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A generalized central sets theorem in partial semigroups

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Abstract

The most powerful formulation of the Central Sets Theorem in an arbitrary semigroup was proved in the work of De, Hindman, and Strauss. The sets which satisfy the conclusion of the above Central Sets Theorem are called C-sets. The original Central Sets Theorem was extended by J. McLeod for adequate commutative partial semigroups. In this work, we will extend the Central Sets Theorem obtained by taking all possible adequate sequences in a commutative adequate partial semigroup. We shall also discuss a sufficient condition for being a set C-set in our context.

Introduction

The notion of the central subset of \({\mathbb {N}}\) was originally introduced by Furstenberg [3] in terms of a topological dynamical system. Before defining central sets let us start with original Central Sets Theorem due to Furstenberg [3].

Theorem 1.1

(The Original Central Sets Theorem) Let \(l\in {\mathbb {N}}\) and for each \(i\in \{1,2,\ldots ,l\}\), and let \(\langle y_{i,n}\rangle _{n=1}^{\infty }\) be a sequence in \(\mathbb {Z}\). Let C be a central subset of \({\mathbb {N}}\). Then there exist sequences \(\langle a_{n}\rangle _{n=1}^{\infty }\) in \({\mathbb {N}}\) and \(\langle H_{n}\rangle _{n=1}^{\infty }\) in \({\mathcal {P}}_{f}({\mathbb {N}})\) such that

  1. (1)

    For all n, \(\max H_{n} < \min H_{n+1}\) and

  2. (2)

    For all \(F\in {\mathcal {P}}_{f}({\mathbb {N}})\) and all \(i\in \{1,2,\ldots ,l\},\sum _{n\in F}(a_{n}+\sum _{t\in H_{n}}y_{i,t})\in C\).

Proof

[3, Proposition 8.21]. \(\square \)

It was shown in [1] that this definition was equivalent to a simpler algebraic characterization which we use below. This algebraic characterization is in the setting of the Stone–Čech compactification, \(\beta S\), of a discrete semigroup S. We shall present a brief introduction to this structure later in this section.

Observe that the conclusion of the Central Sets Theorem shows that central sets posses rich combinatorial structures. To present the notion central sets we present a brief introduction of the algebraic structure of \(\beta S\) for a discrete semigroup \((S,+)\). We take the points of \(\beta S\) to be the ultrafilters on S, identifying the principal ultrafilters with the points of S and thus pretending that \(S\subseteq \beta S\). Given \(A\subseteq S\) let us set, \(\overline{A}=\{p\in \beta S\mid A\in p\}\). Then the set \(\{\overline{A}\mid A\subseteq S\}\) is a basis for a topology on \(\beta S\). The operation \(+\) on S can be extended to the Stone–Čech compactification \(\beta S\) of S so that \((\beta S,+)\) is a compact right topological semigroup (meaning that for any \(p\in \beta S\), the function \(\rho _{p}:\beta S\rightarrow \beta S\) defined by \(\rho _{p}(q)=q+p\) is continuous) with S contained in its topological center (meaning that for any \(x\in S\), the function \(\lambda _{x}:\beta S\rightarrow \beta S\) defined by \(\lambda _{x}(q)=x+q\) is continuous). Given \(p,q\in \beta S\) and \(A\subseteq S\), \(A\in p+q\) if and only if \(\{x\in S \mid -x+A\in q\}\in p\), where \(-x+A=\{y\in S \mid x+y\in A\}\).

A nonempty subset I of a semigroup \((T,+)\) is called a left ideal of T if \(T+I\subset I\), a right ideal if \(I+T\subset I\), 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. Similarly, we can define a minimal right ideal.

Any compact Hausdorff right topological semigroup \((T,+)\) has a smallest two sided ideal

$$\begin{aligned} \begin{array}{rl} K(T) = &{} \bigcup \{L \mid L\text { is a minimal left ideal of }T\}\\ = &{}\bigcup \{R \mid R\text { is a minimal right ideal of }T\} \end{array} \end{aligned}$$

Given a minimal left ideal L and a minimal right ideal R, \(L\cap R\) is a group, and in particular, contains an idempotent. An idempotent in K(T) is called a minimal idempotent. If p and q are idempotents in T, we write \(p\le q\) if and only if \(p+q=q+p=p\). An idempotent is minimal with respect to this relation if and only if it is a member of the smallest ideal. See [5] for an elementary introduction to the algebra of \(\beta S\) and for any unfamiliar details.

Definition 1.2

Let S be a discrete semigroup and let C be a subset of S. Then C is central if and only if there is some minimal idempotent \(p \in \beta S\) with \(p \in \overline{C}\).

The original Central Sets Theorem was already strong enough to derive several combinatorial consequences such as Rado’s theorem. Later this subject has been extensively studied by many authors. In [5] the Central Sets Theorem was extended to arbitrary semigroups. The version for commutative semigroups extended the original Central Sets Theorem by allowing the choice of countably many sequences at a time.

Theorem 1.3

Let \((S,\cdot )\) be a commutative semigroup. Let C be a central subset of S and for each \(l\in {\mathbb {N}}\), let \(\left\langle y_{l,n}\right\rangle _{n=1}^{\infty }\) be a sequence in S. Then there exist sequences \(\left\langle a_{n}\right\rangle _{n=1}^{\infty }\) in S and \(\left\langle H_{n}\right\rangle _{n=1}^{\infty }\) in \({\mathcal {P}}_{f}({\mathbb {N}})\) such that \( \max H_{n} < \min H_{n+1}\) for each \(n\in {\mathbb {N}}\) and such that for each \(f\in \Phi \), the set of all functions \(f:{\mathbb {N}}\rightarrow {\mathbb {N}}\) for which \(f(n)\le n\) for all \(n\in {\mathbb {N}}\), \(FP({\left\langle a_{n}.\Pi _{t\in H_{n}}y_{f(n),t}\right\rangle }_{n=1}^{\infty })\subseteq C\).

Here in the above theorem, FP stands for the finite products. Precisely, for a sequence \(\langle x_n\rangle _{n= 1}^{\infty }\) in S, \(FP(\langle x_n \rangle _{n= 1}^{\infty })= \{\prod \nolimits _{n \in F}x_n \mid F \in {\mathcal {P}}_f({\mathbb {N}})\}\).

Later, in [2], the authors extended the Central Sets Theorem considering all sequences at one time.

Theorem 1.4

Let \((S,+)\) be a commutative semigroup and let \({\mathcal {T}}=S^{{\mathbb {N}}}\), the set of sequences in S. Let C be a central subset of S. There exist functions \(\alpha :{\mathcal {P}}_{f}({\mathcal {T}})\rightarrow S\) and \(H:{\mathcal {P}}_{f}({\mathcal {T}})\rightarrow {\mathcal {P}}_{f}({\mathbb {N}})\) such that

(1) if \(F,G\in {\mathcal {P}}_{f}({\mathcal {T}})\) and \(F\subsetneq G\), then \(\max H(F) < \min H(G)\) and

(2) whenever \(m\in {\mathbb {N}},G_{1},G_{2},\ldots ,G_{m}\in {\mathcal {P}}_{f}({\mathcal {T}}),G_{1}\subsetneq G_{2}\subsetneq \cdots \subsetneq G_{m}\) and for each \(i\in \{1,2,\ldots ,m\}\) , \(f_{i}\in G_{i}\), one has \(\sum _{i=1}^{m}(\alpha (G_{i})+\sum _{t\in H(G_{i})}f_i(t))\in C\).

Proof

[2, Theorem 2.2]. \(\square \)

The authors in [6], introduced the notions of C-sets which satisfying the conclusion of the above Central Sets Theorem.

Definition 1.5

Let \((S,+)\) be a commutative semigroup and let \(A\subseteq S\) and \({\mathcal {T}}=S^{{\mathbb {N}}}\), the set of sequences in S. The set A is a C-set if and only if there exist functions \(\alpha :{\mathcal {P}}_{f}({\mathcal {T}})\rightarrow S\) and \(H:{\mathcal {P}}_{f}({\mathcal {T}})\rightarrow {\mathcal {P}}_{f}({\mathbb {N}})\) such that

(1) if \(F,G\in {\mathcal {P}}_{f}({\mathcal {T}})\) and \(F\subsetneq G\), then \( \max H(F) < \min H(G)\) and

(2) whenever \(m\in {\mathbb {N}},G_{1},G_{2},\ldots ,G_{m}\in {\mathcal {P}}_{f}({\mathcal {T}}),G_{1}\subsetneq G_{2}\subsetneq \cdots \subsetneq G_{m}\) and for each \(i\in \{1,2,\ldots ,m\}\) , \(f_{i}\in G_{i}\), one has \(\sum _{i=1}^{m}(\alpha (G_{i})+\sum _{t\in H(G_{i})}f_i(t))\in A\).

The notion of partial semigroups is another important notion in the study of Ramsey theory. In [7], the author extended the Central Sets Theorem for commutative adequate partial semigroup with a finite number of adequate sequences. Before stating the theorem we need to introduce some definitions.

Definition 1.6

(Partial semigroup) A partial semigroup is defined as a pair \((G, *)\) where \(*\) is an operation defined on a subset X of \(G \times G\) and satisfies the statement that for all x, y, z in G, \((x *y) *z = x *(y *z) \) in the sense that if either side is defined, so is the other and they are equal. A partial semigroup is commutative if \({x*y=y*x}\) for every \({(x,y)\in X}\).

Example 1.7

Every semigroup is a partial semigroup.

Example 1.8

Let us consider \({G={\mathcal {P}}_f({{\mathbb {N}}})}= \{F \mid \emptyset \ne F \subseteq {\mathbb {N}}\) and F is finite\(\}\) and let \({X=\{(\alpha ,\beta )\in G \times G \mid \alpha \cap \beta =\emptyset \}}\) be the family of all pairs of disjoint sets, and let \({*:X\rightarrow G}\) be the union. It is easy to check that this is a commutative partial semigroup. We shall denote this partial semigroup as \(({\mathcal {P}}_f({{\mathbb {N}}}),\uplus )\).

Definition 1.9

Let \((S,*)\) be a partial semigroup.

(a) For \(s\in S\), \(\phi (s)=\{t\in S \mid s*t\) is defined}.

(b) For \(H\in {\mathcal {P}}_{f}(S)\), \(\sigma (H)=\bigcap _{s \in H}\phi (s)\).

(c) \((S,*)\) is adequate if and only if \(\sigma (H)\ne \emptyset \) for all \(H\in {\mathcal {P}}_{f}(S)\).

(d) \(\delta S=\bigcap _{x\in S}cl_{\beta S}(\phi (x))=\bigcap _{H\in {\mathcal {P}}_{f}(S)}cl_{\beta S}(\sigma (H))\).

So, the partial semigroup \(({\mathcal {P}}_f({{\mathbb {N}}}),\uplus )\) is adequate. We are specifically interested in adequate partial semigroups as they lead to an interesting subsemigroup \(\delta S\) of \(\beta S\), the Stone–Čech compactification of S which is itself a compact right topological semigroup. Notice that adequacy of S is exactly what is required to guarantee that \(\delta S\ne \emptyset \). If S is, in fact, a semigroup, then \(\delta S=\beta S\).

Now we recall some of the basic properties of the operation \(*\) in \(\delta S\).

Definition 1.10

Let \((S,*)\) be a partial semigroup. For \(s\in S\) and \(A\subseteq S\), \(s^{-1}A=\{t\in \phi (s) \mid s*t\in A\}\).

Lemma 1.11

Let \((S,*)\) be a partial semigroup, let \(A\subseteq S\) and let \(a,b,c\in S\). Then \(c\in b^{-1}(a^{-1}A)\) if and only if both \(b\in \phi (a)\) and \(c\in (a*b)^{-1}A\). In particular, if \(b\in \phi (a)\), then \(b^{-1}(a^{-1}A)=(a*b)^{-1}A\).

Proof

[4, Lemma 2.3]. \(\square \)

Definition 1.12

Let \((S,*)\) be an adequate partial semigroup.

(a) For \(a\in S\) and \(q\in \overline{\phi (a)}\), \(a*q=\{A\subseteq S \mid a^{-1}A\in q\}\).

(b) For \(p\in \beta S\) and \(q\in \delta S\), \(p*q=\{A\subseteq S \mid \{a^{-1}A\in q\}\in p\}\).

Lemma 1.13

Let \((S,*)\) be an adequate partial semigroup.

(a) If \(a\in S\) and \(q\in \overline{\phi (a)}\), then \(a*q\in \beta S\).

(b) If \(p\in \beta S\) and \(q\in \delta S\), then \(p*q\in \beta S\).

(c) Let \(p\in \beta S,q\in \delta S\), and \(a\in S\). Then \(\phi (a)\in p*q\) if and only if \(\phi (a)\in p\).

(d) If \(p,q\in \delta S\), then \(p*q\in \delta S\).

Proof

[4, Lemma 2.7]. \(\square \)

Lemma 1.14

Let \((S,*)\) be an adequate partial semigroup and let \(q\in \delta S\). Then the function \(\rho _{q}:\beta S\rightarrow \beta S\) defined by \(\rho _{q}(p)=p*q\) is continuous.

Proof

[4, Lemma 2.8]. \(\square \)

Lemma 1.15

Let \(p\in \beta S\) and let \(q,r\in \delta S\). Then \(p*(q*r)=(p*q)*r\).

Proof

[4, Lemma 2.9]. \(\square \)

Definition 1.16

Let \(p=p *p \in \delta S\) and let \(A \in p\). Then \(A^{*} = \{x \in A \mid x^{-1}A \in p\}\).

Given an idempotent \(p \in \delta S\) and \(A \in p\), it is immediate that \(A^{*} \in p\).

Lemma 1.17

Let \(p= p *p \in \delta S\), let \(A \in p\), and let \(x \in A^{*}\). Then \(x^{-1}A^{*} \in p\).

Proof

[4, Lemma 2.12]. \(\square \)

As a consequence of the above results, we have that if \((S, *)\) is an adequate partial semigroup, then \((\delta S, *)\) is a compact right topological semigroup. Being a compact right topological semigroup, \(\delta S\) contains idempotents, left ideals, a smallest two-sided ideal, and minimal idempotents. Thus \(\delta S\) provides a suitable environment for considering the notion of central sets.

Definition 1.18

Let \((S,*)\) be an adequate partial semigroup. A set \(C\subseteq S\) is central if and only if there is an idempotent \(p\in \overline{C}\cap K(\delta S)\).

Definition 1.19

Let \((S,*)\) be an adequate partial semigroup and let\(\left\langle y_{n}\right\rangle _{n=1}^{\infty }\) be a sequence in S. Then \(\left\langle y_{n}\right\rangle _{n=1}^{\infty }\) is adequate if and only if \(\prod _{n\in F}y_{n}\) is defined for each \(F\in {\mathcal {P}}_{f}({\mathbb {N}})\) and for every \(K\in {\mathcal {P}}_{f}(S)\), there exists \(m\in {\mathbb {N}}\) such that \(FP(\left\langle y_{n}\right\rangle _{n=m}^{\infty })\subseteq \sigma (K)\).

The following is the Central Sets Theorem for commutative adequate partial semigroup with a finite number of sequences proved in [7].

Theorem 1.20

Let \((S,*)\) be a commutative adequate partial semigroup and let C be a central subset of S. Let \(k \in {\mathbb {N}}\) and for each \(l\in \{1,2,\ldots ,k\}\), let \(\left\langle y_{l,n}\right\rangle _{n=1}^{\infty }\) be an adequate sequence in S. There exists a sequence \(\left\langle a_{n}\right\rangle _{n=1}^{\infty }\) in S and a sequence \(\left\langle H_{n}\right\rangle _{n=1}^{\infty }\) in \({\mathcal {P}}_{f}({\mathbb {N}})\) such that \(\max H_{n} < \min H_{n+1}\) for each \(n\in {\mathbb {N}}\) and such that for each \(f:{\mathbb {N}}\rightarrow \{1,2,\ldots ,k\}\),

$$\begin{aligned} FP\left( \left\langle a_{n}*\prod _{t\in H_{n}}y_{f(n),t}\right\rangle _{n=1}^{\infty }\right) \subseteq C. \end{aligned}$$

Proof

[7, Theorem 3.4]. \(\square \)

Section 2 of this paper deals with a version of Central Sets Theorem for commutative adequate partial semigroup which deals with all adequate sequences in S at once and we show that, as long as S has infinitely many adequate sequences, this result generalizes Theorem 1.20. In Sect. 3, we define a C-set in an adequate partial semigroup to be a set satisfying the conclusion of this new Central Sets Theorem and establish a sufficient condition for being a C-set.

Notation 1.21

Through out this document \(\overline{A}\) stands for the closure of A with respect to \(\beta S\). Also, we use the notation \(cl_{Y}(A)\) to denote the closure of A with respect to Y, whenever Y is other than \(\beta S\).

New central sets theorem for commutative adequate partial semigroups

In this section, we establish a version of the Central Sets Theorem for commutative adequate partial semigroups that applies to all adequate sequences at once. We start with some relevant definitions.

Definition 2.1

Let \(W_1, W_2 \in {\mathcal {P}}_f(S)\), then define \(W_1 *W_2= \{w_1 *w_2 \mid w_1 \in W_1, w_2 \in W_2 \; \text {and} \; w_1 *w_2 \; \text {is defined}\}\).

Now we recall the following lemma from [7, Theorem 3.4]. For the sake of completeness we include the proof here.

Lemma 2.2

Let \((S, *)\) be a commutative adequate partial semigroup, let m, \(r\in {\mathbb {N}}\), and for each \(i \in \{1,2,\ldots ,m\}\), let \(f_i\) be an adequate sequence in S. Let p be a minimal idempotent in \(\delta S\), let \(B\in p\), and let \(W\in {\mathcal {P}}_{f}(S)\). There exist \(a\in \sigma (W)\) and \(L\in {\mathcal {P}}_{f}({\mathbb {N}})\) such that \(\min L>r\), and for each \(i\in \{1,2,\ldots ,m\}\), \(\prod _{t\in L} f_i(t)\in \phi (a)\) and \(a*\prod _{t\in L}f_i(t)\in B\).

Proof

Let \(\mathcal {D}={\mathcal {P}}_f(S)\times \{r+1,r+2, \ldots \}\) be a directed set with ordering defined by \((W_2,n_2) \ge (W_1,n_1)\) if \(W_1 \subseteq W_2\) and \(n_1 \le n_2\). For each \((W,n) \in \mathcal {D}\) we define to be the set of elements of the form \(\big (a *\prod _{t \in L}f_1(t), a *\prod _{t \in L}f_2(t), \ldots , a *\prod _{t \in L}f_{m}(t)\big )\), such that \(a \in \sigma (W)\) and \(L \in {\mathcal {P}}_f({\mathbb {N}})\) with \(\min L > n\) and \( \prod _{t \in L}f_i(t) \in \sigma (W *a)\) for every \(i \in \{1,2,\ldots ,m\}\). Define also the set \(E_{(W,n)}=I_{(W,n)} \cup \{(a, a, \ldots , a) : a \in \sigma (W)\}\).

Let . Let

$$\begin{aligned} I= \bigcap _{(W,n)\in {D}}cl_Y I_{(W,n)} \text { and let }E= \bigcap _{(W,n)\in {D}}cl_Y E_{(W,n)}\,. \end{aligned}$$

To see that \(I\ne \emptyset \), note that if \((W_1,n_1)\) and \((W_2,n_2)\) are in \({\mathcal D}\), then \(I_{(W_1 \cup W_2, \max \{n_1, n_2\})} \subseteq I_{(W_1,n_1)} \cap I_{(W_2,n_2)}\). It thus suffices to let \((W,n)\in {\mathcal D}\) and show that \(I_{(W,n)}\ne \emptyset \). Pick \(a \in \sigma (W)\) and for each \(i \in \{1,2,\ldots ,m \}\), pick a natural number \(k_i\) such that \(FP(\langle f_i(t) \rangle _{t=k_i}^{\infty }) \subseteq \sigma (W *a)\). Let \(k=\max \big \{k_i\mid i\in \{1,2,\ldots ,m\}\big \}\). Then \(\big (a*f_1(k),a*f_2(k),\ldots ,a*f_m(k)\big )\in I_{(W,n)}\). Since , that implies . Therefore, we have .

Let \(\vec {q}\), \(\vec {r}\in E\). We show that \(\vec {q} *\vec {r} \in E\) and if either \(\vec {q} \in I\) or \(\vec {r} \in I\), then \(\vec {q} *\vec {r} \in I\). Let \((W, n) \in \mathcal {D}\). Let U be a neighbourhood of \(\vec {q} *\vec {r}\) in Y. So for each \(i \in \{1,2, \ldots , m\}\), pick \(A_i \in q_i *r_i\) such that . By Lemma 1.14 pick for each i, \(B_i \in q_i\) such that \(\rho _{r_i}[\,\overline{B_i}\,] \subseteq \overline{A_i}\). Let . Now pick \(\vec {x} \in E_{(W,n)} \cap V\) with \(\vec {x} \in I_{(W,n)}\) if \(\vec {q} \in I\).

If \(\vec {x} \in I_{(W,n)}\), take \(\vec {x}=(a *\prod _{t \in L}f_1(t), a *\prod _{t \in L}f_2(t), \ldots , a *\prod _{t \in L}f_{m}(t))\), where \(a \in \sigma (W)\) and \(L \in {\mathcal {P}}_f({\mathbb {N}})\) such that \(\min L > n\) and \(\prod _{t \in L}f_i(t) \in \sigma (W *a)\) for every \(i \in \{1,2,\ldots ,m\}\). Let \(W^{\prime }=W *F\), where

$$\begin{aligned} F=\left\{ a, a *\prod _{t \in L}f_1(t), a *\prod _{t \in L}f_2(t), \ldots , a *\prod _{t \in L}f_{m}(t)\right\} \,, \end{aligned}$$

and let \(n^{\prime }= \max L\). Then \((W^{\prime } , n^{\prime }) \in \mathcal {D}\). We claim that \(\vec {x} *E_{(W^{\prime }, n^{\prime })} \subseteq I_{(W,n)}\). Let \(\vec {y} \in \vec {x} *E_{(W^{\prime }, n^{\prime })}\). So, \(\vec {y}= \vec {x} *\vec {z}\) for some \(\vec {z} \in E_{(W^{\prime }, n^{\prime })}=I_{(W^{\prime }, n^{\prime })} \cup \{(a^{\prime }, a^{\prime }, \ldots , a^{\prime }) \mid a^{\prime } \in \sigma (W^{\prime })\}\). First we consider \(\vec {z}=(a^{\prime }, a^{\prime }, \ldots , a^{\prime })\) where \(a^{\prime } \in \sigma (W^{\prime })\). Thus we obtain \(\vec {y}= (a *\prod _{t \in L}f_1(t) *a^{\prime }, a *\prod _{t \in L}f_2(t) *a^{\prime }, \ldots ,a *\prod _{t \in L}f_m(t) *a^{\prime })\). Since \(a^{\prime } \in \sigma (W^{\prime })= \sigma (W *F)\), therefore, for all \(w \in W\) and for each \(i \in \{1,2, \ldots ,m\}\), \(w *a *\prod _{t \in L}f_i(t) *a^{\prime }\) is defined. This gives \(a *a^{\prime } *\prod _{t \in L}f_i(t)=a *\prod _{t \in L}f_i(t) *a^{\prime } \in \sigma (W)\) and \(a *a^{\prime } \in \sigma (W)\). Thus we obtain \(\vec {y} \in I_{(W,n)}\). Next, we take \(\vec {z} \in I_{(W^{\prime }, n ^{\prime })}\). Thus we assume \(\vec {z} = (a^{\prime } *\prod _{t \in L^{\prime }}f_1(t), a^{\prime } *\prod _{t \in L^{\prime }}f_2(t), \ldots , a^{\prime } *\prod _{t \in L^{\prime }}f_m(t))\), where \(a^{\prime } \in \sigma (W^{\prime })\) and \(L^{\prime } \in {\mathcal {P}}_f({\mathbb {N}})\) satisfying \(\min L^{\prime } > n^{\prime }\) and \(\prod _{t \in L^{\prime }}f_i(t) \in \sigma (W^{\prime } *a^{\prime })\) for each \(i \in \{1,2, \ldots , m\}\). Then \(\vec {y}= (a *\prod _{t \in L}f_1(t) *a^{\prime } *\prod _{t \in L^{\prime }}f_1(t), a *\prod _{t \in L}f_2(t) *a^{\prime } *\prod _{t \in L^{\prime }}f_2(t), \ldots , a *\prod _{t \in L}f_m(t) *a^{\prime } *\prod _{t \in L^{\prime }}f_m(t))\), i.e.,

$$\begin{aligned} \vec {y}= \left( a *a^{\prime } *\prod _{t \in L \cup L^{\prime }}f_1(t), a *a^{\prime } *\prod _{t \in L \cup L^{\prime }}f_2(t), \ldots , a *a^{\prime } *\prod _{t \in L \cup L^{\prime }}f_m(t)\right) \,. \end{aligned}$$

For all \(w \in W\), and for each \(i \in \{1,2, \ldots ,m\}\), \(w *a *a^{\prime } *\prod _{t \in L \cup L^{\prime }}f_i(t)\) is defined and so \(a *a^{\prime } *\prod _{t \in L \cup L^{\prime }}f_i(t) \in \sigma (W)\) and \(a *a^{\prime } \in \sigma (W)\). Also note that \(\min (L \cup L^{\prime }) > n\). Thus \(\vec {y} \in I_{(W,n)}\). Therefore, we have \(\vec {x} *E_{(W^{\prime }, n^{\prime })} \subseteq I_{(W,n)}\).

If \(\vec {x} \in E_{(W,n)}{\setminus } I_{(W,n)}\), we take \(\vec {x}=(a, a, \ldots , a)\) where \(a \in \sigma (W)\). Let \(W^{\prime }=W *a \) and \(n^{\prime }=n\). Then \((W^{\prime } , n^{\prime }) \in \mathcal {D}\). Our next claim is that \(\vec {x} *E_{(W^{\prime } , n^{\prime })} \subseteq E_{(W,n)}\) and \(\vec {x} *I_{(W^{\prime } , n^{\prime })} \subseteq I_{(W,n)}\). Let \(\vec {y} \in \vec {x} *E_{(W^{\prime }, n^{\prime })}\), then, \(\vec {y}= \vec {x} *\vec {z}\), for some \(\vec {z} \in E_{(W^{\prime }, n^{\prime })}= I_{(W^{\prime }, n^{\prime })} \cup \{(a^{\prime }, a^{\prime }, \ldots , a^{\prime } ) \mid a^{\prime } \in \sigma (W^{\prime })\}\). If \(\vec {z} = (a^{\prime }, a^{\prime }, \ldots , a^{\prime })\) for some \(a^{\prime } \in \sigma (W^{\prime })\). Readily, \(\vec {y}=(a *a^{\prime }, a *a^{\prime }, \ldots , a *a^{\prime })\) and \(a *a^{\prime } \in \sigma (W)\) by the construction of \(W^{\prime }\). Now consider \(\vec {z} \in I_{(W^{\prime }, n^{\prime })}\). Then \(\vec {z}= (b *\prod _{t \in L^{\prime }} f_1(t), b *\prod _{t \in L^{\prime }} f_2(t), \ldots , b *\prod _{t \in L^{\prime }} f_m(t)) \), where \(b \in \sigma (W^{\prime })\) and \(L^{\prime } \in {\mathcal {P}}_f({\mathbb {N}})\) satisfying \(\min L^{\prime } > n^{\prime }\) and \(b *\prod _{t \in L^{\prime }} f_i(t) \in \sigma (W^{\prime })\) for all \(i \in \{1,2, \ldots , m\}\). Now \(\vec {y}= (a *b *\prod _{t \in L^{\prime }}f_1(t), a *b *\prod _{t \in L^{\prime }}f_2(t), \ldots , a *b *\prod _{t \in L^{\prime }}f_m(t ))\). Since for each \(i \in \{1,2,\ldots , m\}\), \(b *\prod _{t \in L^{\prime }}f_i(t) \in \sigma (W *a)\) then \(a *b *\prod _{t \in L^{\prime }}f_i(t) \in \sigma (W) \) and \(a *b \in \sigma (W)\). Therefore, \(\vec {x} *E_{(W^{\prime } , n^{\prime })} \subseteq E_{(W,n)}\) and \(\vec {x} *I_{(W^{\prime } , n^{\prime })} \subseteq I_{(W,n)}\).

Since , we have for each \(i \in \{1,2, \ldots ,m\}\) that \(x_i *r_i \in \overline{A_i}\) so \(x_i^{-1} A_i \in r_i\). Then is a neighbourhood of \(\vec {r}\) so pick \(\vec {y} \in O \cap E_{(W^{\prime }, n^{\prime })}\) with \(\vec {y} \in I_{(W^{\prime }, n^{\prime })}\) if \(\vec {r} \in I\). Then \(\vec {x} *\vec {y} \in U \cap E_{(W,n)}\) and if either \(\vec {q} \in I\) or \(\vec {r} \in I\), then \(\vec {x} *\vec {y} \in U \cap I_{(W,n)}\). This gives E is a subsemigroup of and I is an ideal of E.

Now \(p \in K(\delta S)\). Our next claim is that \(\overline{p}=(p, p, \ldots , p) \in E\). Let U be a neighbourhood of \(\overline{p}\) in Y and let \((W,n)\in {\mathcal D}\). Pick \(C \in p\) such that . Now \(p \in \delta S \subseteq cl_{\beta S} \sigma (W)\), therefore, \(\sigma (W) \in p\). So, we can pick \(a \in C \cap \sigma (W)\), and therefore, \((a, a, \ldots , a) \in U \cap E_{(W,n)}\). By [5, Theorem 2.23] we have that , so . Now by [5, Theorem 1.65], we have that \(\overline{p} \in K(E)\) and, since I is an ideal of E, we have \(\overline{p} \in I\). Since is a neighbourhood of \(\overline{p}\), then we have . \(\square \)

Now we are in a situation to prove a stronger version of the Central Sets Theorem for commutative adequate partial semigroups.

Definition 2.3

Let \((S,*)\) be a commutative adequate partial semigroup. Then \({\mathcal {T}}\) is the set of all adequate sequences in S.

Theorem 2.4

Let \((S,*)\) be a commutative adequate partial semigroup and let C be a central subset of S. There exist functions \(\alpha :{\mathcal {P}}_{f}({\mathcal {T}})\rightarrow S\) and \(H:{\mathcal {P}}_{f}({\mathcal {T}})\rightarrow {\mathcal {P}}_{f}({\mathbb {N}})\) such that

  1. (1)

    \(F,G\in {\mathcal {P}}_{f}({\mathcal {T}})\) and \(F\subsetneq G\), then \(\max H(F) < \min H(G)\) and

  2. (2)

    Whenever \(m\in {\mathbb {N}}\), \(G_{1},G_{2},\ldots ,G_{m}\in {\mathcal {P}}_{f}({\mathcal {T}})\), \(G_{1}\subsetneq G_{2}\subsetneq \cdots \subsetneq G_{m}\), and for each \(i\in \{1,2,\ldots ,m\}\), \(f_i \in G_{i}\), one has\(\prod _{i=1}^{m}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_i(t))\in C\).

Proof

Let p be a minimal idempotent in \(\delta S\), let \(C\in p\), and let \(C^{\star }=\{x\in C:x^{-1}C\in p\}\). Then \(C^{\star }\in p\) and by Lemma 1.17, if \(x\in C^{\star }\), then \(x^{-1}C^{\star }\in p\).

We define \(\alpha (F)\in S\) and \(H(F)\in {\mathcal {P}}_{f}({\mathbb {N}})\) for \(F\in {\mathcal {P}}_{f}({\mathcal {T}})\) by using induction on |F|, the cardinality of F, satisfying the following inductive hypotheses:

  1. (1)

    If \(\emptyset \ne G\subsetneq F\), then \(\max H(G) < \min H(F)\) and

  2. (2)

    If \(n\in {\mathbb {N}}\), \(\emptyset \ne G_{1}\subsetneq G_{2}\subsetneq \cdots \subsetneq G_{n}=F\), and , then \(\prod _{i=1}^{n}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_i(t))\in C^{\star }\).

This will suffice to prove the theorem.

First we assume that \(|F|=1\) so that \(F=\{f\}\) for some \(f \in {\mathcal {T}}\). By Lemma 2.2 pick \(a\in S\) and \(L\in {\mathcal {P}}_{f}({\mathbb {N}})\) such that \(\prod _{t\in L}f(t)\in \phi (a)\) and \(a*\prod _{t\in L}f(t)\in C^{\star }\). Put \(\alpha (F)=a\) and \(H(F)=L\). Hypothesis (1) holds vacuously. To verify hypothesis (2), let \(n \in {\mathbb {N}}\), \(\emptyset \ne G_1 \subsetneq G_2 \subsetneq \cdots \subsetneq G_n=F\), and . Then necessarily \(n=1\) and \(f_n=f\), so \( \prod _{i=1}^{n}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_i(t)) =a *\prod _{t \in L} f(t) \in C^\star \).

Now assume that \(|F|>1\) and \(\alpha (G)\) and H(G) have been defined for every nonempty proper subset G of F. Let \(K=\bigcup \{H(G):\emptyset \ne G\subsetneq F\}\) and let \(r=\max K\). Define

Then by hypothesis (2), M is a finite subset of \(C^\star \). Let \(A=C^{\star }\cap \bigcap _{x\in M}x^{-1}C^{\star }\). Then \(A\in p\). Using Lemma 2.2, pick \(a\in \sigma (M)\) and \(L\in {\mathcal {P}}_{f}({\mathbb {N}})\) such that \(\min L>r\), and for each \(f\in F\), \(\prod _{t\in L}f(t)\in \phi (a)\) and \(a*\prod _{t\in L}f(t)\in A\). Let \(\alpha (F)=a\) and \(H(F)=L\).

Since \(\min L > r\), hypothesis (1) holds. To verify hypothesis (2), let \(n\in {\mathbb {N}}\), let \(\emptyset \ne G_{1}\subsetneq G_{2}\subsetneq \cdots \subsetneq G_{n}=F\), and let . If \(n=1\), then \(\prod _{i=1}^{n}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_{i}(t)) =a*\prod _{t\in L}f(t)\in A \subseteq C^\star \).

Now assume that \(n>1\) and let \(y = \prod _{i=1}^{n-1}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_{i}(t))\). Then \(y\in M\), so \(a \in \sigma (M) \subseteq \phi (y)\) and \(a *\prod _{t \in L} f_n(t) \in A \subseteq y^{-1}C^\star \). Therefore, \(\prod _{i=1}^{n}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_{i}(t))= y *a *\prod _{t \in L}f_n(t) \in C^\star \). \(\square \)

We show now that Theorem 2.4 does generalize [7, Theorem 3.4], assuming only that there are infinitely many adequate sequences in S. Unless \(|S|=1\), there are certainly infinitely many sequences. And it is easy to show that if S is countably infinite, then for each \(a\in S\), there is an adequate sequence f with \(f(1)=a\).

Corollary 2.5

Let \((S, *)\) be a commutative adequate partial semigroup and assume that \({{\mathcal {T}}}\) is infinite. Let \(k \in {\mathbb {N}}\), for each \(l \in \{1,2, \ldots , k\}\) let \(f_l \in {\mathcal {T}}\), and let C be central in S. There exist a sequence \(\langle a_n \rangle _{n=1}^{\infty }\) in S and a sequence \(\langle H_n \rangle _{n=1}^{\infty }\) in \({\mathcal {P}}_f({\mathbb {N}})\) such that \(\max H_n < \min H_{n+1}\) for all n and for each \(g : {\mathbb {N}} \rightarrow \{1,2, \ldots , k\}\), \(FP(\langle a_n *\prod _{t \in H_n}f_{g(n)}(t) \rangle _{n=1}^{\infty }) \subseteq C\).

Proof

Pick functions \(\alpha \) and H as guaranteed by Theorem 2.4 for C. Now choose for each \(n\in {{\mathbb {N}}}\), \(\gamma _n \in {\mathcal {T}} {\setminus } \{f_1, \ldots , f_k \}\) such that \(\gamma _n \ne \gamma _m\) if \(n \ne m\). For \(n \in {\mathbb {N}}\) define \(G_n = \{f_1,f_2 \ldots , f_k \} \cup \{\gamma _1, \gamma _2,\ldots , \gamma _n\}\). Also, define \(a_n = \alpha (G_n)\) and \(H_n = H(G_n)\) for \(n \in {\mathbb {N}}\). Since \(G_n\subsetneq G_{n+1}\) for each n, we have \(\max H_n<\min H_{n+1}\) for each n.

Now let \(g : {\mathbb {N}} \rightarrow \{1, 2, \ldots , k \}\) and let \(F\in {\mathcal P}_f({{\mathbb {N}}})\). Enumerate F in order as \(\langle i(j)\rangle _{j=1}^m\). Then \(\emptyset \ne G_{i(1)}\subsetneq G_{i(2)}\subsetneq \cdots \subsetneq G_{i(m)}\), and for each \(j\in \{1,2,\ldots ,m\}\), \(f_{g(i(j))}\in G_{i(j)}\) so

$$\begin{aligned} \prod _{n\in F}\left( a_n *\prod _{t \in H_n}f_{g(n)}(t)\right) = \prod _{j=1}^m\left( \alpha (G_{i(j)})*\prod _{t\in H(G_{i(j)})}f_{g(i(j))}(t)\right) \in C\,. \end{aligned}$$

\(\square \)

Remark 2.6

The author has recently learned that a result nearly identical to Theorem 2.4 has been independently proved by Pleasant [8].

Sets satisfying the new central sets theorem

We start this section by defining a new notion of sets which satisfy the conclusion of Theorem 2.4.

Definition 3.1

Let \((S,*)\) be a commutative adequate partial semigroup. Then a set A is said to be a C-set if and only if there exist functions \(\alpha :{\mathcal {P}}_{f}({\mathcal {T}})\rightarrow S\) and \(H:{\mathcal {P}}_{f}({\mathcal {T}})\rightarrow {\mathcal {P}}_{f}({\mathbb {N}})\) such that

  1. (1)

    If \(F,G\in {\mathcal {P}}_{f}({\mathcal {T}})\) and \(F\subsetneq G\), then \(\max H(F) < \min H(G)\) and

  2. (2)

    Whenever \(m\in {\mathbb {N}}\), \(G_{1},G_{2},\ldots ,G_{m}\in {\mathcal {P}}_{f}({\mathcal {T}})\), \(G_{1}\subsetneq G_{2}\subsetneq \cdots \subsetneq G_{m}\) and for each \(i\in \{1,2,\ldots ,m\}\), \(f_i \in G_{i}\), one has\(\prod _{i=1}^{m}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_i(t)\in A\).

Definition 3.2

Let \((S,*)\) be a commutative adequate partial semigroup.

(a) A set \(A\subseteq S\) is a \(J_{\delta }\)-set if and only if whenever \(F\in {\mathcal {P}}_{f}({\mathcal {T}})\), \(W\in {\mathcal {P}}_{f}(S)\), there exist \(a\in \sigma (W)\) and \(H\in {\mathcal {P}}_{f}({\mathbb {N}})\) such that for each \(f\in F\), \(\prod _{t\in H}f(t)\in \sigma (W*a)\) and \(a*\prod _{t\in H}f(t)\in A\).

(b) \(J_{\delta }(S)=\{p\in \delta S:\) for all \(A\in p\,,\,A\) is a \(J_{\delta }\)-set}.

Lemma 3.3

Let \((S,*)\) be a commutative adequate partial semigroup and let A be a \(J_{\delta }\)-set in S. Then for all \(F\in {\mathcal {P}}_{f}({\mathcal {T}})\), and all \(r\in {\mathbb {N}}\) and \(W\in {\mathcal {P}}_{f}(S)\), there exist \(a\in \sigma (W)\) and \(H\in {\mathcal {P}}_{f}({\mathbb {N}})\) such that \(\min H>r\) and for all \(f\in F\), \(\prod _{t\in H}f(t)\in \sigma (W*a)\) and \(a*\prod _{t\in H}f(t)\in A\).

Proof

Consider \(F\in {\mathcal {P}}_{f}({{\mathcal {T}}})\), \(r\in {\mathbb {N}}\) and \(W\in {\mathcal {P}}_{f}(S)\). For \(f\in F\), define \(g_{f}\in {\mathcal {T}}\) by \(g_{f}(t)=f(r+t)\), for all \(t\in {\mathbb {N}}\). Since A is a \(J_{\delta }\)-set, then there exist \(a\in \sigma (W)\) and \(K\in {\mathcal {P}}_{f}({\mathbb {N}})\) such that \(\prod _{t\in K}g_{f}(t)\in \sigma (W*a)\) and \(a*\prod _{t\in K}g_{f}(t)\in A\). Using the first condition we get \(\prod _{t\in K}f(r+t)\in \sigma (W*a)\). By translating the variable we obtain \(\prod _{t\in r+K}f(t)\in \sigma (W*a)\). Similarly the second condition yields \(a*\prod _{t\in K}f(r+t)\in A\) which turns out to be \(a*\prod _{t\in H}f(t)\in A \) where \(H=r+K\). \(\square \)

Theorem 3.4

Let \((S,*)\) be a commutative adequate partial semigroup, let \(A\subseteq S\). If there is an idempotent \(p\in \overline{A}\cap J_{\delta }(S)\), then A is a C-set.

Proof

Let p be an idempotent in \(\delta S\) and let \(p\in \overline{A}\cap J_{\delta }(S)\). Let \(A^{\star }=\{a\in A:a^{-1}A\in p\}\), then \(A^{\star }\in p\) and by Lemma 1.17, for every \(a\in A^{\star }\), \(a^{-1}A^{\star }\in p\).

We define \(\alpha (F) \in S\) and \(H(F) \in {\mathcal {P}}_f({\mathbb {N}})\) for \(F \in {\mathcal {P}}_f({\mathcal {T}})\) by using induction on |F| satisfying the following inductive hypotheses:

  1. (1)

    If \(\emptyset \ne G\subsetneq F\), then \(\max H(G)<\min H(F)\) and

  2. (2)

    If \(m\in {\mathbb {N}}\), \(G_{1},G_{2},\ldots ,G_{m}\in {\mathcal {P}}_{f}({\mathcal {T}})\), \(G_{1}\subsetneq G_{2}\subsetneq \cdots \subsetneq G_{m}=F\) and for each \(i\in \{1,2,\ldots ,m\}\), \(f_{i}\in G_{i}\), then\( \prod _{i=1}^{m}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_i(t))\in A^{\star }\).

First assume that \(F=\{f\}\). Since \(A^\star \in p\), \(A^\star \) is a \(J_{\delta }\)-set. Pick \(W\in {\mathcal {P}}_{f}(S)\). (W does not enter in to the argument at this stage.) Since \(A^{\star }\) is a \(J_{\delta }\)-set, pick some \(a\in \sigma (W)\) and \(L\in {\mathcal {P}}_{f}({\mathbb {N}})\) such that \(\prod _{t\in L}f(t)\in \sigma (W*a)\) and \(a*\prod _{t \in L}f(t)\in A^{\star }\). Now let \(\alpha (F)=a\) and \(H(F)=L\). Then hypothesis (1) holds vacuously. To verify hypothesis (2), let \(n \in {\mathbb {N}}\), and assume that \(\emptyset \ne G_1 \subsetneq G_2 \subsetneq \cdots \subsetneq G_n=F\), and . Then \(n=1\) and \(f_n=f\) so \(\prod _{i=1}^{n}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_i(t))= a *\prod _{t \in L} f(t) \in A^\star \).

Now assume that \(|F|>1\) and \(\alpha (K)\) and H(K) have been chosen for all K with \(\emptyset \ne K\subsetneq F\). Let \(r=\max \bigcup \{H(K):\emptyset \ne K\subsetneq F\}\). Let

Then by hypothesis (2), M is finite and \(M\subseteq A^{\star }\). Let \(B=A^{\star }\cap \big (\bigcap _{x\in M}x^{-1}A^{\star }\big )\). Then \(B\in p\) and thus B is a \(J_{\delta }\)-set. Pick by Lemma 3.3, \(a \in \sigma (M)\) and \(L \in {\mathcal {P}}_{f}({\mathbb {N}})\) such that \(\min L>r\), and for all \(f\in F\), \(\prod _{t \in L}f(t) \in \sigma (M *a)\) and \(a *\prod _{t\in L}f(t)\in B\). Let \(\alpha (F)= a\) and let \(H(F)= L\).

Since \(\min L > r\), hypothesis (1) holds. To verify hypothesis (2), let \(n \in {\mathbb {N}}\), let \(\emptyset \ne G_1 \subsetneq G_2 \subsetneq \cdots \subsetneq G_n=F\), and let . If \(n=1\), then \(\prod _{i=1}^{n}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_{i}(t))= a *\prod _{t \in L}f(t) \in B \subseteq A^\star \). Now assume that \(n > 1\) and let \(y= \prod _{i=1}^{n-1}(\alpha (G_{i})*\prod _{t\in H(G_{i})}f_{i}(t))\). Then \(y \in M\). Therefore, \(a \in \sigma (M) \subseteq \phi (y)\) and \(\prod _{t \in L} f_n(t) \in \sigma (M *a) \) and \(a *\prod _{t \in L} f_n(t) \in B \subseteq y^{-1}A^\star \). Therefore, \(\prod _{i=1}^n(\alpha (G_i)\prod _{t \in H(G_i)} f_i(t))= y *a *\prod _{t \in L} f_n(t) \in A^\star \). \(\square \)

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Acknowledgements

The author would like to thank the referee for careful reading of the draft and enormous valuable suggestions to the improvements of the article. She also thanks her advisor Prof. Dibyendu De for his guidance and suggestions. Finally, the author acknowledges support received from the UGC-NET research grant.

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Correspondence to Arpita Ghosh.

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Communicated by Anthony To-Ming Lau.

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Ghosh, A. A generalized central sets theorem in partial semigroups. Semigroup Forum 100, 169–179 (2020). https://doi.org/10.1007/s00233-018-9977-7

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Keywords

  • Central sets theorem
  • Partial semigroups
  • Algebraic structure of Stone–Čech compactification