with the discrete topology. Definition 1.4. Let S be a dense subset of ((0, ∞), +). Then 0+ (S) = {p ∈ βSd : (∀ > 0) (0, ) ∩ S ∈ p}. By Lemma 2.5 in [7], 0+ (S) is a compact right topological subsemigroup of (βSd , +), and 0+ (S) ∩ K(βSd ) = ∅. Since 0+ (S) is a compact right topological semigroup, so 0+ (S) contains idempotents. The set 0+ (S) of all non-principal ultrafilters on S = ((0, ∞), +) that are convergent to 0 is a semigroup under the restriction of the usual ‘+’ on βSd , the Stone–Čech compactification of the discrete semigroup S = ((0, ∞), +), see [7]. In [2], the authors used the algebraic structure of 0+ (S) in their investigation of image partition regularity near 0 of finite and infinite matrices. In [5], the algebraic structure of 0+ (R) was used to investigate image partition regularity of matrices with real entries from R. Central sets near zero were introduced by N. Hindman and I. Leader in [7] as
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central sets. Central sets near zero, too, enjoy rich combinatorial structure. Central sets are ideal objects for Ramsey theory. The central sets theorem was first introduced by H. Furstenberg (see [6]) for the semigroups N and Z. The most general version of Central Sets Theorem is available in [1], [3] and [9]. In section 3, we will introduce the Central Sets Theorem near zero and define C-set near zero, J-set near zero and derive some results for dense subsemigroups of ((0, ∞), +). 2. Preliminary Central subsets of a discrete semigroup S have very strong combinatorial properties, which are consequences of the Central Sets Theorem. There is an elementary description of central sets, which was given in [8]. In this paper, the collection of all functions f : N → S is denoted by T =N S. Also, A ⊂ B means that A ⊆ B and A = B. Theorem 2.1 (Central Sets Theorem [1]). Let (S, +) be a commutative semigroup. Let A be a central subset of S. There exist functions α : Pf (T ) → S and H : Pf (T ) → Pf (N), such that (1) if F, G ∈ Pf (T ) and F ⊂ G, then maxH(F ) < minH(G), and (2) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m α(Gi ) + i=1
fi (t) ∈ A.
t∈H(Gi )
We define a set to be a C-set if and only if it satisfies the conclusion of the Central Sets Theorem. Definition 2.2. Let S be a commutative semigroup, A ⊆ S, and T =N S. The set A is a C-set if and only if there exist functions α : Pf (T ) → S and H : Pf (T ) → Pf (N), such that (1) if F, G ∈ Pf (T ) and F ⊂ G, then maxH(F ) < minH(G), and (2) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m i=1
α(Gi ) +
fi (t) ∈ A.
t∈H(Gi )
C-sets are the important objects in combinatorial theory. In many semigroups, they are the objects that behavior like central sets, see [10]. In [1], the authors obtained a simple characterization of C-sets in an arbitrary discrete semigroup. Definition 2.3. Let (S, +) be a commutative semigroup, T =N S, and A ⊆ S is a J-set if and only if whenever F ∈ Pf (T ), there exist a ∈ S and H ∈ Pf (N), such that for each f ∈ F , a + t∈H f (t) ∈ A. Now we recall the notions of syndetic near zero and piecewise syndetic near zero in [8]. Definition 2.4. A subset B of (0, 1) is syndetic near zero if and only if for every > 0 there exist some F ∈ Pf (0, ) and some δ > 0 such that (0, δ) ⊆ t∈F (−t + B). Definition 2.5. a) A subset A of (0, 1) is piecewise syndetic near zero if and only if there exist sequences ∞ {Fn }∞ n=1 and {δn }n=1 such that
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1 1 (1) for each n ∈ N, Fn ∈ Pf ((0, )) and δn ∈ (0, ), and n n 1 (2) for all G ∈ Pf ((0, )) and all μ > 0 there is some x ∈ (0, μ) such that for all n ∈ N, (G ∩ (0, δn )) + x ⊆ n t∈Fn (−t + A). A is a central set near zero if and only if there exists an idempotent p in the smallest ideal of 0+ (S) with A ∈ p. In this paper, the minimal ideal in 0+ (S) is denoted by K. Theorem 2.6. Let S be a dense subsemigroup of ((0, +∞), +) and p ∈ 0+ (S). The following statements are equivalent. (a) p ∈ K. (b) For every A ∈ p, {x ∈ S : −x + A ∈ p} is syndetic near 0. (c) For every r ∈ 0+ (S), p ∈ 0+ (S) + r + p. Proof. See [7] or see Theorem 3.4 in [11]. 2 Theorem 2.7. Let A ⊆ S, then K ∩ A = ∅ if and only if A is piecewise syndetic near 0. Proof. See Theorem 13.35 in [8]. 2 3. Concepts near zero The results and theorems of this section correspond to the results of [8] in the same order they appear in [8]. In [8], concepts of central set and their relation with the algebraic structure of βS is thoroughly stated. So, we use this to investigate concepts near zero, frequently. Some proofs are similar to the discrete case. However, we try to keep the process of stating theorems on track. In this section we define J-sets near zero and C-sets near zero. Definition 3.1. Let S be a dense subsemigroup of (0, ∞), + . The set of sequences in S converging to 0 is denoted by T0 . Definition 3.2. Let S be a dense subsemigroup of ((0, ∞), +) and let A ⊆ S. Then A is a J-set near zero if and only if whenever F ∈ Pf (T0 ) and δ > 0, there exist a ∈ S ∩ (0, δ) and H ∈ Pf (N) such that for each f ∈ F , a + t∈H f (t) ∈ A. Of course, we can say that A ⊆ S is a J-set near zero if and only if for each F ∈ Pf (T0 ) and for each δ > 0, there exist a ∈ S and H ∈ Pf (N) such that a + t∈H f (t) ∈ A ∩ (0, δ) for each f ∈ F , i.e. for each δ > 0, A ∩ (0, δ) is a J-set. It is obvious that every J-set near zero with respect to this definition is a J-set near zero by Definition 3.2. So we focus on Definition 3.2. Lemma 3.3. Let S be a dense subsemigroup of (0, ∞), + and A ⊆ S be a J-set near zero. Whenever m ∈ N, F ∈ Pf (T0 ) and δ > 0, there exist a ∈ S ∩ (0, δ) and H ∈ Pf (N) such that minH > m and for each f ∈ F , a + t∈H f (t) ∈ A. Proof. See Lemma 14.8.2 in [8].
2
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Theorem 3.4. Let S be a dense subsemigroup of (0, ∞), + and A be a subset of S. If A is piecewise syndetic near zero, then A is a J-set near zero. l Proof. Let F ∈ Pf (T0 ), and l = |F |, and enumerate F as {f1 , · · · , fl }. Let Y = t=1 0+ (S). Then by l → − s ∈ t=1 S, then λ→ Theorem 2.22 in [8], Y is a compact right topological semigroup and if − s is continuous. For i ∈ N and δ > 0, let Ii,δ =
a+
f1 (t), · · · , a +
t∈H
fl (t) : a ∈ S ∩ (0, δ), H ∈ Pf (N), and minH > i
t∈H
and let Ei,δ = Ii,δ ∪ {(a, · · · , a) : a ∈ S ∩ (0, δ)}.
Let E = i∈N,δ>0 Ei,δ and I = i∈N,δ>0 Ii,δ . It is obvious that E ⊆ Y and I ⊆ Y . We claim that E is a subsemigroup of Y and I is an ideal of E. To this end, let p, q ∈ E. We will show that p + q ∈ E and if either p ∈ I or q ∈ I, then p + q ∈ I. Pick δ > 0, then U = clβSd (0, δ) ∩ S is an open neighborhood of p + q and let i ∈ N. Since ρq is continuous, pick a neighborhood V of p such that V + q ⊆ U . Pick → − → → → x ∈ Ei, δ ∩ V with − x ∈ Ii, δ if p ∈ I. If − x ∈ Ii, δ so that − x = (a + t∈H f1 (t), · · · , a + t∈H fl (t)) for 3 3 3 − some a ∈ S ∩ (0, 3δ ) and some H ∈ Pf (N) with minH > i, let j = maxH. Otherwise, let j = i. Since λ→ x is → → → x + W ⊆ U . Pick − y ∈ Ej, δ ∩ W with − y ∈ Ij, δ if q ∈ I. continuous, pick a neighborhood W of q such that − 3 3 → → → → x +− y ∈ Ei,δ ∩ U and if either p ∈ I or q ∈ I, then − x +− y ∈ Ii,δ ∩ U . Then − l By Theorem 2.23 in [8], K(Y ) = t=1 K(0+ (S)). Pick by Theorem 2.7 some p ∈ K(0+ (S)) ∩ A. Then p = (p, · · · , p) ∈ K(Y ). We claim that p ∈ E. To see this, let U be a neighborhood of p, i ∈ N, and
l l pick C1 , · · · , Cl ∈ p such that t=1 Ct ⊆ U . Pick a ∈ t=1 Ct . Then a = (a, · · · , a) ∈ U ∩ Ei,δ . Thus p ∈ K(Y ) ∩ E and consequently K(Y ) ∩ E = ∅. Then by Theorem 1.65 in [8], we have K(E) = K(Y ) ∩ E l l → z ∈ I1,δ ∩ t=1 A, a ∈ S ∩ (0, δ), and and so p ∈ K(E) ⊆ I. Then I1,δ ∩ t=1 A = ∅ for each δ > 0, so let − H ∈ Pf (N) such that − → z = a+
f1 (t), · · · , a +
t∈H
fl (t) .
2
t∈H
Theorem 3.5 (Central Sets Theorem near zero). Let S be a dense subsemigroup of (0, ∞), + . Let A be a central subset of S near zero. Then for each δ ∈ (0, 1), there exist functions αδ : Pf (T0 ) → S and Hδ : Pf (T0 ) → Pf (N) such that (1) αδ (F ) < δ for each F ∈ Pf (T0 ), (2) if F, G ∈ Pf (T0 ) and F ⊂ G, then maxHδ (F ) < minHδ (G) and (3) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m i=1
αδ (Gi ) +
fi (t) ∈ A.
t∈Hδ (Gi )
Proof. Pick a minimal idempotent p of 0+ (S) such that A ∈ p. Let A∗ = {x ∈ A : −x + A ∈ p}, so A∗ ∈ p. Also by Lemma 4.14 in [8], if x ∈ A∗ , then −x + A∗ ∈ p. We define αδ (F ) ∈ S and Hδ (F ) ∈ Pf (N) for F ∈ Pf (T0 ) by induction on |F | satisfying the following inductive hypotheses: (1) αδ (G) < δ for each G ∈ T0 , (2) if F, G ∈ Pf (T0 ) and F ⊂ G, then maxHδ (F ) < minHδ (G) and
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(3) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m αδ (Gi ) + i=1
fi (t) ∈ A∗ .
t∈Hδ (Gi )
Assume that F = {f }. Since A∗ is piecewise syndetic near zero, let by Theorem 3.4, for δ > 0, a ∈ S∩(0, δ) and L ∈ Pf (N), a + t∈L f (t) ∈ A∗ . Let αδ ({f }) = a and Hδ ({f }) = L. Let |F | > 1, αδ (G) and Hδ (G) be defined for all proper subsets G of F and for each δ > 0. Pick δ > 0, let Kδ =
{Hδ (G) : G is a non-empty proper subset of F }
and let m = maxKδ . Let Mδ =
n i=1
αδ (Gi ) +
n fi (t) : n ∈ N, ∅ = G1 ⊂ · · · ⊂ Gn ⊂ F, and {fi }ni=1 ∈ Gi . i=1
t∈Hδ (Gi )
Then Mδ is finite and by hypothesis (3), Mδ ⊆ A∗ . Let B = A∗ ∩ x∈Mδ (−x + A∗ ). Then B ∈ p so pick by Theorem 3.4 and Lemma 3.3, a ∈ S ∩ (0, δ) and L ∈ Pf (N), a + t∈L f (t) ∈ B for each f ∈ F . Let αδ (F ) = a and Hδ (F ) = L. The hypothesis (1) is obvious. Since minL > m, we have the hypothesis (2) is satisfied. To verify n hypothesis (3), pick δ > 0 and n ∈ N, let ∅ ⊂ G1 ⊂ · · · ⊂ Gn = F , and let {fi }ni=1 ∈ i=1 Gi . If n = 1, then α(G1 ) + t∈H(G1 ) f1 (t) = a + t∈L f1 (t) ∈ B ⊆ A∗ . So assume that n > 1 and let y = n n−1 αδ (Gi ) + t∈Hδ (Gi ) fi (t) . Then y ∈ Mδ so a + t∈L f1 (t) ∈ B ⊆ (−y+A∗ ) and thus i=1 αδ (Gi ) + i=1 ∗ t∈Hδ (Gi ) fi (t) = y + a + t∈L f1 (t) ∈ A as required. 2 Definition 3.6. Let S be a dense subsemigroup of (0, ∞), + and let A ⊆ S. We say A is a C-set near zero if and only if for each δ ∈ (0, 1), (a) there exist functions αδ : Pf (T0 ) → S and Hδ : Pf (T0 ) → Pf (N), such that (1) αδ (F ) < δ for each F ∈ Pf (T0 ), (2) if F, G ∈ Pf (T0 ) and F ⊂ G, then maxHδ (F ) < minHδ (G) and (3) whenever m ∈ N, G1 , · · · , Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gm , and for each i ∈ {1, 2, · · · , m}, fi ∈ Gi , one has m
αδ (Gi ) +
i=1
fi (t) ∈ A.
t∈Hδ (Gi )
b) J0 (S) = {p ∈ 0+ (S) : for all A ∈ p, A is a J − set near zero}. Let Φ be the set of all functions f : N → N for which f (n) ≤ n for each n ∈ N. Theorem 3.7. Let S be a dense subsemigroup of (0, ∞), + , and let A be a C-set near zero in S, and for each l ∈ N, let {yl,n }n∈N ∈ T0 . There exist a sequence {an }n∈N in S and a sequence {Hn }n∈N in Pf (N) such that an → 0, maxHn < minHn+1 for each n ∈ N and for each f ∈ Φ F S {an + yf (n),t }n∈N ⊆ A. t∈Hn
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In particular, the above conclusion applies if A is a central set near zero in S. Proof. Give α and H as guaranteed by Definition 3.6. We may assume that the sequences {yl,n }n∈N are distinct. For n ∈ N, let Fn = {{yl,t }t∈N , {y2,t }t∈N , · · · , {yn,t }t∈N } and let an = α n1 (Fn ) < n1 and Hn = H n1 (Fn ). Let f ∈ Φ be given. To see that F S {an + yf (n),t }n∈N ⊆ A, t∈Hn
let K ∈ Pf (N). Let K = {n1 , · · · , nm } where n1 < n2 , · · · < nm . Then Fn1 ⊂ Fn2 ⊂ · · · ⊂ Fnm and for each i ∈ {1, · · · , m}, {yf (ni ),t }t∈N ∈ Fni so
(an +
n∈K
yf (n),t ) =
m i=1
t∈Hn
(α n1 (Fni ) + i
t∈H
1 ni
yf (ni ),t ) ∈ A.
(Fni )
The “in particular” part is obvious. 2 Lemma 3.8. Let S be a dense subsemigroup of (0, ∞), + , and A1 and A2 be subsets of S. If A1 ∪ A2 is a J-set near zero, then either A1 or A2 is a J-set near zero. Proof. Suppose not and pick F1 and F2 in Pf (T◦ ) and δ > 0 such that for each i ∈ {1, 2}, each H ∈ Pf (N), each a ∈ S ∩ (0, δ), there is some f ∈ Fi such that a + t∈H f (t) ∈ / Ai . Let F = F1 ∪ F2 , k =| F |, and write F = {f1 , f2 , ..., fk }. Pick by Exercise 14.2.1., in [8], some n ∈ N such that whenever length n words over the alphabet {1, 2, ..., k} are 2-colored, there is a variable word w(v) such that {w(l) : l ∈ {1, 2, ..., k}} is monochromatic. 1 Let h ∈ T0 . For each t ∈ N, we can choose st ∈ N such that h(st ) < 2t+1 δ. Since we could replace each ∞ fi by the sequence fi , defined by fi (t) = fi (st ), we may suppose that t=1 h(t) < 12 δ. Let W be the set of length n words over {1, 2, ..., k}. It is obvious that W is a finite set. If w = b1 b2 · · · bn ∈ n W (where each bi ∈ {1, 2, · · · , k}), define gw (t) = i=1 fbi (nt + i). Since {gw : w ∈ W } is finite, there exists c ∈ (0, 12 δ) and K ∈ Pf (N) such that c+ t∈K gw (t) ∈ A for each w ∈ W . Define ϕ : W → {1, 2} by ϕ(w) = 1 if c + t∈K gw (t) ∈ A1 and ϕ(w) = 2 otherwise. We can choose a variable word w(v) of length n, such that ϕ is constant on {w(l) : ł ∈ {1, 2, ..., k}}. We may suppose that ϕ(w(j)) = 1 for every j ∈ {1, 2, ..., k}. Suppose that w(v) = b1 b2 · · · bn . Let C = {i ∈ {1, · · · , n} : bi ∈ {1, · · · , k}} and let I = {i ∈ {1, · · · , n} : bi = v}. Then, for each j ∈ {1, 2, · · · , k}, we have c+
t∈K
gw(j) (t) = c +
fbi (nt + i) +
t∈K i∈C
fj (nt + i).
t∈K i∈I
Let a = c + t∈K i∈C fbi (nt + i). Then a ∈ (0, δ) and a + t∈K i∈I fj (nt + i) ∈ A1 for every j ∈ {1, 2, · · · , k}. Putting H = nK + I, we have t∈K i∈I
fj (nt + i) =
fj (t)
t∈H
because the sets of the form nK + i, with i ∈ I, are pairwise disjoint. It follows that a + for every j ∈ {1, 2, · · · , k} – a contradiction. 2
t∈H
fj (t) ∈ A1
Theorem 3.9. Let S be a dense subsemigroup of (0, ∞), + . Then J0 (S) is a compact two sided ideal of 0+ .
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Proof. Trivially J0 (S) is closed subset of βS. Let p ∈ J0 (S) and let q ∈ βS. We show q + p ∈ J0 (S) and p + q ∈ J0 (S). To see q + p ∈ J0 (S), let A ∈ q + p, F ∈ Pf (T0 ) and pick δ > 0. Then {b ∈ S : −b + A ∈ p} ∈ q so pick b ∈ S ∩ (0, 2δ ) such that −b + A ∈ p. Pick H ∈ Pf (N) and a ∈ S ∩ (0, 2δ ) such that for each f ∈ F , a + t∈H f (t) ∈ −b + A. Therefore for each F ∈ P( T0 ) and each δ > 0, there exists H ∈ Pf (N) and c = a + b ∈ S ∩ (0, δ) such that a + b + t∈H f (t) ∈ A. To see p + q ∈ J0 (S), let A ∈ p + q and let B = {x ∈ S : −x + A ∈ q}. Then B ∈ p so for F ∈ Pf (T0 ) and δ > 0, pick H ∈ Pf (N) and a ∈ S ∩ (0, 2δ ) such that for each f ∈ F , a + t∈H f (t) ∈ B. Then
δ
(0, 2 ). Therefore for each f ∈F (−(a + t∈H f (t)) + A) ∈ q so pick b ∈ f ∈F − (a + t∈H f (t)) + A F ∈ Pf (T0 ) and each δ > 0, there exists H ∈ Pf (N) and c = a +b ∈ S ∩(0, δ) such that a +b + t∈H f (t) ∈ A. This implies that p + q ∈ J0 (S). 2 Theorem 3.10. Let S be a dense subsemigroup of (0, ∞), + , let A ⊆ S. Then A ∩ J0 (S) = ∅ if and only if A is a J-set near zero. Proof. The necessity is trivial. By Lemma 3.8, J-sets near zero are partition regular. So, if A is a J-set near zero, by Theorem 1.2, there is some p ∈ βS such that A ∈ p and for every B ∈ p, B is a J-set near zero. 2 As a consequence, we have the following corollary similar to Theorem 3.4. Corollary 3.11. Let S be a dense subsemigroup of (0, ∞), + , and let A be a piecewise syndetic near zero subset of S. Then A is a J-set near zero. Proof. By Theorem 2.7, A∩K(0+ (S)) = ∅. Since K(0+ (S)) ⊆ J0 (S), thus A∩J0 (S) = ∅ so by Theorem 3.10, A is a J-set near zero. 2 Corollary 3.12. Let S be a dense subsemigroup of ((0, ∞), +) and A be a central set near zero in S. Then A is a C-set near zero. Proof. It is obvious. 2 Lemma 3.13. Let J be a set, (D, ≤) be a directed set, and S be a dense subsemigroup of ((0, ∞), +). Let {Ti }i∈D be a decreasing family of nonempty subsets of S, such that 1) 0 ∈ clR Ti ,
2) i∈D Ti = ∅, and 3) for each i ∈ D and each x ∈ Ti , there is some j ∈ D such that x + Tj ⊆ Ti .
Let Q = i∈D clβSd Ti . Then Q is a compact subsemigroup of 0+ (S). Let {Ei }i∈D and {Ii }i∈D be decreasing families of nonempty subsets of Πt∈J S with the following properties: (a) for each i ∈ D, Ii ⊆ Ei ⊆ Πt∈J Ti , → − x ∈ Ii , there exists j ∈ D such that → x + Ej ⊆ Ii , and (b) for each i ∈ D and each − → − − x ∈ Ei \ Ii , there exists j ∈ D such that → x + Ej ⊆ Ei and → x + Ij ⊆ Ii . (c) for each i ∈ D and each −
Let Y = Πt∈J 0+ (S), E = i∈D clY Ei , and I = i∈D clY Ii . Then E is a subsemigroup of Πt∈J Q and I is an ideal of E. If, in addition, either (d) for each i ∈ D, Ti = S and {a ∈ S : a ∈ / Ei } is not piecewise syndetic near zero,
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or (e) for each i ∈ D and each a ∈ Ti , a ∈ Ei , then given any p ∈ K(Q), one has p ∈ E ∩ K(Πt∈J Q) = K(E) ⊆ I. Proof. By Theorem 4.20 in [8], Q is a subsemigroup of 0+ (S). For the proof that E is a subsemigroup of Πt∈J Q and I is an ideal of E, see the proof of Lemma 14.9 in [8]. To complete the proof, assume that (d) or (e) holds. It suffices to establish if p ∈ K(Q), then p ∈ E.
(∗)
Indeed, assume we have established (∗). Then p ∈ E ∩ Πt∈J K(Q) and Πt∈J K(Q) = K(Πt∈J Q) by Theorem 2.23 in [8]. Then by Theorem 1.65 in [8], K(E) = E ∩ K(Πt∈J Q) and, since I is an ideal of E, K(E) ⊆ I. To establish (∗), let p ∈ K(Q) be given. To see that p ∈ E, let i ∈ D be given and let U be a neighborhood
of p. Pick F ∈ Pf (J) and for each t ∈ F , pick some At ∈ p such that t∈F πt−1 [clβSd At ] ⊆ U , where πt is a projection for t ∈ J. Assume now that (d) holds. Since p ∈ K(0+ (S)) and {a ∈ S : a ∈ / Ei } are not piecewise syndetic near zero, so by Theorem 2.7, {a ∈ S : a ∈ / Ei } ∈ / p and hence {a ∈ S : a ∈ Ei } ∈ p. Now pick
a∈ t∈F At ∩ {a ∈ S : a ∈ Ei }. Then a ∈ U ∩ Ei . If (e) holds, see the proof of Lemma 14.9 in [8]. 2 The proof of the following theorem is similar to the proof of Theorem 2.7 in [4]. Theorem 3.14. Let S be a dense subsemigroup of ((0, ∞), +) and A ⊆ S. Then A is a C-set near zero if and only if there is an idempotent in A ∩ J0 (S). Proof. Sufficiency. Let A∗ = {x ∈ A : −x + A ∈ p}. By Lemma 4.14 in [8] if x ∈ A∗ then −x + A∗ ∈ p. For every δ ∈ (0, 1) we define αδ (F ) and Hδ (F ) for F ∈ Pf (T0 ) by induction on |F | such that (1) if F, G ∈ Pf (T0 ) and F ⊂ G, then maxHδ (F ) < minHδ (G) and (2) if m ∈ N, G1 , ..., Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gn = F , and for all i ∈ {1, 2, ..., m}, fi ∈ Gi , then m i=1
(αδ (Gi ) +
fi (t)) ∈ A∗ .
t∈Hδ (Gi )
If F = {f }, since A∗ is a J-set near zero, pick αδ (F ) ∈ S ∩ (0, δ) and Hδ (F ) ∈ Pf (N) such that αδ (F ) + t∈Hδ (Gi ) fi (t) ∈ A∗ . Now assume that |F | > 1 and αδ (K) and Hδ (K) have been chosen for all K with ∅ = K ⊂ F . Let
m r = max {Hδ (K) : ∅ = K ⊂ F }. Let B = A∗ ∩ {−( i=1 (αδ (Gi ) + t∈Hδ (Gi ) fi (t))) + A∗ : m ∈ N, G1 , G2 , ..., Gm ∈ Pf (T0 ), G1 ⊂ G2 ⊂ · · · ⊂ Gn = F, and (∀i ∈ {1, 2, ..., m}) (fi ∈ Gi )}. Then B ∈ p so pick by Lemma 3.3, αδ (F ) ∈ S ∩ (0, δ) and Hδ (F ) ∈ Pf (N) such minHδ (F ) > r and for each f ∈ F , αδ (F ) + t∈Hδ (F ) f (t) ∈ B. Necessity. For each k ∈ N, pick α k1 and H k1 as guaranteed by the definition of C-set near zero. For F ∈ Pf (T0 ) and k ∈ N, let
E. Bayatmanesh, M. Akbari Tootkaboni / Topology and its Applications 210 (2016) 70–80
m (α k1 (Gi ) +
TF,k = {
i=1
79
fi (t)) : m ∈ N, G1 , G2 , ..., Gm ∈ Pf (T0 ),
t∈H 1 (Gi ) k
F ⊂ G1 ⊂ G2 ⊂ · · · ⊂ Gm , and (∀i ∈ {1, 2, ..., m}) (fi ∈ Gi )}.
Then each TF,k = ∅. Let Qk = F ∈Pf (T0 ) TF,k . We show first that Qk is semigroup. Given F, G ∈ Pf (T0 ), TF ∪G,k ⊆ TF,k ∩ TG,k and so Qk = ∅. By Theorem 4.20 in [8], (∀F ∈ Pf (T0 ))(∀x ∈ TF,k )(∃X ∈ Pf (T0 ))(TX,k ⊆ −x + TF,k ) so let F ∈ Pf (T0 ) and let x ∈ TF,k . Pick m, G1 , G2 , ..., Gm , and f1 , f2 , ..., fm as in the definition of TF,k so m that x = i=1 (α k1 (Gi ) + t∈H 1 (Gi ) fi (t)). Then TGm ,k ⊆ −x + TF,k . k
We show that K(Qk ) ⊆ A ∩ J0 (S), so that any idempotent in K(Qk ) establishes the result. We have that each TF,k ⊆ A so Qk ⊆ A. Now let p ∈ K(Qk ) and let B ∈ p. We need to show that B is J-set near zero, so let F ∈ Pf (T0 ) be given. Let k = |F | and write F = {f1 , f2 , ..., fk }. Let D = {G ∈ Pf (T0 ) : F ⊆ G} and
note that Qk = G∈D TG,k . → Let Y = Πki=1 0+ (S). By Theorem 2.22 in [8], Y is a right topological semigroup and if − x ∈ Πki=1 S, then − λ→ x is continuous. For G ∈ D and for k ∈ N, let 1 → IG,k = {− x ∈ Πki=1 TG,k : (∃d ∈ S ∩ (0, ))(∃L ∈ Pf (N)) k → (− x = (d + f (t), ..., d + f (t)))} 1
t∈L
k
t∈L
and let EG,k = IG,k ∪ {(d, d, ..., d) : d ∈ TG,k }. Let Ik = G∈D clY IG,k and let Ek = G∈D clY EG,k . We claim that Ek is a subsemigroup of Πli=1 Qk and that Ik is an ideal of Ek . Trivially Ek ⊆ Πli=1 Qk . Given G1 , G2 ∈ D we have that IG1 ∪G2 ,k ⊆ IG1 ,k ∩ IG2 ,k so to see that Ik = ∅ it suffices to let G ∈ D and show that IG,k = ∅. Pick G1 , G2 ∈ D such that G ⊂ G1 ⊂ G2 . Let L = H k1 (G2 ). Let d = α k1 (G1 ) + t∈H 1 (G1 ) f1 (t) + α k1 (G2 ). Then (d + t∈L f1 (t), ..., d + t∈L fk (t)) ∈ IG,k . k → → → → → → → → q ,− r ∈ Ek . We show that − q +− r ∈ Ek and, if either − q ∈ Ik or − r ∈ Ik , then − q +− r ∈ Ik . To Now let − → − → − → − this end, let G ∈ D and let U be an open neighborhood of q + r . Pick a neighborhood V of q such that − → → → x ∈ V ∩ EG,k , with − x ∈ IG,k if − q ∈ Ik . For each i ∈ {1, 2, ..., k}, we have that xi ∈ TG,k V +− r ⊆ U . Pick → k → so pick Xi ∈ D such that TXi ,k ⊆ −xi + TG,k and let X = i=1 Xi . Pick a neighborhood W of − r such that → − → − → − → − → − → − → − → − x + W ⊆ U . Pick y ∈ W ∩ EX,k with y ∈ IX,k if r ∈ Ik . Then x + y ∈ EG,k and, if x ∈ IG,k or → − → → y ∈ I , then − x +− y ∈I . X,k
G,k
Recall that we have chosen p ∈ K(Qk ) and B ∈ p. We claim that p = (p, p, ..., p) ∈ Ek . To see this let G ∈ D be given and let U be a neighborhood of p in Y. Pick C ∈ p such that Πki=1 C ⊆ U and pick d ∈ C ∩ TG,k . Then (d, d, ..., d) ∈ U ∩ EG,k . By Theorem 2.23 in [8], we have that K(Πki=1 Qk ) = Πki=1 K(Qk ) so p ∈ Ek ∩ K(Πki=1 Qk ). Therefore by Theorem 1.65 in [8] we have that p ∈ K(Ek ) and, since Ik is an ideal → x ∈ Πki=1 B ∩ Πki=1 TF,k . of Ek , we have that p ∈ Ik . Since Πki=1 B is a neighborhood of p, we have some − Thus B is a J-set near zero as required. 2 Acknowledgements The authors are very grateful to the anonymous referee for his or her comments and suggestions, which have been very helpful in improving the presentation of this paper. We would like to express our thanks to the referee for bringing the Proof of Lemma 3.8 to our attention.
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