Finite semigroups and periodic sums systems in $\beta \mathbb {N}$ and their Ramsey theoretic consequences

Abstract

Let $m,n\ge 2$ and define $\nu :\omega \rightarrow \{0,\ldots ,m-1\}$ by $\nu (k)\equiv k\pmod {m}$. We construct some new finite semigroups in $\beta \mathbb {N}$, in particular, a semigroup generated by m elements of order n with cardinality $m^n+m^{n-1}+\cdots +m$. We also show that, for $n\ge m$, there is a sequence $p_0,\ldots ,p_{m-1}$ in $\beta \mathbb {N}$ such that all sums $\sum _{j=i}^{i+k}p_{\nu (j)}$, where $i\in \{0,\ldots ,m-1\}$ and $k\in \{0,\ldots ,n-1\}$, are distinct and $\sum _{j=i}^{i+n}p_{\nu (j)}=\sum _{j=i}^{i+n-m}p_{\nu (j)}$ for each i. As consequences we derive some new Ramsey theoretic results. In particular, we show that, for $n\ge m$, there is a partition $\{A_{i,k}:(i,k)\in \{0,\ldots ,m-1\}\times \{0,\ldots ,n-1\}\}$ of $\mathbb {N}$ such that, whenever for each (i, k), $\mathscr {B}_{i,k}$ is a finite partition of $A_{i,k}$, there exist $B_{i,k}\in \mathscr {B}_{i,k}$ and a sequence $(x_j)_{j=0}^\infty$ such that for every finite sequence $j_0<\ldots <j_s$ such that $j_{t+1}\equiv j_t+1\pmod {m}$ for each $t<s$, one has $x_{j_0}+\cdots +x_{j_s}\in B_{i_0,k_0}$, where $i_0=\nu (j_0)$ and $k_0$ is s if $s\le n-1$ and $n-m+\nu (s-n)$ otherwise.

1 Introduction

The addition of the discrete semigroup $\mathbb {N}$ of natural numbers extends to the Stone–Čech compactification $\beta \mathbb {N}$ of $\mathbb {N}$ so that for each $a\in \mathbb {N}$, the left translation $\lambda _a:\beta \mathbb {N}\ni x\mapsto a+x\in \beta \mathbb {N}$ is continuous, and for each $q\in \beta \mathbb {N}$, the right translation $\rho _q:\beta \mathbb {N}\ni x\mapsto x+q\in \beta \mathbb {N}$ is continuous.

We take the points of $\beta \mathbb {N}$ to be the ultrafilters on $\mathbb {N}$, identifying the principal ultrafilters with the points of $\mathbb {N}$. For every $A\subseteq \mathbb {N}$, $\overline{A}=\{p\in \beta \mathbb {N}:A\in p\}$ and $A^*=\overline{A}\setminus A$. The subsets $\overline{A}$, where $A\subseteq \mathbb {N}$, form a base for the topology of $\beta \mathbb {N}$, and $\overline{A}$ is the closure of A. For $p,q\in \beta \mathbb {N}$, the ultrafilter $p+q$ has a base consisting of subsets of the form $\bigcup _{x\in A}(x+B_x)$, where $A\in p$ and for each $x\in A$, $B_x \in q$.

Being a compact Hausdorff right topological semigroup, $\beta \mathbb {N}$ has a smallest two sided ideal $K(\beta \mathbb {N})$ which is a disjoint union of minimal right ideals and a disjoint union of minimal left ideals. Every right (left) ideal of $\beta \mathbb {N}$ contains a minimal right (left) ideal, the intersection of a minimal right ideal and a minimal left ideal is a group, and the idempotents in a minimal right (left) ideal form a right (left) zero semigroup, that is, $x+y=y$ ($x+y=x$) for all x, y.

The semigroup $\beta \mathbb {N}$ has important applications to Ramsey theory and to topological dynamics. The first application to Ramsey theory was the proof of Hindman’s theorem: whenever $\mathbb {N}$ is finitely colored, there is an infinite sequence all of whose sums are monochrome. An elementary introduction to $\beta \mathbb {N}$ can be found in [4].

In 1979, E. van Douwen asked (in [3], published much later) whether there are topological and algebraic copies of $\beta \mathbb {N}$ contained in $\mathbb {N}^*=\beta \mathbb {N}\setminus \mathbb {N}$. This question was answered in the negative by D. Strauss in [7], where it was in fact established that continuous homomorphisms from $\beta \mathbb {N}$ to $\mathbb {N}^*$ have finite images. It follows that if $\varphi :\beta \mathbb {N}\rightarrow \mathbb {N}^*$ is a continuous homomorphism, then $\varphi (\beta \mathbb {N})$ is a finite cyclic semigroup generated by $p=\varphi (1)$. That is, there are $n\ge 1$ and $1\le m\le n$ called the order and the period of p (and of the cyclic semigroup) such that all $ip=\underbrace{p+\cdots +p}_{i}$, where $i\in \{1,\ldots ,n\}$, are distinct and $(n+1)p=(n+1-m)p$. Conversely, every element $p\in \mathbb {N}^*$ of finite order determines a continuous homomorphism $\varphi :\beta \mathbb {N}\rightarrow \mathbb {N}^*$ by $\varphi (1)=p$. In 1996, the author proved that $\beta \mathbb {N}$ contains no nontrivial finite groups (see [4, Theorem 7.17]). Since the periodic part of a cyclic semigroup is a group, it follows that if $p\in \beta \mathbb {N}$ is an element of order n, then $(n+1)p=np$, that is, p has period 1.

As distinguished from finite groups, $\beta \mathbb {N}$ does contain bands (semigroups of idempotents): for example, left zero semigroups, right zero semigroups, chains of idempotents (with respect to the order $x\le y$ if and only if $x+y=y+x=x$), and even rectangular bands (direct products of a left zero semigroup and a right zero semigroup). To ask whether $\beta \mathbb {N}$ contains a finite semigroup distinct from bands is the same as asking whether $\beta \mathbb {N}$ contains an element of order 2 which is the same as asking whether there exists a nontrivial continuous homomorphism from $\beta \mathbb {N}$ to $\mathbb {N}^*$ [4, Question 10.19]. If the answer to this question is positive, then there is a subset A of $\mathbb {N}$ with the following Ramsey theoretic property: whenever A is finitely colored, there is an infinite sequence in the complement of A, all of whose sums two or more terms at a time are monochrome [2].

The question whether $\beta \mathbb {N}$ contains an element of order 2 was solved in the affirmative in [8, Theorem 1]. In [9], some further finite semigroups in $\beta \mathbb {N}$ consisting of idempotents and elements of order 2 were constructed, in particular, null semigroups ($x+y=0$ for all x, y), and a connection of finite semigroups in $\beta \mathbb {N}$ with Ramsey theory was discussed, see also [1]. In [12], it was shown that for every $m\ge 1$, the direct product of the m-element null semigroup and the rectangular band $2^\mathfrak {c}\times 2^\mathfrak {c}$ has copies in $\beta \mathbb {N}$ (that the rectangular band $2^\mathfrak {c}\times 2^\mathfrak {c}$ has copies in $\beta \mathbb {N}$ was established in [5]).

The question whether $\beta \mathbb {N}$ contains an element of finite order $n\ge 3$ was solved in the affirmative in [10, Theorem 3]. In fact, it was shown that for any $m\ge 1$ and $n\ge 2$, $\beta \mathbb {N}$ contains copies of the semigroup $C_{m,n}$ generated by the elements $q=q_1$, $q_2,\ldots ,q_m$ with defining relations $(n+1)q=nq$ and $q_s+q_t=2q$, where $s,t\in \{1,\ldots ,m\}$. (If $m=1$, this is the cyclic semigroup of order n and period 1, and if $n=2$, this is the m-element null semigroup.) In [13], it was shown that for any $m\ge 1$ and $n\ge 2$, the direct product of the semigroup $C_{m,n}$ and the left zero semigroup $2^\mathfrak {c}$ has copies in $\beta \mathbb {N}$.

Let $m,n\ge 2$ and define $\nu :\omega \rightarrow \{0,\ldots ,m-1\}$ by $\nu (k)\equiv k\pmod {m}$.

In [6], it was shown that there is a sequence $p_0,\ldots ,p_{m-1}$ in $\beta \mathbb {N}$ such that all sums $\sum _{j=i}^{i+k}p_{\nu (j)}$, where $i\in \{0,\ldots ,m-1\}$ and $k\in \{0,\ldots ,mn-1-i\}$ for each i, are distinct and $\sum _{j=i}^{mn}p_{\nu (j)}=\sum _{j=i}^{mn-m}p_{\nu (j)}$ for each i.

In this paper, we construct some new finite semigroups in $\beta \mathbb {N}$, in particular, a semigroup generated by m elements of order n with cardinality $m^n+m^{n-1}+\cdots +m$. In fact, we construct large locally finite semigroups. The construction is given in Sect. 2.

In Sect. 3, using those semigroups, we show that, for $n\ge m$, there is a sequence $p_0,\ldots ,p_{m-1}$ in $\beta \mathbb {N}$ such that all sums $\sum _{j=i}^{i+k}p_{\nu (j)}$, where $i\in \{0,\ldots ,m-1\}$ and $k\in \{0,\ldots ,n-1\}$, are distinct and $\sum _{j=i}^{i+n}p_{\nu (j)}=\sum _{j=i}^{i+n-m}p_{\nu (j)}$ for each i. We also discuss all possible finite systems of such periodic sums.

And in Sect. 4, we derive some new Ramsey theoretic results. In particular, we show that, for $n\ge m$, there is a partition $\{A_{i,k}:(i,k)\in \{0,\ldots ,m-1\}\times \{0,\ldots ,n-1\}\}$ of $\mathbb {N}$ such that, whenever for each (i, k), $\mathscr {B}_{i,k}$ is a finite partition of $A_{i,k}$, there exist $B_{i,k}\in \mathscr {B}_{i,k}$ and a sequence $(x_j)_{j=0}^\infty$ such that for every finite sequence $j_0<\ldots <j_s$ such that $j_{t+1}\equiv j_t+1\pmod {m}$ for each $t<s$, one has $x_{j_0}+\cdots +x_{j_s}\in B_{i_0,k_0}$, where $i_0=\nu (j_0)$ and $k_0$ is s if $s\le n-1$ and $n-m+\nu (s-n)$ otherwise.

2 Construction of semigroups

Let $m\ge 1$, $n\ge 2$, and $l=m+n-1$. For every $x\in \mathbb {N}$, $\hbox {supp }x$ is a unique finite nonempty subset of $\omega =\mathbb {N}\cup \{0\}$ such that

$$\begin{aligned} x=\sum _{k\in \hbox {supp }x}2^k. \end{aligned}$$

Pick an increasing sequence $I_0\subseteq I_1\subseteq \ldots \subseteq I_l=\omega$ of subsets of $\omega$ such that $I_i\setminus I_{i-1}$ is infinite for each $i\in \{0,1,\ldots ,l\}$ (with $I_{-1}=\emptyset$). Define a function h from $\mathbb {N}$ onto the decreasing chain $0>1>\ldots >l$ of idempotents (with the operation $i*j=\max \{i,j\}$) by

$$\begin{aligned} h(x)=\min \{i\le l:\hbox {supp }x\subseteq I_i\}=\max \{i\le l:(\hbox {supp }x)\cap (I_i{\setminus } I_{i-1})\ne \emptyset \} \end{aligned}$$

and let the same letter h denote its continuous extension $\beta \mathbb {N}\rightarrow \{0,1,\ldots ,l\}$. If $x,y\in \mathbb {N}$ and $\max \hbox {supp }x<\min \hbox {supp }y$, then $h(x+y)=h(x)*h(y)$. It then follows (see [4, Theorem 4.21]) that for any $u\in \beta \mathbb {N}$ and $v\in \mathbb {H}$, where

$$\begin{aligned} \mathbb {H}=\bigcap _{n=0}^\infty \overline{2^n\mathbb {N}}, \end{aligned}$$

one has $h(u+v)=h(u)*h(v)$, in particular, the restriction of h to $\mathbb {H}$ is a homomorphism. For each $i\in \{0,1,\ldots ,l\}$, let

$$\begin{aligned} T_i=h^{-1}(\{0,1,\ldots ,i\})\cap \mathbb {H}. \end{aligned}$$

Then $T_0\subseteq T_1\subseteq \ldots \subseteq T_l=\mathbb {H}$ is an increasing sequence of closed subsemigroups of $\mathbb {H}$ such that $h(K(T_i))=\{i\}$ for each $i\le l$, and so $T_i\cap \overline{K(T_{i+1})}=\emptyset$ for each $i<l$ and $K(T_l)=K(\beta \mathbb {N})\cap T_l$ [9, Lemma 3.1], in particular, all $K(T_0),K(T_1),\ldots ,K(T_l)$ are pairwise disjoint. Moreover, $h(K(\beta \mathbb {N}))=\{l\}$, and so $T_{l-1}\cap \overline{K(\beta \mathbb {N})}=\emptyset$.

To see this, let $u\in K(\beta \mathbb {N})$. Then $u+\beta \mathbb {N}$ is the minimal right ideal of $\beta \mathbb {N}$ containing u and $\beta \mathbb {N}+u$ the minimal left ideal containing u. Let v be the identity of the group $(u+\beta \mathbb {N})\cap (\beta \mathbb {N}+u)$. Then $u=u+v$ and $v\in K(\mathbb {H})$, so $h(u)=h(u+v)=h(u)*h(v)=h(u)*l=l$.

For each $i\in \{0,1,\ldots ,l\}$, let

$$\begin{aligned} X_i=\{x\in \mathbb {N}:(\hbox {supp }x)\cap (I_i\setminus I_{i-1})\ne \emptyset \}. \end{aligned}$$

Notice that for any $v\in \overline{X_i}\cap \mathbb {H}$ and $u\in \beta \mathbb {N}$, $u+v\in \overline{X_i}$, and for any $v\in \overline{X_i}$ and $w\in \mathbb {H}$, $v+w\in \overline{X_i}$.

Define $\phi _i:X_i\rightarrow \omega$ by

$$\begin{aligned} \phi _i(x)=\max ((\hbox {supp }x)\cap (I_i\setminus I_{i-1})) \end{aligned}$$

and let the same letter $\phi _i$ denote its continuous extension $\overline{X_i}\rightarrow \beta \omega$. Notice that $\{2^k:k\in I_i\setminus I_{i-1}\}\subseteq X_i$ and, since $\phi _i(2^k)=k$, $\phi _i$ homeomorphically maps $\overline{\{2^k:k\in I_i\setminus I_{i-1}\}}$ onto $\overline{I_i\setminus I_{i-1}}$. If $x\in \mathbb {N}$, $y\in X_i$ and $\max \hbox {supp }x<\min \hbox {supp }y$, then $x+y\in X_i$ and $\phi _i(x+y)=\phi _i(y)$. And if $y\in X_i$, $z\in \mathbb {N}\setminus X_i$ and $\max \hbox {supp }y<\min \hbox {supp }z$, then $\phi _i(y+z)=\phi _i(y)$. It then follows that for any $v\in \overline{X_i}\cap \mathbb {H}$ and $u\in \beta \mathbb {N}$, $\phi _i(u+v)=\phi _i(v)$, and for any $v\in \overline{X_i}$ and $w\in \mathbb {H}\setminus \overline{X_i}$, $\phi _i(v+w)=\phi _i(v)$.

To see for example the first statement, we first note that for any $x\in \mathbb {N}$ and $v\in \overline{X_i}\cap \mathbb {H}$, $\phi _i(x+v)=\phi _i(v)$ because the continuous functions $\phi _i\circ \lambda _x$ and $\phi _i$ agree on $X_i\cap 2^n\mathbb {N}$, where $n=(\max \hbox {supp }x)+1$. Then for any $v\in \overline{X_i}\cap \mathbb {H}$ and $u\in \beta \mathbb {N}$, $\phi _i(u+v)=\phi _i(v)$ because the continuous function $\phi _i\circ \rho _v$ is constantly equal to $\phi _i(v)$ on $\mathbb {N}$.

Notice that $K(T_i)\subseteq \overline{X_i}\cap \mathbb {H}$ and $T_{i-1}\subseteq \mathbb {H}\setminus \overline{X_i}$ (with $T_{-1}=\emptyset$).

We shall construct

• (i) a chain $e_0>e_1>\ldots >e_l$ of idempotents with $e_i\in K(T_i)$,

• (ii) for each $i\in \{0,1,\ldots ,l\}$, a left zero semigroup $\{e_{i,\alpha }:\alpha <2^\mathfrak {c}\}\subseteq K(T_i)$ such that $e_{i,0}=e_i$ and $e_{i,\alpha }=e_{0,\alpha }+e_i$ for all $\alpha <2^\mathfrak {c}$, and

• (iii) for each $i\in \{1,m+1,\ldots ,l-1\}$ (for $i=1$ if $n=2$), a right zero semigroup $\{e_i(j):j\in \omega \}\subseteq K(T_i)$ such that $e_i(0)=e_i$, $e_i(j)<e_{i-1}$ for all $j\in \omega$, and $\phi _i(e_i(j))\ne \phi _i(e_i(k))$ if $j\ne k$.

Notice that (i) and (ii) imply that

$$\begin{aligned} e_{i,\alpha }+e_{j,\beta }=e_{i*j,\alpha } \end{aligned}$$

for all $i,j\in \{0,1,\ldots ,l\}$ and $\alpha ,\beta <2^\mathfrak {c}$.

Indeed,

$$\begin{aligned} e_{i,\alpha }+e_{j,\beta }&=e_{0,\alpha }+e_i+e_{0,\beta }+e_j=e_{0,\alpha }+(e_i+e_0)+e_{0,\beta }+e_j\\&=e_{0,\alpha }+e_i+(e_0+e_{0,\beta })+e_j=e_{0,\alpha }+e_i+e_0+e_j\\&=e_{0,\alpha }+e_{i*j}=e_{i*j,\alpha }. \end{aligned}$$

The construction goes by induction on $i\in \{0,1,\ldots ,l\}$.

For $i=0$, pick an injective $2^\mathfrak {c}$-sequence $\{r_{0,\alpha }:\alpha <2^\mathfrak {c}\}$ in $\{2^k:k\in I_0\}^*$.

Lemma 2.1

$(r_{0,\alpha }+T_l)\cap (r_{0,\beta }+T_l)=\emptyset$ if $\alpha \ne \beta$.

Proof

Consider the function $\mathbb {N}\ni x\mapsto \min \hbox {supp }x\in \omega$ and let $\theta$ denote its continuous extension $\beta \mathbb {N}\rightarrow \beta \omega$. If $x,y\in \mathbb {N}$ and $\max \hbox {supp }x<\min \hbox {supp }y$, then $\theta (x+y)=\theta (x)$. It then follows that for any $u\in \beta \mathbb {N}$ and $v\in \mathbb {H}$, $\theta (u+v)=\theta (u)$. Consequently, $\theta (r_{0,\alpha }+T_l)=\{\theta (r_{0,\alpha })\}$ and $\theta (r_{0,\beta }+T_l)=\{\theta (r_{0,\beta })\}$. Since $\theta (2^k)=k$, $\theta (r_{0,\alpha })\ne \theta (r_{0,\beta })$, so $(r_{0,\alpha }+T_l)\cap (r_{0,\beta }+T_l)=\emptyset$. $\square$

For every $\alpha <2^\mathfrak {c}$, choose a minimal right ideal $R_{0,\alpha }$ of $T_0$ contained in $r_{0,\alpha }+T_0$. Pick a minimal left ideal $L_0$ of $T_0$, and for every $\alpha <2^\mathfrak {c}$, let $e_{0,\alpha }$ be the identity of the group $R_{0,\alpha }\cap L_0$. By Lemma 2.1, $e_{0,\alpha }\ne e_{0,\beta }$ if $\alpha \ne \beta$. Put $e_0=e_{0,0}$.

For $i=1$, choose a minimal right ideal $R_1$ of $T_1$ contained in $e_0+T_1$. Pick an injective sequence $(r_{1,j})_{j=0}^\infty$ in $\{2^k:k\in I_1\setminus I_0\}^*$, and for every $j\in \omega$, choose a minimal left ideal $L_{1,j}$ of $T_1$ contained in $T_1+r_{1,j}+e_0$. For every $j\in \omega$, let $e_1(j)$ be the identity of the group $R_1\cap L_{1,j}$. Then $\phi _1(e_1(j))=\phi _1(r_{1,j}+e_0)=\phi _1(r_{1,j})$, so $\phi _1$ is injective on $\{e_1(j):j\in \omega \}$. Since $e_1(j)\in e_0+T_1$, one has $e_0+e_1(j)=e_1(j)$, and since $e_1(j)\in T_1+r_{1,j}+e_0$, one has $e_1(j)+e_0=e_1(j)$, so $e_1(j)<e_0$. Put $e_1=e_1(0)$. For every $\alpha <2^\mathfrak {c}$, put $e_{1,\alpha }=e_{0,\alpha }+e_1$. Then $e_{1,\alpha }+e_{1,\beta }=e_{0,\alpha }+e_1+e_{0,\beta }+e_1=e_{0,\alpha }+(e_1+e_0)+e_{0,\beta }+e_1=e_{0,\alpha }+e_1+(e_0+e_{0,\beta })+e_1=e_{0,\alpha }+e_1+e_0+e_1=e_{0,\alpha }+e_1=e_{1,\alpha }$, so $\{e_{1,\alpha }:\alpha <2^\mathfrak {c}\}$ is a left zero semigroup (in $K(T_1)$). Since $e_{1,\alpha }=e_{0,\alpha }+e_1\in r_{0,\alpha }+T_0+e_1\in r_{0,\alpha }+T_1$, by Lemma 2.1, $e_{1,\alpha }\ne e_{1,\beta }$ if $\alpha \ne \beta$.

For $i\in \{2,\ldots ,m\}$, pick a minimal right ideal $R_i$ of $T_i$ contained in $e_{i-1}+T_i$ and a minimal left ideal $L_i$ of $T_i$ contained in $T_i+e_{i-1}$ and let $e_i$ be the identity of the group $R_i\cap L_i$. For every $\alpha <2^\mathfrak {c}$, let $e_{i,\alpha }=e_{0,\alpha }+e_i$. Then $\{e_{l,\alpha }:\alpha <2^\mathfrak {c}\}$ is a left zero semigroup and $e_{i,\alpha }\ne e_{i,\beta }$ if $\alpha \ne \beta$.

For $i\in \{m+1,\ldots ,l-1\}$ (for $n\ge 3$), choose a minimal right ideal $R_i$ of $T_i$ contained in $e_{i-1}+T_i$. Pick an injective sequence $(r_{i,j})_{j=0}^\infty$ in $\{2^k:k\in I_i\setminus I_{i-1}\}^*$, and for every $j\in \omega$, choose a minimal left ideal $L_{i,j}$ of $T_i$ contained in $T_i+r_{i,j}+e_{i-1}$, and let $e_i(j)$ be the identity of the group $R_i\cap L_{i,j}$. Then $\phi _i(e_i(j))=\phi _i(r_{i,j}+e_0)=\phi _i(r_{i,j})$, so $\phi _i$ is injective on $\{e_i(j):j\in \omega \}$, and $e_i(j)<e_{i-1}$ for all j. Put $e_i=e_i(0)$. For every $\alpha <2^\mathfrak {c}$, put $e_{i,\alpha }=e_{0,\alpha }+e_i$. Then $\{e_{i,\alpha }:\alpha <2^\mathfrak {c}\}$ a left zero semigroup and $e_{i,\alpha }\ne e_{i,\beta }$ if $\alpha \ne \beta$.

For $i=l$, pick a minimal right ideal $R_l$ of $T_l$ contained in $e_{l-1}+T_l$ and a minimal left ideal $L_l$ of $T_l$ contained in $T_l+e_{l-1}$ and let $e_l$ be the identity of the group $R_l\cap L_l$. For every $\alpha <2^\mathfrak {c}$, put $e_{l,\alpha }=e_{0,\alpha }+e_l$.

Now for each $\alpha <2^\mathfrak {c}$, let

$$\begin{aligned} D_{l-1,\alpha }={\left\{ \begin{array}{ll}\{e_{l,\alpha }+e_1(j):j\in \mathbb {N}\}&{}\text { if }n=2\\ \{e_{l,\alpha }+e_{l-1}(j):j\in \mathbb {N}\}&{}\text { if }n\ge 3. \end{array}\right. } \end{aligned}$$

Since $\phi _1(e_{l,\alpha }+e_1(j))=\phi _1(e_1(j))$ and $\phi _{l-1}(e_{l,\alpha }+e_{l-1}(j))=\phi _{l-1}(e_{l-1}(j))$, we have that if $n=2$, $\phi _1$ is injective on $D_{l-1,\alpha }$ (and so $|\phi _1(\overline{D_{l-1,\alpha }})|=2^\mathfrak {c}$) and if $n\ge 3$, $\phi _{l-1}$ is injective on $D_{l-1,\alpha }$ (and so $|\phi _{l-1}(\overline{D_{l-1,\alpha }})|=2^\mathfrak {c}$). For every $\alpha <2^\mathfrak {c}$, pick inductively $q_{l-1,\alpha }\in \overline{D_{l-1,\alpha }}\setminus D_{l-1,\alpha }$ such that

if $n=2$, $\phi _1(q_{l-1,\alpha })\ne \phi _1(e_1)$ and all $\phi _1(q_{l-1,\alpha })$ are distinct, and

if $n\ge 3$, $\phi _{l-1}(q_{l-1,\alpha })\ne \phi _{l-1}(e_{l-1})$ and all $\phi _{l-1}(q_{l-1,\alpha })$ are distinct.

Then by downward induction on $i\in \{m+1,\ldots ,l-2\}$ (for $n\ge 4$), for each $\alpha <2^\mathfrak {c}$, let

$$\begin{aligned} D_{i,\alpha }=\{e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j):j\in \mathbb {N}\}. \end{aligned}$$

Since $\phi _i(e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j))=\phi _i(e_i(j))$, $\phi _i$ is injective on $D_{i,\alpha }$. For every $\alpha <2^\mathfrak {c}$, pick inductively $q_{i,\alpha }\in \overline{D_{i,\alpha }}\setminus D_{i,\alpha }$ such that

$\phi _i(q_{i,\alpha })\ne \phi _i(e_i)$ and all $\phi _i(q_{i,\alpha })$ are distinct.

For $i=m$ (for $n\ge 3$), for each $\alpha <2^\mathfrak {c}$, let

$$\begin{aligned} D_{m,\alpha }=\{e_{m+1,\alpha }+q_{m+1,\alpha }+e_1(j):j\in \mathbb {N}\}. \end{aligned}$$

Since $\phi _1(e_{m+1,\alpha }+q_{m+1,\alpha }+e_1(j))=\phi _1(e_1(j))$, $\phi _1$ is injective on $D_{m,\alpha }$. For every $\alpha <2^\mathfrak {c}$, pick inductively $q_{m,\alpha }\in \overline{D_{m,\alpha }}\setminus D_{m,\alpha }$ such that

$\phi _1(q_{m,\alpha })\ne \phi _1(e_m)$ and all $\phi _1(q_{m,\alpha })$ are distinct.

Since $e_{l,\alpha }\in K(\beta \mathbb {N})$ and $\overline{K(\beta \mathbb {N})}$ is an ideal of $\beta \mathbb {N}$ [4, Theorem 4.44], we have by downward induction that for each $i\in \{m,\ldots ,l-1\}$, $D_{i,\alpha }\subseteq \overline{K(\beta \mathbb {N})}$ and $q_{i,\alpha }\in \overline{K(\beta \mathbb {N})}$.

For each $s\in \{0,1,\ldots ,l\}$, $e_{l,\alpha }=e_{s,\alpha }+e_{l,\alpha }$ and $e_{s,\alpha }\in \overline{X_s}$, so $e_{l,\alpha }\in \overline{X_s}$. It then follows by downward induction that for each $i\in \{m,\ldots ,l-1\}$, $D_{i,\alpha }\subseteq \overline{X_s}\cap \mathbb {H}$ and $q_{i,\alpha }\in \overline{X_s}\cap \mathbb {H}$. We also have that $\phi _1$ is injective on $D_{m,\alpha }$ and for each $i\in \{m+1,\ldots ,l-1\}$ (for $n\ge 3$), $\phi _i$ is injective on $D_{i,\alpha }$.

An ultrafilter $q\in \mathbb {N}^*$ is right cancelable (in $\beta \mathbb {N}$) if the right translation of $\beta \mathbb {N}$ by q is injective. An ultrafilter $q\in \mathbb {N}^*$ is right cancelable if and only if $q\notin \mathbb {N}^*+q$ [4, Theorem 8.18]. From the next lemma we obtain that all $q_{i,\alpha }$, where $i\in \{m,\ldots ,l-1\}$ and $\alpha <2^\mathfrak {c}$, are right cancelable.

Lemma 2.2

Let $i\in \{0,1,\ldots ,l\}$, let D be a countable subset of $\overline{X_i}\cap \mathbb {H}$, and suppose that $\phi _i$ is injective on D. Then every $q\in \overline{D}\setminus D$ is right cancelable.

Proof

This is [10, Lemma 5]. $\square$

The next lemma gives us relations between $q_{i,\alpha }$ and $e_{s,\beta }$.

Lemma 2.3

For any $\alpha ,\beta <2^\mathfrak {c}$,

• (1) $q_{l-1,\alpha }+e_{l-1,\beta }=e_{l,\alpha }$,

• (2) if $n=2$, then for each $s\in \{1,\ldots ,l\}$, $q_{l-1,\alpha }+e_{s,\beta }=e_{l,\alpha }$,

• (3) if $n\ge 3$, then for each $i\in \{m+1,\ldots ,l-1\}$, $q_{i,\alpha }+e_{i-1,\beta }=q_{i,\alpha }$,

• (4) if $n\ge 3$, then for each $i\in \{m,\ldots ,l-2\}$, $q_{i,\alpha }+e_{i,\beta }=e_{i+1,\alpha }+q_{i+1,\alpha }$, and

• (5) if $n\ge 3$, then for each $s\in \{1,\ldots ,m\}$, $q_{m,\alpha }+e_{s,\beta }=e_{m+1,\alpha }+q_{m+1,\alpha }$.

Proof

• (1) For $n\ge 3$, $(e_{l,\alpha }+e_{l-1}(j))+e_{l-1,\beta }=e_{l,\alpha }+(e_{l-1}(j)+e_{l-1,\beta })=e_{l,\alpha }+((e_{l-1}(j)+e_{l-2,0})+e_{l-1,\beta })=e_{l,\alpha }+(e_{l-1}(j)+(e_{l-2,0}+e_{l-1,\beta }))=e_{l,\alpha }+e_{l-1}(j)+e_{l-1,0}=e_{l,\alpha }+e_{l-1,0}=e_{l,\alpha }$, and since $\rho _{e_{l-1,\beta }}$ is constantly equal to $e_{l,\alpha }$ on $D_{l-1,\alpha }$, $\rho _{e_{l-1,\beta }}(q_{l-1,\alpha })=e_{l,\alpha }$, so $q_{l-1,\alpha }+e_{l-1,\beta }=e_{l,\alpha }$. The case $n=2$ is included in (2).

• (2) $(e_{l,\alpha }+e_1(j))+e_{s,\beta }=e_{l,\alpha }+(e_1(j)+e_{0,0})+e_{s,\beta }=e_{l,\alpha }+e_1(j)+(e_{0,0}+e_{s,\beta })=e_{l,\alpha }+e_1(j)+e_{s,0}=e_{l,\alpha }+e_1(j)+(e_{1,0}+e_{s,0})=e_{l,\alpha }+(e_1(j)+e_{1,0})+e_{s,0}=e_{l,\alpha }+e_{1,0}+e_{s,0}=e_{l,\alpha }+e_{s,0}=e_{l,\alpha }$.

• (3) For $i=l-1$, $(e_{l,\alpha }+e_{l-1}(j))+e_{l-2,\beta }=e_{l,\alpha }+(e_{l-1}(j)+e_{l-2,0})+e_{l-2,\beta }=e_{l,\alpha }+e_{l-1}(j)+(e_{l-2,0}+e_{l-2,\beta })=e_{l,\alpha }+e_{l-1}(j)+e_{l-2,0}=e_{l,\alpha }+e_{l-1}(j)$, and since $\rho _{e_{l-2,\beta }}$ is the identity on $D_{l-1,\alpha }$, $\rho _{e_{l-2,\beta }}(q_{l-1,\alpha })=q_{l-1,\alpha }$, so $q_{l-1,\alpha }+e_{l-2,\beta }=q_{l-1,\alpha }$. For $i\le l-2$, $(e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j))+e_{i-1,\beta }=e_{i+1,\alpha }+q_{i+1,\alpha }+(e_i(j)+e_{i-1,0})+e_{i-1,\beta }=e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j)+(e_{i-1,0}+e_{i-1,\beta })=e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j)+e_{i-1,0}=e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j)$.

• (4) For $i\ge m+1$, $(e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j))+e_{i,\beta }=e_{i+1,\alpha }+q_{i+1,\alpha }+(e_i(j)+e_{i-1,0})+e_{i,\beta }=e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j)+(e_{i-1,0}+e_{i,\beta })=e_{i+1,\alpha }+q_{i+1,\alpha }+e_i(j)+e_{i,0}=e_{i+1,\alpha }+q_{i+1,\alpha }+e_{i,0}=e_{i+1,\alpha }+q_{i+1,\alpha }$ because $q_{i+1,\alpha }+e_{i,0}=q_{i+1,\alpha }$ by (3). The case $i=m$ is included in (5).

• (5) $e_{m+1,\alpha }+q_{m+1,\alpha }+e_1(j)+e_{s,\beta }=e_{m+1,\alpha }+q_{m+1,\alpha }+(e_1(j)+e_{0,0})+e_{s,\beta }=e_{m+1,\alpha }+q_{m+1,\alpha }+e_1(j)+(e_{0,0}+e_{s,\beta })=e_{m+1,\alpha }+q_{m+1,\alpha }+e_1(j)+e_{s,0}=e_{m+1,\alpha }+q_{m+1,\alpha }+e_1(j)+(e_{1,0}+e_{s,0})=e_{m+1,\alpha }+q_{m+1,\alpha }+(e_1(j)+e_{1,0})+e_{s,0}=e_{m+1,\alpha }+q_{m+1,\alpha }+e_{1,0}+e_{s,0}=e_{m+1,\alpha }+q_{m+1,\alpha }+e_{s,0}=e_{m+1,\alpha }+q_{m+1,\alpha }$ because by (3), $q_{m+1,\alpha }+e_{s,0}=(q_{m+1,\alpha }+e_{m,0})+e_{s,0}=q_{m+1,\alpha }+(e_{m,0}+e_{s,0})=q_{m+1,\alpha }+e_{m,0}=q_{m+1,\alpha }$.

$\square$

From Lemma 2.3 we obtain that for each $i\in \{m,\ldots ,l-1\}$ and each $s\in \{1,\ldots ,l\}$,

$$\begin{aligned} q_{i,\alpha }+e_{s,\beta }={\left\{ \begin{array}{ll}e_{l,\alpha }&{}\text { if }m=l-1\\ e_{m+1,\alpha }+q_{m+1,\alpha }&{}\text { if }s\le i=m\le l-2\\ q_{i,\alpha }\text { if }i\ge m+1&{}\text { and }s<i\\ e_{s+1,\alpha }+q_{s+1,\alpha }&{}\text { if }i\le s\le l-2\\ e_{l,\alpha }&{}\text { if }l-1\le s\le l. \end{array}\right. } \end{aligned}$$

Indeed, the first and the second cases are Lemma 2.3(2) and Lemma 2.3(5) respectively.

In the third case, using Lemma 2.3(3), $q_{i,\alpha }+e_{s,\beta }=(q_{i,\alpha }+e_{i-1,0})+e_{s,\beta }=q_{i,\alpha }+(e_{i-1,0}+e_{s,\beta })=q_{i,\alpha }+e_{i-1,0}=q_{i,\alpha }$.

The fourth case for $i=s$ is Lemma 2.3(4). Then by downward induction on $i\in \{m,m+1,s\}$, for $i<s$, $q_{i,\alpha }+e_{s,\beta }=q_{i,\alpha }+(e_{i,\beta }+e_{s,\beta })=(q_{i,\alpha }+e_{i,\beta })+e_{s,\beta }=e_{i+1,\alpha }+q_{i+1,\alpha }+e_{s,\beta }=e_{i+1,\alpha }+(q_{i+1,\alpha }+e_{s,\beta })=e_{i+1,\alpha }+e_{s+1,\alpha }+q_{s+1,\alpha }=e_{s+1,\alpha }+q_{s+1,\alpha }$.

The fifth case for $i=s=l-1$ is Lemma 2.3(1). For $i\le l-2$, using the already established fourth case, $q_{i,\alpha }+e_{l-1,\beta }=q_{i,\alpha }+e_{l-2,\beta }+e_{l-1,\beta }=e_{l-1,\alpha }+q_{l-1,\alpha }+e_{l-1,\beta }=e_{l-1,\alpha }+e_{l,\alpha }=e_{l,\alpha }$. Then for each i, $q_{i,\alpha }+e_{l,\beta }=q_{i,\alpha }+e_{l-1,\beta }+e_{l,\beta }=e_{l,\alpha }+e_{l,\beta }=e_{l,\alpha }$.

Now consider the subsemigroup Q of $\mathbb {H}$ generated algebraically by the elements $e_{i,\alpha }$ and $q_{s,\beta }$, where $i\in \{1,\ldots ,l\}$, $s\in \{m,\ldots ,l-1\}$, and $\alpha ,\beta <2^\mathfrak {c}$ (we have interchanged i and s, and so are $\alpha$ and $\beta$). It follows from the formula above that Q consists of elements of the form

$$\begin{aligned} e_{i,\alpha }, \ q_{s_1,\beta _1}+\ldots +q_{s_t,\beta _t},\text { and }e_{i,\alpha }+q_{s_1,\beta _1}+\ldots +q_{s_t,\beta _t}, \end{aligned}$$

where $i\in \{1,\ldots ,l\}$, $t\in \mathbb {N}$, $s_1,\ldots ,s_t\in \{m,\ldots ,l-1\}$, and $\alpha ,\beta _1,\ldots ,\beta _t<2^\mathfrak {c}$.

Lemma 2.4

All elements

$$\begin{aligned} e_{i,\alpha }, \ q_{s_1,\beta _1}+\ldots +q_{s_t,\beta _t},\text { and }e_{i,\alpha }+q_{s_1,\beta _1}+\ldots +q_{s_t,\beta _t}, \end{aligned}$$

where $i\in \{1,\ldots ,l\}$, $t\in \mathbb {N}$, $s_1,\ldots ,s_t\in \{m,\ldots ,l-1\}$, and $\alpha ,\beta _1,\ldots ,\beta _t<2^\mathfrak {c}$, are distinct.

Proof

Assume on the contrary that some two distinct expressions represent the same element. Then canceling the equality by q-s we arrive at one of the following cases:

• (1) $u+q_{i,\alpha }=v+q_{s,\beta }$ for some $u,v\in \beta \mathbb {N}$ and $(i,\alpha )\ne (s,\beta )$,

• (2) $u+q_{i,\alpha }=q_{s,\beta }$ for some $u\in \beta \mathbb {N}$,

• (3) $u+q_{i,\alpha }=e_{s,\beta }$ for some $u\in \beta \mathbb {N}$,

• (4) $e_{i,\alpha }=e_{s,\beta }$ with $(i,\alpha )\ne (s,\beta )$.

The last one is obviously impossible.

In (1), we have that $\phi _i(q_{i,\alpha })=\phi _i(u+q_{i,\alpha })=\phi _i(v+q_{s,\beta })=\phi _i(q_{s,\beta })$. If $i=s$, then $\alpha \ne \beta$ and $\phi _i(q_{i,\alpha })=\phi _i(q_{i,\beta })$, a contradiction. If $i\ne s$, say $i<s$, then $\phi _i(q_{s,\beta })=\phi _i(q_{s,\beta }+e_{i,0})=\phi _i(e_{i,0})$ and $\phi _i(q_{i,\alpha })\ne \phi _i(e_{i,0})$, again a contradiction.

In (2), since $q_{s,\beta }$ is right cancelable, one has $s\ne i$. Suppose $i<s$. Then $\phi _i(q_{i,\alpha })=\phi _i(q_{s,\beta })$. But $\phi _i(q_{s,\beta })=\phi _i(e_{i,0})$ (as in (1)) and $\phi _i(q_{i,\alpha })\ne \phi _i(e_{i,0})$, a contradiction. The case $s<i$ is essentially the same, since applying $\phi _s$ to $q_{s,\beta }=u+q_{i,\alpha }$ gives us $\phi _s(q_{s,\beta })=\phi _s(q_{i,\alpha })$.

In (3), since $q_{i,\alpha }\in \overline{K(\beta \mathbb {N})}$, $e_1,\ldots ,e_{l-1}\in T_{l-1}$ and $T_{l-1}\cap \overline{K(\beta \mathbb {N})}=\emptyset$, one has $s=l$. Then $\phi _i(q_{i,\alpha })=\phi _i(e_{l,\beta })$. But $\phi _i(e_{l,\beta })=\phi _i(e_{l,\beta }+e_{i,0})=\phi _i(e_{i,0})$ and $\phi _i(q_{i,\alpha })\ne \phi _i(e_{i,0})$, a contradiction. $\square$

From Lemma 2.4 we obtain that

Corollary 2.5

As an abstract semigroup, Q is generated by the chain of left zero semigroups $\{e_{i,\alpha }:\alpha <2^\mathfrak {c}\}$, where $i\in \{1,\ldots ,l\}$ and for each $i\le l-1$, $e_{i,\alpha }+e_{i+1,\beta }=e_{i+1,\alpha }$ and $e_{i+1,\beta }+e_{\alpha ,i}=e_{i+1,\beta }$, and elements $q_{s,\beta }$, where $s\in \{m,\ldots ,l-1\}$ and $\beta <2^\mathfrak {c}$, with the defining relations (1)-(5) in Lemma 2.3.

Now consider the subsemigroup P of Q generated by the elements

$$\begin{aligned} p_{s,\alpha ,\beta }=e_{s,\alpha }+q_{m,\beta }, \end{aligned}$$

where $s\in \{1,\ldots ,m\}$ and $\alpha ,\beta <2^\mathfrak {c}$.

Lemma 2.6

For all $i\ge 2$, $s_1,\ldots ,s_i\in \{1,\ldots ,m\}$, and $\alpha _1,\beta _1\ldots ,\alpha _i,\beta _i<2^\mathfrak {c}$,

$$\begin{aligned} p_{s_i,\alpha _i,\beta _i}+\ldots +p_{s_1,\alpha _1,\beta _1}={\left\{ \begin{array}{ll}e_{m+i-1,\alpha _i}+q_{m+i-1,\beta _i}+\ldots +q_{m,\beta _1}&{}\text { if }i\le n-1\\ e_{l,\alpha _i}+q_{l-1,\beta _{n-1}}+\ldots +q_{m,\beta _1}&{}\text { otherwise.} \end{array}\right. } \end{aligned}$$

Proof

We use Lemma 2.3. If $n=2$, then

$$\begin{aligned} p_{s_2,\alpha _2,\beta _2}+p_{s_1,\alpha _1,\beta _1}&=e_{s_2,\alpha _2}+q_{m,\beta _2}+e_{s_1,\alpha _1}+q_{m,\beta _1}\\&=e_{s_2,\alpha _2}+(q_{m,\beta _2}+e_{s_1,\alpha _1})+q_{m,\beta _1}\\&=e_{s_2,\alpha _2}+e_{l,\beta _2}+q_{m,\beta _1}=e_{l,\alpha _2}+q_{m,\beta _1}\text { and}\\ p_{s_3,\alpha _3,\beta _3}+p_{s_2,\alpha _2,\beta _2}+p_{s_1,\alpha _1,\beta _1}&=(p_{s_3,\alpha _3,\beta _3}+p_{s_2,\alpha _2,\beta _2})+p_{s_1,\alpha _1,\beta _1}\\&=e_{l,\alpha _3}+q_{m,\beta _2}+e_{s_1,\alpha _1}+q_{m,\beta _1}\\&=e_{l,\alpha _3}+(q_{m,\beta _2}+e_{s_1,\alpha _1})+q_{m,\beta _1}\\&=e_{l,\alpha _3}+e_{l,\beta _2}+q_{m,\beta _1}=e_{l,\alpha _3}+q_{m,\beta _1}. \end{aligned}$$

Let $n\ge 3$. We first notice that for each $j\in \{1,\ldots ,n-2\}$,

$$\begin{aligned} q_{m+j-1,\beta _j}+\ldots +q_{m,\beta _1}+e_{s,\alpha }&=e_{m+j,\beta _j}+q_{m+j,\beta _j}+\ldots +q_{m+1,\beta _1}\text { and }\\ q_{l-1,\beta _{n-1}}+\ldots +q_{m,\beta _1}+e_{s,\alpha }&=e_{l,\beta _{n-1}}+q_{l-1,\beta _{n-2}}+\ldots +q_{m+1,\beta _1}. \end{aligned}$$

Indeed, inductively, $q_{m,\beta _1}+e_{s,\alpha }=e_{m+1,\beta _1}+q_{m+1,\beta _1}$, and for $j\ge 2$,

$$\begin{aligned} q_{m+j-1,\beta _j}+\ldots +q_{m,\beta _1}+e_{s,\alpha }&=q_{m+j-1,\beta _j}+(q_{m+j-2,\beta _{j-1}}+\ldots +q_{m,\beta _1}+e_{s,\alpha })\\&=q_{m+j-1,\beta _j}+e_{m+j-1,\beta _{j-1}}+q_{m+j-1,\beta _{j-1}}\\&\quad +\ldots +q_{m+1,\beta _1}\\&=e_{m+j,\beta _j}+q_{m+j,\beta _j}+q_{m+j-1,\beta _{j-1}}+\ldots \\&\quad +q_{m+1,\beta _1}, \end{aligned}$$

and then

$$\begin{aligned} q_{l-1,\beta _{n-1}}+\ldots +q_{m,\beta _1}+e_{s,\alpha }&=q_{l-1,\beta _{n-1}}+(q_{l-2,\beta _{n-2}}+\ldots +q_{m,\beta _1}+e_{s,\alpha })\\&=q_{l-1,\beta _{n-1}}+e_{l-1,\beta _{n-2}}+q_{l-1,\beta _{n-2}}+\ldots +q_{m+1,\beta _1}\\&=e_{l,\beta _{n-1}}+q_{l-1,\beta _{n-2}}+\ldots +q_{m+1,\beta _1}. \end{aligned}$$

Now by induction on $i\in \{2,\ldots ,n-1\}$,

$$\begin{aligned} p_{s_2,\alpha _2,\beta _2}+p_{s_1,\alpha _1,\beta _1}&=e_{s_2,\alpha _2}+q_{m,\beta _2}+e_{s_1,\alpha _1}+q_{m,\alpha _1}=e_{s_2,\alpha _2}\\&\quad +(q_{m,\beta _2}+e_{s_1,\alpha _1})+q_{m,\beta _1}\\&=e_{s_2,\alpha _2}+e_{m+1,\beta _2}+q_{m+1,\beta _2}+q_{m,\beta _1}=e_{m+1,\alpha _2}\\&\quad +q_{m+1,\beta _2}+q_{m,\beta _1}, \end{aligned}$$

and for $i\ge 2$,

$$\begin{aligned} p_{s_i,\alpha _i,\beta _i}+\ldots +p_{s_1,\alpha _1,\beta _1}&=(p_{s_i,\alpha _i,\beta _i}+\ldots +p_{s_2,\alpha _2,\beta _2})+p_{s_1,\alpha _1,\beta _1}\\&=e_{m+i-2,\alpha _i}+q_{m+i-2,\beta _i}+\ldots +q_{m,\beta _2}+e_{s_1,\alpha _1}+q_{m,\beta _1}\\&=e_{m+i-2,\alpha _i}+e_{m+i-1,\beta _i}+q_{m+i-1,\beta _i}+\ldots \\&\quad +q_{m+1,\beta _2}+q_{m,\beta _1}\\&=e_{m+i-1,\alpha _i}+q_{m+i-1,\beta _i}+\ldots +q_{m,\beta _1}, \end{aligned}$$

and then

$$\begin{aligned} p_{s_n,\alpha _n,\beta _n}+\ldots +p_{s_1,\alpha _1,\beta _1}&=(p_{s_n,\alpha _n,\beta _n}+\ldots +p_{s_2,\alpha _2,\beta _2})+p_{s_1,\alpha _1,\beta _1}\\&=e_{l-1,\alpha _n}+q_{l-1,\beta _n}+\ldots +q_{m,\beta _2}+e_{s_1,\alpha _1}+q_{m,\beta _1}\\&=e_{l-1,\alpha _n}+e_{l,\beta _n}+q_{l-1,\beta _{n-1}}+\ldots \\&\quad +q_{m+1,\beta _2}+q_{m,\beta _1}\\&=e_{l,\alpha _n}+q_{l-1,\beta _{n-1}}+\ldots +q_{m,\beta _1} \end{aligned}$$

and

$$\begin{aligned} p_{s_{n+1},\alpha _{n+1},\beta _{n+1}}+\ldots +p_{s_1,\alpha _1,\beta _1}&=(p_{s_{n+1},\alpha _{n+1},\beta _{n+1}}+\ldots +p_{s_2,\alpha _2,\beta _2})+p_{s_1,\alpha _1,\beta _1}\\&=e_{l,\alpha _{n+1}}+q_{l-1,\beta _n}+\ldots +q_{m,\beta _2}+e_{s_1,\alpha _1}+q_{m,\beta _1}\\&=e_{l,\alpha _{n+1}}+e_{l,\beta _n}+q_{l-1,\beta _{n-1}}+\ldots \\&\quad +q_{m+1,\beta _2}+q_{m,\beta _1}\\&=e_{l,\alpha _{n+1}}+q_{l-1,\beta _{n-1}}+\ldots +q_{m,\beta _1}. \end{aligned}$$

$\square$

It follows from Lemma 2.6 that the subsemigroup P consists of the elements

$$\begin{aligned} p_{s,\alpha ,\beta }, \ e_{m+i-1,\alpha }+q_{m+i-1,\beta _i}+\ldots +q_{m,\beta _1},\text { and }e_{l,\alpha }+q_{l-1,\beta _{n-1}}+\ldots +q_{m,\beta _1}, \end{aligned}$$

where $s\in \{1,\ldots ,m\}$, $2\le i\le n-1$, and $\alpha ,\beta _1\ldots ,\beta _{n-1}<2^\mathfrak {c}$, and by Lemma 2.4, all these elements are distinct. Notice that the elements $e_{l,\alpha }+q_{l-1,\beta _{n-1}}+\ldots +q_{m,\beta _1}$ form K(P). Since all $q_{j,\beta }$ are in $\overline{K(\beta \mathbb {N})}$, $P\subseteq \overline{K(\beta \mathbb {N})}$, and since $e_{l,\alpha }\in K(\beta \mathbb {N})$, $K(P)\subseteq K(\beta \mathbb {N})$. Also notice that the subsemigroup generated by $p_{s_1,\alpha _1,\beta _1},\ldots ,p_{s_i,\alpha _i,\beta _i}$ is finite. It then follows that P is locally finite, that is, every finitely generated subsemigroup is finite.

Given cardinals $\kappa \ge 1$ and $\lambda \ge 1$ and integers $m\ge 1$ and $n\ge 2$, let $S(\kappa ,\lambda ,m,n)$ denote the semigroup whose elements are the words $s\alpha \beta$, $\alpha \beta _i\ldots \beta _1$, and $*\alpha \beta _{n-1}\ldots \beta _1$, where $s\in \{1,\ldots ,m\}$, $2\le i\le n-1$, $\alpha \in \kappa$, and $\beta ,\beta _1,\ldots ,\beta _{n-1}\in \lambda$, and defining relations are, for $j\ge 2$,

$$\begin{aligned} s_j\alpha _j\beta _j+\ldots +s_1\alpha _1\beta _1={\left\{ \begin{array}{ll}\alpha _j\beta _j\ldots \beta _1&{}\text { if }j\le n-1\\ *\alpha _j\beta _{n-1}\ldots \beta _1&{}\text { otherwise,} \end{array}\right. } \end{aligned}$$

so $\alpha \beta _i\ldots \beta _1=1\alpha \beta _i+\ldots +1\alpha \beta _1$, and $*\alpha \beta _{n-1}\ldots \beta _1=1\alpha \beta _{n-1}+1\alpha \beta _{n-1}+\ldots +1\alpha \beta _1$. If $m=1$, we write $\alpha \beta$ instead of $1\alpha \beta$.

It is easy to see that the mapping $g:P\rightarrow S(2^\mathfrak {c},2^\mathfrak {c},m,n)$ defined by

$$\begin{aligned} g(p_{s,\alpha ,\beta })=s\alpha \beta , \ g(e_{m+i-1,\alpha }+q_{m+i-1,\beta _k}+\ldots +q_{m,\beta _1})=\alpha \beta _i\ldots \beta _1,\text { and }\\ g(e_{m+n-1,\alpha }+q_{m+n-2,\beta _{n-1}}+\ldots +q_{m,\beta _1})=*\alpha \beta _{n-1}\ldots \beta _1 \end{aligned}$$

is an isomorphism.

We thus have proved the following result.

Theorem 2.7

Let $m\ge 1$ and $n\ge 2$ and let $S=S(2^\mathfrak {c},2^\mathfrak {c},m,n)$. There is an isomorphic embedding $\varepsilon :S\rightarrow \mathbb {H}$. Furthermore, $\varepsilon$ can be chosen so that $\varepsilon (S)\subseteq \overline{K(\beta \mathbb {N})}$ and $\varepsilon (K(S))\subseteq K(\beta \mathbb {N})$.

For each $(\alpha ,\beta )\in \kappa \times \lambda$, the subsemigroup of $S(\kappa ,\lambda ,m,n)$ consisting of the elements $s\alpha \beta$, where $s\in \{1,\ldots ,m\}$, and $\alpha \beta \beta ,\ldots ,\alpha \underbrace{\beta \ldots \beta }_{n-1}, *\alpha \underbrace{\beta \ldots \beta }_{n-1}$ is isomorphic to the semigroup $C_{m,n}$. The semigroup $S(\kappa ,1,m,n)$ consists of the elements $s\alpha 0$ and

$$\begin{aligned} \alpha 00,\ldots ,\alpha \underbrace{0\ldots 0}_{n-1}, *\alpha \underbrace{0\ldots 0}_{n-1}, \end{aligned}$$

where $s\in \{1,\ldots ,m\}$ and $\alpha \in \kappa$, and is isomorphic to the direct product of $C_{m,n}$ and the left zero semigroup $\kappa$. The semigroup $S(\kappa ,\lambda ,m,2)$ consists of the elements $s\alpha \beta$ and $*\alpha \beta$, where $s\in \{1,\ldots ,m\}$ and $(\alpha ,\beta )\in \kappa \times \lambda$, and is isomorphic to the direct product of $C_{m,2}$ (the m-element null semigroup) and the rectangular band $\kappa \times \lambda$.

Now consider the subsemigroup T of $S=S(\kappa ,\kappa ,1,n)$ generated by the elements $\beta \beta$, where $\beta \in \kappa$. Since

$$\begin{aligned} \beta _j\beta _j+\ldots +\beta _1\beta _1={\left\{ \begin{array}{ll}\beta _j\beta _j\ldots \beta _1&{}\text { if }j\le n-1\\ *\beta _j\beta _{n-1}\ldots \beta _1&{}\text { otherwise,} \end{array}\right. } \end{aligned}$$

T consists of the words $\beta _i\beta _i\ldots \beta _1$ and $*\alpha \beta _{n-1}\ldots \beta _1$, where $1\le i\le n-1$ and $\alpha ,\beta _1,\ldots ,\beta _{n-1}\in \kappa$. Notice that $K(T)=K(S)$.

Given a cardinal $\kappa \ge 1$ and an integer $n\ge 2$, let $F(\kappa ,n)$ denote the semigroup whose elements are the words $\beta _i\ldots \beta _1$, where $1\le i\le n$ and $\beta _1,\ldots ,\beta _i\in \kappa$, and defining relations are

$$\begin{aligned} \beta _j+\ldots +\beta _1={\left\{ \begin{array}{ll}\beta _j\ldots \beta _1&{}\text { if }j\le n\\ \beta _j\beta _{n-1}\ldots \beta _1&{}\text { otherwise,} \end{array}\right. } \end{aligned}$$

so the operation of $F(\kappa ,n)$ is defined by

$$\begin{aligned} \beta _{i+t}\ldots \beta _{i+1}+\beta _i\ldots \beta _1={\left\{ \begin{array}{ll}\beta _{i+t}\ldots \beta _1&{}\text { if }i+t\le n\\ \beta _{i+t}\beta _{n-1}\ldots \beta _1&{}\text { otherwise.} \end{array}\right. } \end{aligned}$$

It is easy to see that the mapping $f:T\rightarrow F(\kappa ,n)$ defined by

$$\begin{aligned} f(\beta _i\beta _i\ldots \beta _1)=\beta _i\ldots \beta _1\text { and }f(*\alpha \beta _{n-1}\ldots \beta _1)=\alpha \beta _{n-1}\ldots \beta _1 \end{aligned}$$

is an isomorphism.

Thus, we obtain from Theorem 2.7 the following result.

Theorem 2.8

Let $n\ge 2$ and let $F=F(2^\mathfrak {c},n)$. There is an isomorphic embedding $\epsilon :F\rightarrow \mathbb {H}$. Furthermore, $\epsilon$ can be chosen so that $\epsilon (F)\subseteq \overline{K(\beta \mathbb {N})}$ and $\epsilon (K(F))\subseteq K(\beta \mathbb {N})$.

The semigroup $F(\kappa ,n)$ is generated by the 1-letter words $\beta$, where $\beta \in \kappa$, each of which is an element of order n and each $m\ge 1$ of which generate a subsemigroup of cardinality $m^n+m^{n-1}+\ldots +m$.

3 Periodic sums systems

Let $m\ge 2$ and define $\nu =\nu _m:\omega \rightarrow \{0,\ldots ,m-1\}$ by $\nu (k)\equiv k\pmod {m}$. Given a sequence $p_0,\ldots ,p_{m-1}$ in an additive semigroup, the periodic sums are sums of the form $\sum _{j=i}^{i+k}p_{\nu (j)}$, where $i\in \{0,\ldots ,m-1\}$ and $k\ge 0$, and $(\sum _{j=i}^{i+k}p_{\nu (j)})_{k=0}^\infty$ is the sequence of periodic sums with initial term $p_i$. Suppose that $\{\sum _{j=i}^{i+k}p_{\nu (j)}:k\ge 0\}$ is finite. Then $\sum _{j=i}^{i+m-1}p_{\nu (j)}$ is an element of finite order, say of order $s_i$ and period $t_i$, that is, all elements $k\sum _{j=i}^{i+m-1}p_{\nu (j)}$, where $k\in \{1,\ldots ,s_i\}$, are distinct and $(s_i+1)\sum _{j=i}^{i+m-1}p_{\nu (j)}=(s_i+1-t_i)\sum _{j=i}^{i+m-1}p_{\nu (j)}$. Notice that $k\sum _{j=i}^{i+m-1}p_{\nu (j)}=\sum _{j=i}^{i+km-1}p_{\nu (j)}$. It follows that there is a smallest $l_i$ in $\{s_im,\ldots ,(s_i+1)m-1\}$ such that $\sum _{j=i}^{i+l_i}p_{\nu (j)}=\sum _{j=i}^{i+l_i-t_im}p_{\nu (j)}$. We call $l_i$ and $t_im$ the order and the period of the sequence $(\sum _{j=i}^{i+k}p_{\nu (j)})_{k=0}^\infty$. If in addition all elements $\sum _{j=i}^{i+k}p_{\nu (j)}$, where $k\in \{0,\ldots ,l_i-1\}$, are distinct, then we call the sequence cyclic of order $l_i$ and period $t_im$.

Lemma 3.1

• (i) $t_i$ is the smallest $t\ge 1$ such that $\sum _{j=i}^{i+l}p_{\nu (j)}=\sum _{j=i}^{i+l-tm}p_{\nu (j)}$ for some $l\ge tm$,

• (ii) $l_i$ is the smallest $l\ge m$ such that $\sum _{j=i}^{i+l}p_{\nu (j)}=\sum _{j=i}^{i+l-tm}p_{\nu (j)}$ for some $t\ge 1$ with $tm\le l$.

Proof

• (i) Assume on the contrary that there is $t<t_i$ such that $\sum _{j=i}^{i+l'}p_{\nu (j)}=\sum _{j=i}^{i+l'-tm}p_{\nu (j)}$ for some $l'\ge tm$. It then follows that $\sum _{j=i}^{i+l}p_{\nu (j)}=\sum _{j=i}^{i+l-tm}p_{\nu (j)}$ for all $l\ge l'$. Pick $l=km-1\ge l'$ with $k\ge s_i+1$. Then $k\sum _{j=i}^{i+m-1}p_{\nu (j)}=\sum _{j=i}^{i+km-1}p_{\nu (j)}=\sum _{j=i}^{i+km-1-tm}p_{\nu (j)}=(k-t)\sum _{j=i}^{i+m-1}p_{\nu (j)}$. But we also have that $k\sum _{j=i}^{i+m-1}p_{\nu (j)}=(k-t_i)\sum _{j=i}^{i+m-1}p_{\nu (j)}$, because $\sum _{j=i}^{i+m-1}p_{\nu (j)}$ is an element of order $s_i$ and period $t_i$ and $k\ge s_i+1$. Consequently, $(k-t)\sum _{j=i}^{i+m-1}p_{\nu (j)}=(k-t_i)\sum _{j=i}^{i+m-1}p_{\nu (j)}$ and $(k-t)-(k-t_i)=t_i-t<t_i$, a contradiction.

• (ii) Assume on the contrary that there is $l'<l_i$ such that $\sum _{j=i}^{i+l'}p_{\nu (j)}=\sum _{j=i}^{i+l'-tm}p_{\nu (j)}$ for some t, and consequently, $\sum _{j=i}^{i+l}p_{\nu (j)}=\sum _{j=i}^{i+l-tm}p_{\nu (j)}$ for all $l\ge l'$. Then by (i), $t\ge t_i$. If $t>t_i$, then taking $l=(s_i+1)m-1$ gives us $(s_i+1)\sum _{j=i}^{i+m-1}p_{\nu (j)}=(s_i+1-t)\sum _{j=i}^{i+m-1}p_{\nu (j)}$, a contradiction. And if $t=t_i$, then $l'<s_im$, so taking $l=s_im-1$ gives us $s_i\sum _{j=i}^{i+m-1}p_{\nu (j)}=(s_i-t_i)\sum _{j=i}^{i+m-1}p_{\nu (j)}$, again a contradiction.

$\square$

The periodic sums system generated by the sequence $p_0,\ldots ,p_{m-1}$ is the subset S of the semigroup consisting of all periodic sums $\sum _{j=i}^{i+k}p_{\nu (j)}$, where $i<m$ and $k\ge 0$.

Lemma 3.2

Suppose that for some $i_0<m$, $\{\sum _{j=i_0}^{i_0+k}p_{\nu (j)}:k\ge 0\}$ is finite. Then

• (1) S is finite,

• (2) there are $t\ge 1$ and $l_i\ge tm$ for each $i<m$ such that $(\sum _{j=i}^{i+k}p_{\nu (j)})_{k=0}^\infty$ has order $l_i$ and period tm and $l_i\le l_{\nu (i+1)}+1$,

• (3) for each $i<m$, $\sum _{j=i}^{i+m-1}p_{\nu (j)}$ is an element of order $s_i=[\frac{l_i}{m}]$ and period t.

Proof

For (1) and (2), write $i_0=\nu (i_1+1)$ and suppose that $(\sum _{j=i_0}^{i_0+k}p_{\nu (j)})_{k=0}^\infty$ has order $l_{i_0}$ and period tm. From $\sum _{j=i_0}^{i_0+l_{i_0}}p_{\nu (j)}=\sum _{j=i_0}^{i_0+l_{i_0}-tm}p_{\nu (j)}$ we obtain that

$$\begin{aligned} \sum _{j=i_1}^{i_0+l_{i_0}}p_{\nu (j)}=p_{i_1}+\sum _{j=i_0}^{i_0+l_{i_0}}p_{\nu (j)}=p_{i_1}+\sum _{j=i_0}^{i_0+l_{i_0}-tm}p_{\nu (j)}=\sum _{j={i_1}}^{i_0+l_{i_0}-tm}p_{\nu (j)}. \end{aligned}$$

It follows that $\{\sum _{j=i_1}^{i_1+k}p_{\nu (j)}:k\ge 0\}$ is finite, and by Lemma 3.1, $(\sum _{j=i_1}^{i_1+k}p_{\nu (j)})_{k=0}^\infty$ has order $l_{i_1}\le l_{i_0}+1$ and period $t'm$ for some $t'\le t$. From

$$\begin{aligned}{} & {} \sum _{j=i_0}^{m-1+i_1+l_{i_1}}p_{\nu (j)}=\sum _{j=i_0}^{i_0+m-1}p_{\nu (j)}+\sum _{j=i_1}^{i_1+l_{i_1}}p_{\nu (j)}=\sum _{j=i_0}^{i_0+m-1}p_{\nu (j)}+\sum _{j=i_1}^{i_1+l_{i_1}-t'm}p_{\nu (j)}\\{} & {} \quad =\sum _{j=i_0}^{m-1+i_1+l_{i_1}-t'm}p_{\nu (j)}, \end{aligned}$$

we obtain that $t'\ge t$. Hence $t'=t$. Then write $i_1=\nu (i_2+1)$ and so on.

For (3), if s is the order of $\sum _{j=i}^{i+m-1}p_{\nu (j)}$, then $l_i\in \{sm,\ldots ,(s+1)m-1\}$, and since $s_im\in \{l_i-m+1,\ldots ,l_i\}$, one has $s=s_i$. $\square$

It follows from Lemma 3.2 that $|l_i-l_r|\le m-1$ and $|s_i-s_r|\le 1$ for all $i,r\in \{0,\ldots ,m-1\}$.

We call the m-tuple $(l_0,\ldots ,l_{m-1})$ and the number tm the order and the period of S.

Let S and $S'$ be two periodic sums systems generated by sequences $p_0,\ldots ,p_{m-1}$ and $q_0,\ldots ,q_{m-1}$ respectively. A mapping $h:S\rightarrow S'$ is a homomorphism if there is $s<m$ such that for each $i<m$ and each $k\ge 0$, $h(\sum _{j=i}^{i+k}p_{\nu (j)})=\sum _{i+s}^{i+s+k}q_{\nu (j)}$. An isomorphism is a bijective homomorphism. If S is finite of order $(l_0,l_1,\ldots ,l_{m-1})$ and period tm and $S'$ is isomorphic to S, then $S'$ is finite of order $(l_s,l_{\nu (s+1)},\ldots ,\ldots ,l_{\nu (s+m-1)})$ for some $s<m$ and period tm. If for each $i<m$, $(\sum _{j=i}^kp_{\nu (j)})_{k=i}^\infty$ is a cyclic sequence of order $l_i$ and period tm, and all these sequences are pairwise disjoint, then S is said to be a free finite periodic sums system of order $(l_0,l_1,\ldots ,l_{m-1})$ and period tm.

Lemma 3.3

Let any $m,l_0,\ldots ,l_{m-1},t\ge 1$ be given such that $tm\le l_i\le l_{\nu (i+1)}+1$ for each $i<m$ and consider the semigroup Q generated by elements $p_0,\ldots ,p_{m-1}$ with defining relations $\sum _{j=i}^{i+l_i}p_{\nu (j)}=\sum _{j=i}^{i+l_i-tm}p_{\nu (j)}$, where $i<m$. Then the periodic sums system in Q generated by the sequence $p_0,\ldots ,p_{m-1}$ is free of order $(l_0,\ldots ,l_{m-1})$ and period tm.

Proof

Let F be the free semigroup over the alphabet $\{0,\ldots ,m-1\}$ and let W be the subset of F consisting of words $i_0\ldots i_k$ such that $k\ge 0$ and $i_{s+1}=\nu (i_s+1)$ for each $s\le k-1$. For each $i\in \{0,\ldots ,m-1\}$ and $k\ge 0$, let w(i, k) denote the word $i_0\ldots i_k$ in W with $i_0=i$. Let V be the subset of W consisting of words w(i, k), where $i\in \{0,\ldots ,m-1\}$ and $k\le l_i-1$ for each i, and K(V) the subset of V consisting of words w(i, k), where $i\in \{0,\ldots ,m-1\}$ and $l_i-tm\le k\le l_i-1$ for each i.

Let $\delta$ be the smallest congruence on F generated by the relations $w(i,l_i)=w(i,l_i-tm)$, where $i\le m-1$ (that is, for all $v,w\in F$, $v\delta w$ if and only if v is derivable from w under those relations). Then $Q=F/\delta$ with $p_i=\overline{w(i,0)}$, where $\overline{w}$ denotes the congruence class of w, and $\sum _{j=i}^{i+k}p_{\nu (j)}=\overline{w(i,k)}$. Clearly, for every $w\in W$, $\overline{w}\subseteq W$ and $\overline{w}\cap V\ne \emptyset$. Also for every $v\in \overline{w}$, v and w have the same first and last letters and $|v|\equiv |w|\pmod {tm}$. It then follows that for all distinct $v,w\in K(V)$, $\overline{v}\cap \overline{w}=\emptyset$. We claim that for each $w\in V\setminus K(V)$, $\overline{w}=\{w\}$, and consequently, for all distinct $v,w\in V$, $\overline{v}\cap \overline{w}=\emptyset$.

To show this notice that if $w=i_0\ldots i_k\in W$ and $\overline{w}\ne \{w\}$, then there is $s\in \{0,\ldots ,k\}$ such that $k-s\ge l_{i_s}-tm$. Therefore, it suffices to prove the following statement:

For each $w=i_0\ldots i_k\in W$ and each $s\in \{0,\ldots ,k\}$, if $k-s\ge l_{i_s}-tm$, then $k\ge l_{i_0}-tm$.

We proceed by induction on s. If $s=0$, it is obviously true. Fix $r\ge 0$ and suppose that the statement holds for $s=r$ and let $s=r+1$. Then considering the subword $i_1\ldots i_k$ the inductive hypothesis gives us that $k-1\ge l_{i_1}-tm$, so $k\ge l_{i_1}+1-tm$. And since $l_{i_1}\ge l_{i_0}-1$, we obtain that $k\ge l_{i_0}-1+1-tm=l_{i_0}-tm$. $\square$

The subset V of W in the proof of Lemma 3.3 may be considered as a free finite periodic sums system of order $(l_0,\ldots ,l_{m-1})$ and period tm, and W itself a free m-generated periodic sums system of infinite order. Then the mapping $\pi :W\rightarrow V$ defined by $\pi (w)=\overline{w}\cap V$ (that is, $\pi (w)=w$ if $w\in V$ and $\pi (w)$ is the word $v\in K(V)$ such that v and w have the same first and last letters otherwise) is a homomorphism. We call W the set of periodic words over $\{0,\ldots ,m-1\}$, V (together with K(V)) the subset of W representing a free finite periodic sums system of order $(l_0,\ldots ,l_{m-1})$ and period tm, and $\pi :W\rightarrow V$ the canonical mapping.

Remark 3.4

One may consider the semigroup $Q'$ generated by idempotents $p'_0,\ldots , p'_{m-1}$ with defining relations $\sum _{j=i}^{i+l_i}p'_{\nu (j)}=\sum _{j=i}^{i+l_i-tm}p'_{\nu (j)}$, where $i<m$. Then the periodic sums system in $Q'$ generated by the sequence $p'_0,\ldots ,p'_{m-1}$ is also free of order $(l_0,\ldots ,l_{m-1})$ and period tm.

The proof is practically the same. Let $\delta '$ be the smallest congruence on F generated by the relations $w(i,l_i)=w(i,l_i-tm)$ and $w(i,1)=w(i,0)$, where $i\le m-1$. Then $Q=F/\delta '$ with $p'_i=\overline{w(i,0)}'$, where $\overline{w}'$ denotes the $\delta '$ congruence class of w, and for every $w\in W$, $\overline{w}'\cap W=\overline{w}$.

Since every element of finite order in $\beta \mathbb {N}$ has period 1, it follows that

Theorem 3.5

Every finite m-generated periodic sums system in $\beta \mathbb {N}$ has period m.

In [6] it was shown that for any $m\ge 2$ and $n\ge 2$, there is a free finite m-generated periodic sums system in $\mathbb {H}$ of order $(mn,mn-1,\ldots ,mn-m+1)$. Now using Theorem 2.8 we prove the following result.

Theorem 3.6

For any $n\ge m\ge 2$, there is a free finite m-generated periodic sums system in $\mathbb {H}$ of order $(n,n,\ldots ,n)$.

Proof

First consider the main case where $n\ge m+1$. Let $n'=n-m+1$ and $F=F(m,n')$. By Theorem 2.8, F has copies in $\mathbb {H}$, so it suffices to construct a free m-generated periodic sums system of order $(n,n,\ldots ,n)$ in F. For each $i\in \{0,\ldots ,m-1\}$, let $p_i$ be the 1-letter word i in F, and for each $k\in \{0,\ldots ,n'+m-1\}$, let $v_{i,k}$ be the word in F representing $\sum _{j=i}^{i+k}p_{\nu (j)}$. Then

$$\begin{aligned} v_{i,k}={\left\{ \begin{array}{ll}i\nu (i+1)\ldots \nu (i+k)&{}\text { if }k\le n'-1\\ i\nu (i+k-n'+2)\nu (i+k-n'+3)\ldots \nu (i+k)&{}\text { otherwise}. \end{array}\right. } \end{aligned}$$

All words $v_{i,k}$, where $i\in \{0,\ldots ,m-1\}$ and $k\in \{0,\ldots ,n'+m-2\}$, are distinct (if $k\le n'-1$, the length of $v_{i,k}$ is $k+1$, and if $n'-1\le k\le n'+m-2$, the length of $v_{i,k}$ is $n'$ and the last letter in $v_{i,k}$ is $\nu (i+k)$), and $v_{i,n'+m-1}=i\nu (i+m+1)\nu (i+m+2)\ldots \nu (i+n'+m-1)=i\nu (i+1)\nu (i+2)\ldots \nu (i+n'-1)=v_{i,n'-1}$.

Now let $n=m$. Consider the rectangular band $\{0,\ldots ,m-1\}\times \{0,\ldots ,m-1\}$, and for each $i\in \{0,\ldots ,m-1\}$, let $p_i=(i,i)$. Then for each $k\in \{0,\ldots ,m\}$, $\sum _{j=i}^{i+k}p_{\nu (j)}=(i,\nu (i+k))$, so all sums $\sum _{j=i}^{i+k}p_{\nu (j)}$, where $i,k\in \{0,\ldots ,m-1\}$, are distinct and $\sum _{j=i}^{i+m}p_{\nu (j)}=(i,i)=p_i$. $\square$

4 Ramsey theoretic consequences

We first prove a general result. It can be deduced from [9, Theorem 4.4], but for convenience of the reader, we give a straight proof. We shall use the fact that every finite subsemigroup S of $\beta \mathbb {N}$ is contained in $\mathbb {H}$ [9, Lemma 4.1], and so for all $p\in S$ and $j\ge 0$, $2^j\mathbb {N}\in p$.

Theorem 4.1

Let S be a finite semigroup in $\beta \mathbb {N}$ generated by elements $p_0,\ldots ,p_{m-1}$, and for each $p\in S$, let $(A_p(j))_{j=0}^\infty$ be a sequence of members of the ultrafilter p. There is a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in A_{p_{\nu (j)}}(j)\cap 2^j\mathbb {N}$ and for every finite sequence $j_0<\ldots <j_s$, if $q=p_{\nu (j_0)}+\ldots +p_{\nu (j_s)}$, then $x_{j_0}+\ldots +x_{j_s}\in A_q(j_0)$.

Proof

We construct inductively a sequence $(x_j)_{j=0}^\infty$ satisfying for every j the following conditions in addition to $x_j\in 2^j\mathbb {N}$:

for each finite sequence $j_0<\ldots <j_s=j$,

$$\begin{aligned} x_{j_0}+\ldots +x_{j_s}\in A_q(j_0), \end{aligned}$$

where $q=p_{\nu (j_0)}+\ldots +p_{\nu (j_s)}$, and for each $p\in S$,

$$\begin{aligned} x_{j_0}+\ldots +x_{j_s}+p\in \overline{A_{q+p}(j_0)}. \end{aligned}$$

To define $x_0$, for each $p\in S$, choose $P(p)\in p_0$ such that $P(p)+p\subseteq \overline{A_{p_0+p}(0)}$. We can do this because the right translation by p is continuous. Pick

$$\begin{aligned} x_0\in A_{p_0}(0)\cap \bigcap _{p\in S}P(p). \end{aligned}$$

Then $x_0\in A_{p_0}(0)$ and for each $p\in S$, $x_0+p\in P(p)+p\subseteq \overline{A_{p_0+p}(0)}$, so $x_0$ is as required.

Fix $j\ge 0$ and suppose that we have defined $x_0,\ldots ,x_j$ as required. To define $x_{j+1}$, let F be the set of all sequences $j_0<\ldots <j_s\le j$ and let $i=\nu (j+1)$. For each $p\in S$, choose $B(p)\in p_i$ such that $B(p)+p\subseteq \overline{A_{p_i+p}(j+1)}$. Then for each $(j_0,\ldots ,j_s)\in F$, choose $C(j_0,\ldots ,j_s)\in p_i$ such that $x_{j_0}+\ldots +x_{j_s}+C(j_0,\ldots ,j_s)\subseteq A_{q+p_i}(j_0)$, where $q=p_{\nu (j_0)}+\ldots +p_{\nu (j_s)}$, and for each $p\in S$, choose $D(j_0,\ldots ,j_s,p)\in p_i$ such that $x_{j_0}+\ldots +x_{j_s}+D(j_0,\ldots ,j_s,p)+p\subseteq \overline{A_{q+p_i+p}(j_0)}$. We can do the first because by the inductive hypothesis $x_{j_0}+\ldots +x_{j_s}+p_i\in \overline{A_{q+p_i}(j_0)}$ and $\lambda _x$, where $x=x_{j_0}+\ldots +x_{j_s}$, is continuous, and the second because $p_i+p\in S$ and by the inductive hypothesis $x_{j_0}+\ldots +x_{j_s}+p_i+p\in \overline{A_{q+p_i+p}(j_0)}$ and $\lambda _x$ and $\rho _p$ are continuous. Now pick

$$\begin{aligned}{} & {} x_{j+1}\in 2^{j+1}\mathbb {N}\cap A_{p_i}(j+1)\cap \bigcap _{p\in S}B(p)\cap \bigcap _{(j_0,\ldots ,j_s)\in F}(C(j_0,\ldots ,j_s)\cap \\{} & {} \bigcap _{p\in S}D(j_0,\ldots ,j_s,p)) \end{aligned}$$

(all those sets are members of $p_i$).

To see that $x_{j+1}$ is as required, let any $j_0<\ldots <j_s=j+1$ be given. If $s=0$, then $x_{j+1}\in A_{p_i}(j+1)$ and for each $p\in S$, $x_{j+1}+p\in B(w)+p\subseteq \overline{A_{p_i+p}(j+1)}$. If $s\ge 1$, then

$$\begin{aligned} x_{j_0}+\ldots +x_{j_s}\in x_{j_0}+\ldots +x_{j_{s-1}}+C(j_0,\ldots ,j_{s-1})\subseteq A_{q+p_i}(j_0), \end{aligned}$$

where $q=p_{\nu (j_0)}+\ldots +p_{\nu (j_{s-1})}$, and for each $p\in S$,

$$\begin{aligned} x_{j_0}+\ldots +x_{j_s}+p\in x_{j_0}+\ldots +x_{j_{s-1}}+D(x_{j_0},\ldots x_{j_{s-1}},p)+p\subseteq \overline{A_{q+p_i+p}(j_0)}. \end{aligned}$$

$\square$

Corollary 4.2

Let S be a finite semigroup generated by elements $p_0,\ldots ,p_{m-1}$ and suppose that S has a copy in $\mathbb {H}$. Then there is a partition $\{A_p:p\in S\}$ of $\mathbb {N}$ such that whenever for each p, $\mathscr {B}_p$ is a finite partition of $A_p$, there exist $B_p\in \mathscr {B}_p$ and a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in B_{p_{\nu (j)}}\cap 2^j\mathbb {N}$ and for every finite sequence $j_0<\ldots <j_s$, if $q=p_{\nu (j_0)}+\ldots +p_{\nu (j_s)}$, then $x_{j_0}+\ldots +x_{j_s}\in B_q$.

Proof

One may suppose that S is in $\beta \mathbb {N}$. Choose a partition $\{A_p:p\in S\}$ of $\mathbb {N}$ such that $A_p\in p$. To see that this partition is as required, for each p, let $\mathscr {B}_p$ be a finite partition of $A_p$. Pick $B_p\in \mathscr {B}_p$ such that $B_p\in p$, and for every $j\ge 0$, put $A_p(j)=B_p$. Let $(x_j)_{j=0}^\infty$ be a sequence guaranteed by Theorem 4.1. For any $j_0<\ldots <j_s$, if $q=p_{\nu (j_0)}+\ldots +p_{\nu (j_s)}$, then $x_{j_0}+\ldots +x_{j_s}\in A_p(j_0)=B_q$. $\square$

Now from Theorem 2.8 and Corollary 4.2 we obtain the following result.

Corollary 4.3

Let $m\ge 1$ and $n\ge 2$ and let F be the set of nonempty words over $\{0,\ldots ,m-1\}$ of length $\le n$. There is a partition $\{A_w:w\in F\}$ of $\mathbb {N}$ such that, whenever for each $w\in F$, $\mathscr {B}_w$ is a finite partition of $A_w$, there exist $B_w\in \mathscr {B}_w$ and a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in 2^j\mathbb {N}$ and for every finite sequence $j_0<\ldots <j_s$, if

$$\begin{aligned} v={\left\{ \begin{array}{ll}\nu (j_0)\ldots \nu (j_s)&{}\text { if }s\le n-1\\ \nu (j_0)\nu (j_{s-n+2})\ldots \nu (j_s)&{}\text { otherwise,} \end{array}\right. } \end{aligned}$$

then $x_{j_0}+\ldots +x_{j_s}\in B_v$.

Proof

Consider F as the semigroup F(m, n). $\square$

Remark 4.4

We have extended the addition of natural numbers to an operation $+$ on $\beta \mathbb {N}$ so as to obtain a right topological semigroup. But one can equally well extend the addition to an operation $*$ on $\beta \mathbb {N}$ so as to obtain a left topological semigroup. The semigroup $(\beta \mathbb {N},*)$ is the opposite of the semigroup $(\beta \mathbb {N},+)$: $p*q=q+p$. There are finite semigroups which have copies in $(\beta \mathbb {N},*)$ and not in $(\beta \mathbb {N},+)$. For example, the 3-element band $\{a,b,c\}$, where $\{a,b\}$ is right zero semigroup and c is zero [11]. At the end of the paper [9] it was wrongly remarked that Theorem 4.4 there, an analogue of Theorem 4.1 here, holds for the semigroup $(\beta \mathbb {N},*)$ as well and so the result can be extended to finite semigroups which have copies in $(\beta \mathbb {N},*)$. In fact Theorem 4.1 holds for $(\beta \mathbb {N},*)$ with a correction:

Let S be a finite semigroup in $(\beta \mathbb {N},*)$ generated by elements $p_0,\ldots ,p_{m-1}$, and for each $p\in S$, let $(A_p(j))_{j=0}^\infty$ be a sequence of members of the ultrafilter p. There is a sequence $(x_j)_{j=0}^\infty$ in $\mathbb {N}$ such that $x_j\in A_{p_{\nu (j)}}(j)\cap 2^j\mathbb {N}$ and for every finite sequence $j_0<\ldots <j_s$, if $q=p_{\nu (j_s)}*\ldots *p_{\nu (j_0)}$, then $x_{j_0}+\ldots +x_{j_s}\in A_q(j_0)$.

And since $p_{\nu (j_s)}*\ldots *p_{\nu (j_0)}=p_{\nu (j_0)}+\ldots +p_{\nu (j_s)}$, this is the result for the semigroup $(S,+)$ in $(\beta \mathbb {N},+)$. Hence, using $(\beta \mathbb {N},*)$ in addition to $(\beta \mathbb {N},+)$ gives no new result.

Theorem 4.5

Let S be a finite periodic sums system in $\mathbb {H}$ generated by a sequence $p_0,\ldots ,p_{m-1}$, and for each $p\in S$, let $(A_p(j))_{j=0}^\infty$ be a sequence of members of p. There is a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in A_{p_{\nu (j)}}(j)\cap 2^j\mathbb {N}$ and for every finite sequence $j_0<\ldots <j_s$ such that $j_{t+1}\equiv j_t+1\pmod {m}$ for each $t<s$, if $q=p_{\nu (j_0)}+\ldots +p_{\nu (j_s)}$, then $x_{j_0}+\ldots +x_{j_s}\in A_q(j_0)$.

Proof

Let $(l_0,\ldots ,l_{m-1})$ be the order of S and let W be the set of periodic words over $\{0,\ldots ,m-1\}$, V the subset of W representing a free finite periodic sums system of order $(l_0,\ldots ,l_{m-1})$ and period m, and $\pi :W\rightarrow V$ the canonical mapping. Also for each $i\in \{0,\ldots ,m-1\}$, let V(i) denote the subset of V consisting of words with first letter i. Define $f:W\rightarrow S$ by $f(i_0\ldots i_k)=p_{i_0}+\ldots +p_{i_k}$. Then $f(w)=f(\pi (w))$ for all $w\in W$ and $f(wv)=f(w)+f(v)$ for all $w,v\in W$ such that $wv\in W$.

We construct inductively a sequence $(x_j)_{j=0}^\infty$ satisfying for every j the following conditions in addition to $x_j\in 2^j\mathbb {N}$:

for each finite sequence $j_0<\ldots <j_s=j$ with $w=\nu (j_0)\ldots \nu (j_s)\in W$,

$$\begin{aligned} x_{j_0}+\ldots +x_{j_s}\in A_{f(w)}(j_0) \end{aligned}$$

and for each $v\in V(\nu (j+1))$,

$$\begin{aligned} x_{j_0}+\ldots +x_{j_s}+f(v)\in \overline{A_{f(wv)}(j_0)}. \end{aligned}$$

To define $x_0$, for each $v\in V(1)$, choose $P(v)\in p_0$ such that $P(v)+f(v)\subseteq \overline{A_{f(0v)}(0)}$. We can do this because $p_0+f(v)=f(0v)$ and $\rho _{f(v)}$ is continuous. Pick

$$\begin{aligned} x_0\in A_0(0)\cap \bigcap _{v\in V(1)}P(v). \end{aligned}$$

Then $x_0\in A_0(0)$ and for each $v\in V(1)$, $x_0+f(v)\in P(v)+f(v)\subseteq \overline{A_{f(0v)}(0)}$, so $x_0$ is as required.

Fix $j\ge 0$ and suppose that we have defined $x_0,\ldots ,x_j$ as required. To define $x_{j+1}$, let F be the set of all sequences $j_0<\ldots <j_s\le j$ with $\nu (j_0)\ldots \nu (j_s)\in W$ and $\nu (j_s)=\nu (j)$ and let $i=\nu (j+1)$ and $r=\nu (j+2)$. For each $v\in V(r)$, choose $B(v)\in p_i$ such that $B(v)+f(v)\subseteq \overline{A_{f(iv)}(j+1)}$. Then for each $(j_0,\ldots ,j_s)\in F$, choose $C(j_0,\ldots ,j_s)\in p_i$ such that $x_{j_0}+\ldots +x_{j_s}+C(j_0,\ldots ,j_s)\subseteq A_{f(wi)}(j_0)$, where $w=\nu (j_0)\ldots \nu (j_s)$, and for each $v\in V(r)$, choose $D(j_0,\ldots ,j_s,v)\in p_i$ such that $x_{j_0}+\ldots +x_{j_s}+D(j_0,\ldots ,j_s,v)+f(v)\subseteq \overline{A_{f(wiv)}(j_0)}$. We can do the first because by the inductive hypothesis $x_{j_0}+\ldots +x_{j_s}+p_i\in \overline{A_{f(wi)}(j_0)}$ and $\lambda _x$, where $x=x_{j_0}+\ldots +x_{j_s}$, is continuous, and the second because $p_i+f(v)=f(iv)=f(\pi (iv))$ and by the inductive hypothesis $x_{j_0}+\ldots +x_{j_s}+f(\pi (iv))\in \overline{A_{f(w\pi (iv))}(j_0)}=\overline{A_{f(wiv)}(j_0)}$ (since $f(wiv)=f(w)+f(iv)=f(w)+f(\pi (iv))=f(w\pi (iv))$) and $\lambda _x$ and $\rho _{f(v)}$ are continuous. Now pick

$$\begin{aligned}{} & {} x_{j+1}\in 2^{j+1}\mathbb {N}\cap A_i(j+1)\cap \bigcap _{v\in V(r)}B(v)\cap \bigcap _{(j_0,\ldots ,j_s)\in F}(C(j_0,\ldots ,j_s)\cap \\{} & {} \bigcap _{v\in V(r)}D(j_0,\ldots ,j_s,v)) \end{aligned}$$

(all those sets are members of $p_i$).

To see that $x_{j+1}$ is as required, let any $j_0<\ldots <j_s=j+1$ with $\nu (j_0)\ldots \nu (j_s)\in W$ be given. If $s=0$, then $x_{j+1}\in A_i(j+1)$ and for each $v\in V(r)$, $x_{j+1}+f(v)\in B(v)+f(v)\subseteq \overline{A_{f(iv)}(j+1)}$. If $s\ge 1$, then

$$\begin{aligned} x_{j_0}+\ldots +x_{j_s}\in x_{j_0}+\ldots +x_{j_{s-1}}+C(j_0,\ldots ,j_{s-1})\subseteq A_{f(wi)}(j_0), \end{aligned}$$

where $w=\nu (j_0)\ldots \nu (j_{s-1})$, and for each $v\in V(r)$,

$$\begin{aligned}{} & {} x_{j_0}+\ldots +x_{j_s}+f(v)\in x_{j_0}+\ldots +x_{j_{s-1}}+D(x_{j_0},\ldots ,x_{j_{s-1},v})\\{} & {} +f(v)\subseteq \overline{A_{f(wiv)}(j_0)}. \end{aligned}$$

$\square$

Corollary 4.6

Let S be a finite periodic sums system generated by a sequence $p_0,\ldots ,p_{m-1}$ and suppose that S has a copy in $\mathbb {H}$. Then there is a partition $\{A_p:p\in S\}$ of $\mathbb {N}$ such that whenever for each p, $\mathscr {B}_p$ is a finite partition of $A_p$, there exist $B_p\in \mathscr {B}_p$ and a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in B_{\nu (j)}\cap 2^j\mathbb {N}$ and for every finite sequence $j_0<\ldots <j_s$ such that $j_{t+1}\equiv j_t+1\pmod {m}$ for each $t<s$, if $q=p_{\nu (j_0)}+\ldots +p_{\nu (j_s)}$, then $x_{j_0}+\ldots +x_{j_s}\in B_q$

Proof

Similar to the proof of Corollary 4.2. $\square$

In [6] it was also deduced from the existence of a free finite m-generated periodic sums system in $\mathbb {H}$ of order $(mn,mn-1,\ldots ,mn-m+1)$ that:

There is a partition

$$\begin{aligned} \{A_{i,k}:i\in \{0,\ldots ,m-1\}\text { and }k\in \{i,\ldots ,mn-1\}\text { for each }i\} \end{aligned}$$

of $\mathbb {N}$ such that, whenever for each (i, k), $\mathscr {B}_{i,k}$ is a finite partition of $A_{i,k}$, there exist $B_{i,k}\in \mathscr {B}_{i,k}$ and a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in 2^j\mathbb {N}$ and for every finite sequence $j_0<\ldots <j_s$ such that $j_{t+1}\equiv j_t+1\pmod {m}$ for each $t<s$, if $i_0=\nu (j_0)$ and

$$\begin{aligned} k_0={\left\{ \begin{array}{ll}i_0+s&{}\text { if }i_0+s\le mn-1\\ mn-m+\nu (i_0+s-mn)&{}\text { otherwise}, \end{array}\right. } \end{aligned}$$

then $x_{j_0}+\ldots +x_{j_s}\in B_{i_0,k_0}$.

Now from Theorem 3.6 and Corollary 4.6 we obtain the following result.

Corollary 4.7

Let $n\ge m\ge 2$. There is a partition

$$\begin{aligned} \{A_{i,k}:(i,k)\in \{0,\ldots ,m-1\}\times \{0,\ldots ,n-1\}\} \end{aligned}$$

of $\mathbb {N}$ such that, whenever for each (i, k), $\mathscr {B}_{i,k}$ is a finite partition of $A_{i,k}$, there exist $B_{i,k}\in \mathscr {B}_{i,k}$ and a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in 2^j\mathbb {N}$ and for every finite sequence $j_0<\ldots <j_s$ such that $j_{t+1}\equiv j_t+1\pmod {m}$ for each $t<s$, if $i_0=\nu (j_0)$ and

$$\begin{aligned} k_0={\left\{ \begin{array}{ll}s&{}\text { if }s\le n-1\\ n-m+\nu (s-n)&{}\text { otherwise}, \end{array}\right. } \end{aligned}$$

then $x_{j_0}+\ldots +x_{j_s}\in B_{i_0,k_0}$.

Proof

Consider $\{0,\ldots ,m-1\}\times \{0,\ldots ,n-1\}$ as a free finite m-generated periodic sums system of order $(n,\ldots ,n)$ with $(i,k)=\sum _{j=i}^{i+k}p_{\nu (j)}$. $\square$

In cases $n=m$ and $n=m+1$, Corollary 4.7 can be strengthened. The free finite m-generated periodic sums systems of orders $(m,\ldots ,m)$ and $(m+1,\ldots ,m+1)$ constructed in Theorem 3.6 are in fact the $m\times m$ rectangular band and the semigroup F(m, 2). Therefore, by Corollary 4.2, the following stronger results hold.

Corollary 4.8

For every $m\ge 2$, there is a partition

$$\begin{aligned} \{A_{i,k}:(i,k)\in \{0,\ldots ,m-1\}\times \{0,\ldots ,m-1\}\} \end{aligned}$$

of $\mathbb {N}$ such that, whenever for each (i, k), $\mathscr {B}_{i,k}$ is a finite partition of $A_{i,k}$, there exist $B_{i,k}\in \mathscr {B}_{i,k}$ and a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in 2^j\mathbb {N}$ and for every finite nonempty $J\subseteq \omega$, if $i_0=\nu (\min J)$ and $k_0=\nu (\max J)$, then $\sum _{j\in J}x_j\in B_{i_0,k_0}$.

Corollary 4.9

For every $m\ge 2$, there is a partition

$$\begin{aligned} \{A_{i,k}:(i,k)\in \{0,\ldots ,m-1\}\times \{0,\ldots ,m\}\} \end{aligned}$$

of $\mathbb {N}$ such that, whenever for each (i, k), $\mathscr {B}_{i,k}$ is a finite partition of $A_{i,k}$, there exist $B_{i,k}\in \mathscr {B}_{i,k}$ and a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in 2^j\mathbb {N}\cap B_{\nu (j),0}$ and for every finite $J\subseteq \omega$ with $|J|\ge 2$, if $i_0=\nu (\min J)$ and $k_0=1+\nu (\max J)$, then $\sum _{j\in J}x_j\in B_{i_0,k_0}$.

In Corollary 4.9, (i, k) is identified with the 1-letter word i of F(m, 2) if $k=0$ and the word $i(k-1)$ otherwise. It is a restatement of case $m\ge n=2$ of Corollary 4.3.

We also notice that a finite periodic sums system generated by two idempotents is a semigroup, and so for such systems, if they have copies in $\beta \mathbb {N}$, also stronger results hold.

For every $n\ge 3$ ($n\ge 2$), a free finite 2-idempotent generated periodic sums system of order $(n,n-1)$ ((n, n)) is the semigroup $S_{n,n-1}$ ($S_{n,n}$) generated by idempotents $p_0,p_1$ with defining relations $\sum _{j=0}^np_{\nu (j)}=\sum _{j=0}^{n-2}p_{\nu (j)}$ and $\sum _{j=1}^np_{\nu (j)}=\sum _{j=1}^{n-2}p_{\nu (j)}$ ($\sum _{j=1}^{n+1}p_{\nu (j)}=\sum _{j=1}^{n-1}p_{\nu (j)}$). Presently $m=2$, so $\nu =\nu _2$. We know only three of those semigroups that have copies in $\beta \mathbb {N}$: $S_{2,2}$ ($2\times 2$ rectangular band), $S_{3,2}$ (the band (10) in [9, Theorem 2.3]), and $S_{4,3}$ (the semigroup (3) in [9, Corollary 3.11]). For all others we do not know whether they have copies in $\beta \mathbb {N}$, in particular, for $S_{3,3}$ which is a free 2-generated band. We also do not know whether a sum of two idempotents in $\beta \mathbb {N}$ can be an element of order $n\ge 3$.

For every finite nonempty subset $J\subseteq \omega$, write the elements of J as $j_0<\ldots <j_s$ and let f(J) be the number of all $t<s$ such that $j_{t+1}\equiv j_t+1\pmod {2}$.

Corollary 4.10

Let $n\ge 3$ and suppose that the semigroup $S_{n,n-1}$ has a copy in $\beta \mathbb {N}$. Then there is a partition

$$\begin{aligned} \{A_{i,k}:i\in \{0,1\}\text { and }k\in \{i,\ldots ,n-1\}\text { for each }i\} \end{aligned}$$

of $\mathbb {N}$ such that, whenever for each (i, k), $\mathscr {B}_{i,k}$ is a finite partition of $A_{i,k}$, there exist $B_{i,k}\in \mathscr {B}_{i,k}$ and a sequence $(x_j)_{j=0}^\infty$ such that $x_j\in 2^j\mathbb {N}$ and for every finite nonempty $J\subseteq \omega$, if $i_0=\nu (\min J)$ and

$$\begin{aligned} k_0={\left\{ \begin{array}{ll}i_0+f(J)&{}\text { if }i_0+f(J)\le n-1\\ n-2+\nu (i_0+f(J)-n)&{}\text { otherwise}, \end{array}\right. } \end{aligned}$$

then $\sum _{j\in J}x_j\in B_{i_0,k_0}$.

Proof

Consider $\{(i,k):i\in \{0,1\}\text { and }k\in \{i,\ldots ,n-1\}\text { for each }i\}$ as the semigroup $S_{n,n-1}$ with $(i,k)=\sum _{j=i}^{k}p_{\nu (j)}$. For any finite nonempty $J\subseteq \omega$, if $i_0=\nu (\min J)$, then $\sum _{j\in J}p_{\nu (j)}=\sum _{j=i_0}^{i_0+f(J)}p_{\nu (j)}$. Apply Corollary 4.2. $\square$

A subset $A\subseteq \mathbb {N}$ is an $\hbox {IP}$ set if it contains an infinite sequence all of whose sums belong to A. By Hindman’s Theorem, whenever $\mathbb {N}$ is partitioned into finitely many cells, at least one of the cells is an $\hbox {IP}$ set.

Remark 4.11

All results of this section extend to $\hbox {IP}$ sets, that is, in the statement of each corollary the partitioning set $\mathbb {N}$ can be replaced with any $\hbox {IP}$ set $A\subseteq \mathbb {N}$.

Indeed, let $(a_n)_{n=0}^\infty$ be a sequence all of whose sums belong to A. Taking a sum subsystem of $(a_n)_{n=0}^\infty$ one may suppose that $\max \hbox {supp }a_n<\min \hbox {supp }a_{n+1}$ (see [4, Exercise 5.2.2]), and also that A coincides with the set of all sums of the sequence. Define a bijection $f:\mathbb {N}\rightarrow A$ by $f(x)=\sum _{n\in \hbox {supp }x}a_n$. Then whenever $\max \hbox {supp }x<\min \hbox {supp }y$, one has $f(x+y)=f(x)+f(y)$.

Now consider say Corollary 4.6. Let $\{A_p^\mathbb {N}:p\in S\}$ be a partition of $\mathbb {N}$ guaranteed by the corollary. Define a partition $\{A_p:p\in S\}$ of A by $A_p=f(A_p^\mathbb {N})$.

To see that this partition is as required, let for each p, $\mathscr {B}_p$ be a finite partition of $A_p$ and let $\mathscr {B}_p^\mathbb {N}=f^{-1}(\mathscr {B}_p)$. Let $B_p^\mathbb {N}\in \mathscr {B}_p^\mathbb {N}$ and $(x_j^\mathbb {N})_{j=0}^\infty$ be as guaranteed by the corollary. One may suppose that $\max \hbox {supp }x_j^\mathbb {N}<\min \hbox {supp }x_{j+1}^\mathbb {N}$. Define $B_p\in \mathscr {B}_p$ and $(x_j)_{j=0}^\infty$ by $B_p=f(\mathscr {B}_p^\mathbb {N})$ and $x_j=f(x_j^\mathbb {N})$.