Force recurrence of semigroup actions

Abstract

We investigate the sets of countable discrete semigroups that force recurrence, that is, the recurrent properties of a point along a subset of a countable semigroup action. We show that a subset of a monoid forces recurrence (resp., forces minimality) if and only if it contains a broken IP-set (resp., broken syndetic set), and forces infinite recurrence implies it is contains a broken infinite IP-sets. As an example, we show that every subset with positive upper Banach density of infinite countable amenable groups forces infinite recurrence.

Appendix: When \(b {\mathcal {P}}_{\mathrm {inf, ip}}\) has the Ramesey property

In this section we utilize the algebraic structure of the Stone–Čech compactification $\beta G$ of a discrete semigroup $(G, \cdot )$. We shall assume that the reader is familiar with the basic facts about this structure. For an elementary introduction see [24,  Part I].

We shall show that if $\beta G \setminus G$ is a subsemigroup of $\beta G$, in particular if G is either right or left cancellative, then both ${\mathcal {P}}_{\mathrm {inf,ip}}$ and $b{\mathcal {P}}_{\mathrm {inf,ip}}$ have the Ramsey property. An exact characterization of when $\beta G \setminus G$ is a subsemigroup of $\beta G$ is given in [24,  Theorem 4.28].

Lemma 5.1

Assume that G is a semigroup, p is an idempotent in $\beta G \setminus G$, and $A \in p$. Then A contains an infinite IP-set. In fact there exists an injective sequence $(x_n)_{n=1}^{\infty }$ in G such that $FP(\{x_n\}_{n=1}^{\infty })\subseteq A$.

Proof

Let $A^{\star }=\{x \in A: x^{-1}A \in p\}$. By [24,  Lemma 4.14] if $x \in A^{\star }$, then $x^{-1}A^{\star } \in p$. Choose $x_1 \in A^{\star }$. Inductively let $n \in {\mathbb {N}}$ and assume we have chosen injective $(x_t)_{t=1}^n$ in G such that $E =FP(\{x_t\}_{t=1}^n) \subseteq A^{\star }$. Since $p \in \beta G \setminus G$, $G \setminus \{x_1, x_2, \ldots , x_n\} \in p$ so

$$\begin{aligned} \left( A^{\star } \cap \bigcap _{y \in E}y^{-1}A^{\star }\right) \setminus \{x_1, x_2, \ldots , x_n\} \in p. \end{aligned}$$

Pick $x_{n+1} \in \left( A^{\star } \cap \bigcap _{y \in E}y^{-1}A^{\star }\right) \setminus \{x_1, x_2, \ldots , x_n\}$. $\square$

Lemma 5.2

Assume that G is a semigroup, $\beta G \setminus G$ is a subsemigroup of $\beta G$, and $A \subseteq G$. If A contains an infinite IP-set $FP(\{x_n\}_{n=1}^{\infty })$, then there is an idempotent $p \in \beta G \setminus G$ such that for every $m \in {\mathbb {N}}$, $FP(\{x_n\}_{n=m}^{\infty }) \in p$.

Proof

We claim that for each $m \in {\mathbb {N}}$, $FP(\{x_n\}_{n=m}^{\infty })$ is infinite. To see this let $m>1$ and let $E=FP(\{x_n\}_{n=1}^{m-1})$. Then

$$\begin{aligned} FP(\{x_n\}_{n=1}^{\infty })=E \cup FP(\{x_n\}_{n=m}^{\infty }) \cup \bigcup _{y \in E} y \cdot FP(\{x_n\}_{n=m}^{\infty }) \end{aligned}$$

so one of the listed sets is infinite and thus $FP(\{x_n\}_{n=m}^{\infty })$ is infinite.

Let ${\mathcal {A}}=\{FP(\{x_n\}_{n=m}^{\infty }): m \in {\mathbb {N}}\}$. Then ${\mathcal {A}}$ is a nested family of infinite sets so by [24,  Corollary 3.14], there is some $q \in \beta G \setminus G$ such that ${\mathcal {A}} \subseteq q$. That is, $(\beta G \setminus G) \cap \bigcap _{m=1}^{\infty } \overline{FP(\{x_n\}_{n=m}^{\infty })} \ne \emptyset$. By [24,  Lemma 5.11], $\bigcap _{m=1}^{\infty } \overline{FP(\{x_n\}_{n=m}^{\infty })}$ is a semigroup so $(\beta G \setminus G) \cap \bigcap _{m=1}^{\infty } \overline{FP(\{x_n\}_{n=m}^{\infty })}$ is a compact right topological semigroup so by [16,  Lemma 1], there is an idempotent $p \in (\beta G \setminus G) \cap \bigcap _{m=1}^{\infty } \overline{FP(\{x_n\}_{n=m}^{\infty })}$. $\square$

Theorem 5.3

Assume that G is a semigroup, $\beta G \setminus G$ is a subsemigroup of $\beta G$, and $A \subseteq G$.

• (1) $A \in {\mathcal {P}}_{\mathrm {inf, ip}}$ if and only if there is an idempotent $p \in \beta G \setminus G$ such that $A \in p$;

• (2) $A \in b {\mathcal {P}}_{\mathrm {inf, ip}}$ if and only if there exists an idempotent $p \in \beta G \setminus G$ and $q \in \beta G$ such that $A \in p \cdot q$.

Proof

(1) The necessity follows from Lemma 5.2 and the sufficiency follows from Lemma 5.1.

(2) To establish the necessity, pick a sequence $(x_n)_{n=1}^{\infty }$ in G such that $FP(\{x_n\}_{n=1}^{\infty })$ is infinite and for each $m \in {\mathbb {N}}$ there exists $s_m \in G$ such that $FP(\{x_n\}_{n=1}^m)\cdot s_m \subseteq A$. Pick by Lemma 5.2 an idempotent $p \in \beta G \setminus G$ such that for every $m \in {\mathbb {N}}$, $FP(\{x_n\}_{n=m}^{\infty }) \in p$. Pick $q \in \beta G$ such that $\left\{ \{s_m: m>n\}: n \in {\mathbb {N}}\right\} \subseteq q$. We claim that $A \in p \cdot q$. To see this, it suffices to show that $FP(\{x_n\}_{n=1}^{\infty }) \subseteq \{y \in G: y^{-1}A \in q\}$ by [24,  Lemma 4.12] so let $z \in FP(\{x_n\}_{n=1}^{\infty })$ and pick $F \in {\mathcal {P}}_f({\mathbb {N}})$ such that $z=\prod _{t \in F} x_t$. Let $n =\max F$. Then $\{s_m: m>n\} \in q$ and $\{s_m: m>n\} \subseteq z^{-1}A$.

For the sufficiency, pick an idempotent $p \in \beta G \setminus G$ and $q \in \beta G$ such that $A \in p \cdot q$. Let $B=\{y \in G: y^{-1}A \in q\}$. Then $B \in p$ by [24,    Lemma 4.12] so pick by Lemma 5.1 an injective sequence $(x_n)_{n=1}^{\infty }$ in G such that $FP(\{x_n\}_{n=1}^{\infty })\subseteq B$. Now let $m \in {\mathbb {N}}$. It suffices to show that there exists $s_m \in G$ such that $FP(\{x_n\}_{n=1}^m) \cdot s_m \subseteq A$. Let $E=FP(\{x_n\}_{n=1}^m)$. Then E is a finite subset of B so we may pick $s_m \in \bigcap _{y \in E} y^{-1}A$. $\square$

Corollary 5.4

Assume that G is a semigroup and $\beta G \setminus G$ is a subsemigroup of $\beta G$. Then ${\mathcal {P}}_{\mathrm {inf,ip}}$ and $b{\mathcal {P}}_{\mathrm {inf,ip}}$ have the Ramsey property. In particular if G is infinite and is either right cancellative or left cancellative, then ${\mathcal {P}}_{\mathrm {inf,ip}}$ and $b{\mathcal {P}}_{\mathrm {inf,ip}}$ have the Ramsey property.

Proof

Let $S_1$ and $S_2$ be subsets of G. If $S_1 \cup S_2 \in {\mathcal {P}}_{\mathrm {inf,ip}}$, pick an idempotent $p \in \beta G \setminus G$ such that $S_1 \cup S_2 \in p$. Since p is an ultrafilter, either $S_1 \in p$ or $S_2 \in p$. If $S_1 \cup S_2 \in b{\mathcal {P}}_{\mathrm {inf,ip}}$, pick an idempotent $p \in \beta G \setminus G$ and $q \in \beta G$ such that $S_1 \cup S_2 \in p \cdot q$. Since $p \cdot q$ is an ultrafilter, either $S_1 \in p \cdot q$ or $S_2 \in p \cdot q$. The “in particular" conclusions follow from [24,  Corollary 4.29]. $\square$