# Reccurence Sets for Partial Inverse
Semigroup Actions and Related Structures

M. Măntoiu  M. Măntoiu 111 2010 Mathematics Subject Classification: Primary
20M18, 37B20, Secondary 22A22.
Key Words: inverse semigroup, groupoid, action, dynamical system, orbit,
recurrence.
Acknowledgements: The author has been supported by the Fondecyt Project
1200884.

###### Abstract

Two types of recurrence sets are introduced for inverse semigroup partial
actions in topological spaces. We explore their connections with similar
notions for related types of imperfect symmetries (prefix inverse semigroup
expansions, partial group and groupoid actions).

##  1 Introduction

In the theory of classical dynamical systems, a special role is played by
recurrence sets (also called dwelling sets); this can be seen in any of the
standard texts, as [1, 4] for example. Suppose that the group ${\sf G}$ acts
continuously on the topological space $\Sigma$ , the action being denoted by
$\alpha$ . For $M,N\subset\Sigma$ one defines

${\sf G}(\alpha)_{M}^{N}:=\big{\\{}g\in{\sf G}\,\big{|}\,\alpha_{g}(M)\cap
N\neq\emptyset\big{\\}}\,.$ (1.1)

Such subsets of ${\sf G}$ are relevant for the behavior of the dynamical
systems in many ways. As a basic example, for singletons $M=\\{\sigma\\}$ and
$N=\\{\tau\\}$ , the set ${\sf G}(\alpha)_{\sigma}^{\tau}\equiv{\sf
G}(\alpha)_{\\{\sigma\\}}^{\\{\tau\\}}$ is non-void if and only if $\sigma$
and $\tau$ belong to the same orbit. This serves to characterize invariant
subsets. Many other important properties may be defined or described in terms
of recurrence sets; we mention only (topological) transitivity, limit sets,
(non-)wandering points, periodic and almost periodic points, minimality,
mixing, but there are others. The versatility of the notion comes both from
the nature and the relative position of the sets $M,N$ and from the
requirements on the size of the correspondent recurrent set.

To study more general and subtle notions of (local or partial) symmetry, in
the last decades groups have been replaced by more general mathematical
objects as groupoids or inverse semigroups. Even in the setting of groups,
partial actions are an important way to encode imperfect symmetries. One
important application (which has been our initial motivation) is towards the
theory of $C^{*}$-algebras, but we will not refer to this in the present
paper.

Dynamical notions of such general type of actions have mostly been introduced
in a non-systematic ad hoc manner. In [8] the topological dynamics of groupoid
actions has been treated in a more consistent way, but only the basic theory
has been developed. We are also interested in a similar project for inverse
semigroup actions. In starting this project, we become aware of a certain
intricacy of the topic and especially of the fact that well-known connections
of the inverse semigroup actions with other mathematical notions may be a
fruitful technical tool. Actually the present short article is devoted merely
to explore these connections in the setting of recurrence, and the development
of the theory itself is postponed to a further publication.

We are interested in triples $(\mathcal{S},\theta,\Sigma)$ , where
$\mathcal{S}$ is a (discrete) inverse semigroup, $\Sigma$ is a topological
space and the action $\theta$ is composed of a family of homeomorphisms
$\\{\theta_{s}\\!\mid\\!s\in\mathcal{S}\\}$ between open subsets of $\Sigma$ ,
having suitable properties. The precise definition and the basic theory is
presented in Section 2, but let us just note three peculiar features that are
not present for groups: (a) The transformations $\theta_{s}$ are not defined
everywhere. (b) Two transformations $\theta_{s}$ and $\theta_{t}$ may act in
the same way on an open subset of the intersection of their domains. (c) The
composition $\theta_{s}\circ\theta_{t}$ has to be defined in a precise
(maximal) way. We speak of a (genuine) action of $\mathcal{S}$ if this
composition equals $\theta_{st}$ . Actually, to include the concept of partial
group action, guided by [2], we allow partial inverse semigroup actions, in
which $\theta_{s}\circ\theta_{t}$ is only required to be a restriction of
$\theta_{st}$ .

By analogy to (1.1), one can define recurrence sets of the form

$\mathcal{S}(\theta)_{M}^{N}:=\big{\\{}s\in\mathcal{S}\,\big{|}\,\theta_{s}(M)\cap
N\neq\emptyset\big{\\}}\,.$ (1.2)

Since it is easier, and since sharp results will be obtained for this
situation, let us restrict to the case when $M=\\{\sigma\\}$ is a one-point
set. As a consequence of (a) , it is understood that only elements $s$ for
which $\theta_{s}$ is defined in $\sigma$ are candidates in (1.2) . On the
other hand, because of (b), the set $\mathcal{S}(\theta)_{\sigma}^{N}$ seems
redundant. This could be important, since for some dynamical properties its
size (infinite, complement of a finite set, syndetic) should play an important
role. This is why, besides the ”naive” set $\mathcal{S}(\theta)_{\sigma}^{N}$
we also define in (2.7) a quotient notion $\mathscr{S}(\theta)_{\sigma}^{N}$,
that is more suited to the developments we have in view in the future.
Comparing these two types of recurrence sets with others, attached to related
mathematical structures, is the main goal of the paper.

In [5], to each group ${\sf G}$ one attaches an inverse semigroup
$\mathcal{S}_{{\sf G}}$ , suitably generated by ${\sf G}$ , such that the
partial actions of ${\sf G}$ transform into actions of $\mathcal{S}_{\sf G}$ ,
the procedure satisfying a certain universal property. In [11] and [3] this is
extended to partial inverse semigroup actions: One starts with an inverse
semigroup $\mathcal{A}$ , generates another inverse semigroup
$\mathcal{S}_{\mathcal{A}}$ satisfying certain relations, and then there is a
one-to-one correspondence between the partial actions of $\mathcal{A}$ and the
genuine actions of $\mathcal{S}_{\mathcal{A}}$ . In Section 3 we study the
fate of orbits and recurrence sets under this correspondence of actions.

In Section 4 we work with quotients $\mathcal{S}/_{\\!\approx}$ of
$\mathcal{S}$ through idempotent pure congruences. When $\mathcal{S}$ is
$E$-unitary, and $\approx$ is the minimum group congruence, then
$\mathcal{S}/_{\\!\approx}$ is actually the maximal homomorphic group image of
$\mathcal{S}$ . By [9], there is a canonical way to transform a partial action
of $\mathcal{S}$ into a partial action of $\mathcal{S}/_{\\!\approx}$ in the
same topological space (the case of genuine actions is simpler). We show that
the two connected actions have the same invariant sets, and we find relations
between the respective recurrence sets.

In Sections 5 and 6 we use well-known connections between inverse semigroups
and étale groupoids, looking for their effect on recurrence sets. In [8] a
rather detailed study of recurrence and related dynamical properties for
continuous groupoid actions can be found, so the results of these two sections
will be combined in a future work to convert available groupoid information in
terms of inverse semigroup action. But more work will be needed to have a
comprehensive theory.

To an inverse semigroup partial action one associates its groupoid of germs;
there is a perfect correspondence of invariant sets. Natural recurrence sets
(see Definition 5.4) may be expressed in pure groupoid lenguaje, as (maybe
non-invariant) reductions, as shown in Proposition 5.5. Their connection with
the two types of recurrence sets of the initial partial action are then
subject of Propositions 5.7 and 5.9.

Section 6. For a groupoid action on a topological space, Definition 6.4
indicates the natural recurrence sets. If the groupoid is étale, there is a
canonical inverse semigroup formed of bisections. The actions of the groupoid
generate actions of this inverse semigroup. Theorem 6.8 and Proposition 6.9
indicate the way the relevant recurrece sets are linked.

The results on recurrent sets, in all the situations we treat, show a clear
preference for $\mathscr{S}(\theta)_{\sigma}^{N}$ upon
$\mathcal{S}(\theta)_{\sigma}^{N}$. The first set is each time in a one-to-one
correspondence with the analog set of the related structure. The second one is
only mapped surjectively. All the maps defining the connections seem
interesting.

##  2 Inverse semigroup partial actions and their recurrence sets

For general facts on inverse semigroups and their partial actions we refer
mainly to [10, 12, 2].

######  Definition 2.1.

An inverse semigroup is a semigroup $\mathcal{S}$ such that for every
$s\in\mathcal{S}$ there is a unique element $s^{\sharp}\in\mathcal{S}$
satisfying

$s^{\sharp}ss^{\sharp}=s^{\sharp},\quad ss^{\sharp}s=s\,.$

We are always going to assume that $\mathcal{S}$ is discrete and infinite. In
an inverse semigroup $\mathcal{S}$ a partial order relation is defined by

$s\leq t\quad{\rm if\ and\ only\ if}\quad s=ts^{\sharp}s\,.$

On the commutative subsemigroup of idempotents $\mathcal{E}(\mathcal{S})$ this
reduces to $e=fe$ . The order may be reformulated:

$s\leq
t\,\Leftrightarrow\,\exists\,e\in\mathcal{E}(\mathcal{S})\,,\,s=et\,\Leftrightarrow\,\exists\,f\in\mathcal{E}(\mathcal{S})\,,\,s=tf\,.$
(2.1)

${\rm PHomeo}(\Sigma)$ , the family of all the homeomorphisms between open
subsets of $\Sigma$ , is an inverse semigroup with obvious composition and
inversion, the order relation being now just the restriction of functions.

######  Definition 2.2.

A partial action of the inverse semigroup $\mathcal{S}$ in the topological
space $\Sigma$ is a family

$\theta:=\Big{\\{}\Sigma_{s^{\sharp}}\\!\overset{\theta_{s}}{\longrightarrow}\Sigma_{s}\,\Big{|}\,s\in\mathcal{S}\Big{\\}}\,,$

where $\theta_{s}:\Sigma_{s^{\sharp}}\\!\equiv{\rm
dom}(\theta_{s})\\!\to\Sigma_{s}\equiv{\rm im}(\theta_{s})$ is a homeomorphism
between open subsets of $\Sigma$ , satisfying for every $s,t\in\mathcal{S}$ :

  1. (i)

$\theta_{s^{\sharp}}=\theta_{s}^{-1}$ ,

  2. (ii)

$\theta_{s}\circ\theta_{t}$ is a restriction of $\theta_{st}$ ,

  3. (iii)

if $s\leq t$ then $\theta_{s}$ is a restriction of $\theta_{t}$ ,

  4. (iv)

$\Sigma=\bigcup_{e\in\mathcal{E}(\mathcal{S})}\\!\Sigma_{e}$ (non-degeneracy).

We speak of a (genuine) action whenever

$\theta_{s}\circ\theta_{t}=\theta_{st}\,,\,\forall\,s,t\in\mathcal{S}.$

Note that (ii) implies that

$\Sigma_{t}\cap\theta_{t}\big{(}\Sigma_{s}\big{)}\subset\Sigma_{ts}\,,\quad\forall\,s,t\in\mathcal{S},$
(2.2)

from which one deduces immediately

$\Sigma_{t^{\sharp}}\subset\Sigma_{t^{\sharp}t}\,,\quad\Sigma_{t}\subset\Sigma_{tt^{\sharp}}\,,\quad\forall\,t\in\mathcal{S}.$

In addition, $\theta_{e}={\rm id}_{\Sigma_{e}}$ for any
$e\in{\mathcal{E}}(\mathcal{S})$ . We will often use the notation

$\theta_{s}(\sigma)\equiv s\diamond\sigma\equiv s\diamond_{\theta}\\!\sigma.$

We define now several dynamical notions. It is useful to set

$\mathcal{S}\\{\theta,\sigma\\}:=\big{\\{}s\in\mathcal{S}\,\big{|}\,\sigma\in\Sigma_{s^{\sharp}}\\!={\rm
dom}(\theta_{s})\big{\\}}$

for the family of points of $\mathcal{S}$ which can be applied to $\sigma$. In
virtue of (2.2)

$t\in\mathcal{S}\\{\theta,\sigma\\}\,,\,s\in\mathcal{S}\\{\theta,t\diamond\sigma\\}\,\Rightarrow\,st\in\mathcal{S}\\{\theta,\sigma\\}\,.$

An equivalence relation is defined in $\Sigma$ by

$\sigma\overset{\theta}{\sim}\tau\quad{\rm if\ and\ only\
if}\quad\exists\,s\in\mathcal{S}\\{\theta,\sigma\\}\ \,{\rm with}\
\,\theta_{s}(\sigma)=\tau.$ (2.3)

Consequently, one has usual notions as orbit, orbit closure, invariant subset,
etc.

######  Definition 2.3.

Let $(\mathcal{S},\theta,\Sigma)$ be an inverse semigroup action. The naïve
recurrence set assigned to $M,N\subset\Sigma$ is

$\displaystyle\mathcal{S}(\theta)^{N}_{M}\\!:$
$\displaystyle=\big{\\{}s\in\mathcal{S}\,\big{|}\,\theta_{s}(M)\cap
N\neq\emptyset\big{\\}}$ (2.4)
$\displaystyle=\big{\\{}s\in\mathcal{S}\,\big{|}\,\exists\,\sigma\in\Sigma_{s^{\sharp}}\cap
M,\ \theta_{s}(\sigma)\in N\big{\\}}\,.$

Clearly, if $s\in\mathcal{S}(\theta)^{N}_{M}$ and $s\leq t$ then
$t\in\mathcal{S}(\theta)^{N}_{M}$ . In the case $M=\\{\sigma\\}$ and
$N=\\{\tau\\}$ (with simplified notations)

$r,s,t\in\mathcal{S}(\theta)^{\tau}_{\sigma}\,\Rightarrow
s^{\sharp}\in\mathcal{S}(\theta)_{\tau}^{\sigma}\ \,{\rm and}\
\,rs^{\sharp}t\in\mathcal{S}(\theta)^{\tau}_{\sigma}\,.$

In particular $\mathcal{S}(\theta)^{\sigma}_{\sigma}$ is an inverse
subsemigroup of $\mathcal{S}$, and $\mathcal{S}(\theta)^{N}_{\sigma}$ is
contained in
$\mathcal{S}\\{\theta,\sigma\\}=\mathcal{S}(\theta)_{\sigma}^{\Sigma}$ for
every $N\subset\Sigma$ .

######  Definition 2.4.

We introduce an equivalence relation on $\mathcal{S}\\{\theta,\sigma\\}$ by

$s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t\quad{\rm if\ and\
only\ if}\quad\exists\,r\in\mathcal{S}\\{\theta,\sigma\\}\,,\ r\leq s,t\,.$
(2.5)

The quotient space is denoted by
$\mathscr{S}\\{\theta,\sigma\\}:=\mathcal{S}\\{\theta,\sigma\\}/_{\\!\overset{\theta}{\underset{\sigma}{\leftrightarrow}}}$
. The equivalence class of $s$ will be denoted by $\langle
s\rangle_{\sigma}^{\theta}$ or by $\pi_{\sigma}^{\theta}(s)$ , giving rise to
the quotient map

$\pi_{\sigma}^{\theta}:\mathcal{S}\\{\theta,\sigma\\}\to\mathscr{S}\\{\theta,\sigma\\}\,.$
(2.6)

######  Remark 2.5.

It follows from the definition that if
$s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$ then
$\theta_{s}(\sigma)=\theta_{t}(\sigma)$ (the converse fails in general).
Concerning the connection between
$\overset{\theta}{\underset{\sigma}{\leftrightarrow}}$ and the minimum group
congruence $\leftrightarrow$ , see below.

The next definition is meant to eliminate redundant contributions to
recurrence.

######  Definition 2.6.

Let $(\mathcal{S},\theta,\Sigma)$ be an inverse semigroup action. The
recurrence set assigned to $\sigma\in\Sigma$ and $N\subset\Sigma$ is

$\mathscr{S}(\theta)^{N}_{\sigma}\\!:=\mathcal{S}(\theta)^{N}_{\sigma}\\!/_{\\!\overset{\theta}{\underset{\sigma}{\leftrightarrow}}}=\big{\\{}\langle
s\rangle^{\theta}_{\sigma}\,\big{|}\,s\in\mathcal{S}(\theta)^{N}_{\sigma}\big{\\}}\,.$
(2.7)

A standard notion [10, 12] is the following:

######  Definition 2.7.

The maximum group homomorphic image of the inverse semigroup $\mathcal{S}$ is
the quotient $\mathcal{S}/_{\\!\leftrightarrow}$ of $\mathcal{S}$ through the
minimum group congruence

$s\leftrightarrow t\,\Leftrightarrow\,\exists\,r\in\mathcal{S}\,,\,r\leq
s,t\,.$ (2.8)

The class of $s\in\mathcal{S}$ is denoted by $\langle s\rangle$ or by $\pi(s)$
.

######  Remark 2.8.

The congruence $\leftrightarrow$ is intrinsic to $\mathcal{S}$. If
$(\mathcal{S},\theta,\Sigma)$ is a partial action, on every set
$\mathcal{S}\\{\theta,\sigma\\}$ we have defined the equivalence relation
$\overset{\theta}{\underset{\sigma}{\leftrightarrow}}$ in (2.5) , which does
depend on the given action. If $s,t\in\mathcal{S}\\{\theta,\sigma\\}$ for some
$\sigma$, then $s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$
implies $s\leftrightarrow t$ , but the opposite implication generally fails,
since in (2.8) $r$ is not bound to belong to $\mathcal{S}\\{\theta,\sigma\\}$
. In general the restriction of the equivalence relation $\leftrightarrow$ to
$\mathcal{S}\\{\theta,\sigma\\}$ is weaker than
$s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$ . However, see Lemma
2.9.

Recall that the inverse semigroup $\mathcal{S}$ is called $E$-unitary if every
element larger than an idempotent must be an idempotent. This can be restated
in several ways, for instance: if $e\leftrightarrow s$ and $e$ is idempotent,
then $s$ must also be idempotent. For us, an important fact is the following
result, that we did not find in the literature:

######  Lemma 2.9.

Let $(\mathcal{S},\theta,\Sigma)$ be the partial action of an $E$-unitary
inverse semigroup. If $\sigma\in\Sigma$ and
$s,t\in\mathcal{S}\\{\theta,\sigma\\}$ , then

$s\leftrightarrow
t\,\Leftrightarrow\,s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t\,.$

###### Proof.

As mentioned in Remark 2.8, we only need to check $\Rightarrow$ . So let
$s,t\in\mathcal{S}\\{\theta,\sigma\\}$ such that $s\leftrightarrow t$ . From
[10, Pag. 25, 66] it follows that they are compatible, i.e. $s^{\sharp}t$ and
$st^{\sharp}$ are idempotents, and $s\wedge t:=st^{\sharp}t=ts^{\sharp}s$ is
the infimum of $s$ and $t$ with respect to $\leq$ . The domain of
$\theta_{s\wedge t}$ contains

${\rm dom}\big{(}\theta_{s}\circ\theta_{t^{\sharp}}\circ\theta_{t}\big{)}={\rm
dom}\big{(}\theta_{s}\big{)}\cap{\rm
dom}\big{(}\theta_{t}\big{)}=\Sigma_{s^{\sharp}}\cap\Sigma_{t^{\sharp}}\ni\sigma\,,$

so $\mathcal{S}\\{\theta,\sigma\\}\ni s\wedge t\leq s,t$ , thus
$s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$ . ∎

######  Remark 2.10.

Let $(\mathcal{S},\theta,\Sigma)$ be the partial action of an inverse
semigroup and let $M,N\subset\Sigma$ . Then the recurrence set
$\mathscr{S}(\theta)^{N}_{M}$ may be introduced directly as the quotient of
the naïve recurrence set $\mathcal{S}(\theta)^{N}_{M}$ through the minimum
group congruence $\leftrightarrow$ :

$\mathscr{S}(\theta)^{N}_{M}\\!:=\mathcal{S}(\theta)^{N}_{M}/_{\\!\leftrightarrow}\,.$

If $\mathcal{S}$ is $E$-unitary, this may be written in terms of (2.7) as

$\mathscr{S}(\theta)^{N}_{M}=\\!\bigcup_{\sigma\in
M}\\!\mathscr{S}(\theta)^{N}_{\sigma}.$

##  3 Inverse semigroup expansions

Following [3], we describe briefly the prefix expansion inverse semigroup
$\mathcal{S}_{\mathcal{A}}$ associated to an inverse semigroup $\mathcal{A}$ .
A different point of view can be found in [11] ; the group case appeared in
[5] (see also [7]). $\mathcal{S}_{\mathcal{A}}$ is generated by elements $[a]$
, where $a\in\mathcal{A}$ , subject to the relations

$(i)\ \,[a^{\sharp}][a][b]=[a^{\sharp}][ab]\,,\quad(ii)\
\,[a][b][b^{\sharp}]=[ab][b^{\sharp}]\,,\quad(iii)\
\,[a][a^{\sharp}][a]=[a]\,.$

Buss and Exel show that there is an involutive anti-automorphism
$\,{}^{\\#}\\!:\mathcal{S}_{\mathcal{A}}\to\mathcal{S}_{\mathcal{A}}$ such
that $[a]^{\\#}\\!=[a^{\sharp}]$ for every $a\in\mathcal{A}$ . In fact
$\mathcal{S}_{\mathcal{A}}$ is an inverse semigroup; it is $E$-unitary if and
only if $\mathcal{A}$ is $E$-unitary (in this case Lemma 2.9 applies). The
elements $\epsilon_{a}:=[a][a^{\sharp}]=[a][a]^{\\#}$ are commuting
idempotents. It is also shown that every element of
$\mathcal{S}_{\mathcal{A}}$ has a unique normal form

$\epsilon_{c_{1}}\dots\epsilon_{c_{n}}[a]\,,\quad c_{1},\dots
c_{n},a\in\mathcal{A},\ n\in\mathbb{N}\,,$ (3.1)

under the conditions
$c_{1}c_{1}^{\sharp}=\dots=c_{n}c_{n}^{\sharp}=aa^{\sharp}$ and
$a,aa^{\sharp}\in\\{c_{1},\dots,c_{n}\\}$ . It is useful to use the notations

$\iota:\mathcal{A}\to\mathcal{S}_{\mathcal{A}}\,,\quad\iota(a):=[a]\,,$
$q:\mathcal{S}_{\mathcal{A}}\to\mathcal{A}\,,\quad
q\big{(}\epsilon_{c_{1}}\dots\epsilon_{c_{n}}[a]\big{)}\\!:=a\,.$

Clearly $\iota$ is a section of $q$ .

The main property of $\mathcal{S}_{\mathcal{A}}$ is its universal property: If
$\mathcal{T}$ is an inverse semigroup and $\psi:\mathcal{A}\to\mathcal{T}$ is
a map satisfying for all elements $a,b\in\mathcal{A}$ the conditions

$\psi(a^{\sharp})\psi(a)\psi(b)=\psi(a^{\sharp})\psi(ab)\,,\quad\psi(a)\psi(b)\psi(b^{\sharp})=\psi(ab)\psi(b^{\sharp})\,,\quad\psi(a)\psi(a^{\sharp})\psi(a)=\psi(a)\,,$

there is a unique inverse semigroup morphism
$\Psi:\mathcal{S}_{\mathcal{A}}\to\mathcal{T}$ such that $\Psi([a])=\psi(a)$
for each $a\in\mathcal{A}$ .

It is then easy to see that there is a one-to-one correspondence between
partial actions of the inverse semigroup $\mathcal{A}$ and (genuine) actions
of its prefix expansion inverse semigroup $\mathcal{S}_{\mathcal{A}}$ (in the
same topological space $\Sigma$).

######  Proposition 3.1.

The two actions have the same invariant sets.

###### Proof.

Start with the partial action $(\mathcal{A},\vartheta,\Sigma)$ and denote by
$\theta:\mathcal{S}_{\mathcal{A}}\to{\rm PHomeo}(\Sigma)$ the associated
inverse semigroup action. For an element in normal form (3.1) one has

$\theta_{\epsilon_{c_{1}}\dots\epsilon_{c_{n}}[a]}=\theta_{\epsilon_{c_{1}}}\\!\circ\dots\circ\theta_{\epsilon_{c_{n}}}\\!\circ\vartheta_{a}\,.$
(3.2)

Since $\epsilon_{c_{k}}$ is an idempotent, $\theta_{\epsilon_{c_{k}}}$ is the
identity map on its domain. Consequently, for every $\sigma\in\Sigma$
belonging to the (common) domain

${\rm
dom}\big{(}\theta_{\epsilon_{c_{1}}\dots\epsilon_{c_{n}}[a]}\big{)}=\Sigma_{a^{\sharp}}\cap\vartheta_{a}^{-1}\big{(}\Sigma_{\epsilon_{c_{1}}}\cap\dots\cap\Sigma_{\epsilon_{c_{n}}}\big{)}$
(3.3)

of the two transformations in (3.2), one has

$\theta_{\epsilon_{c_{1}}\\!\dots\epsilon_{c_{n}}[a]}(\sigma)=\vartheta_{a}(\sigma)\,.$

From this the conclusion can be deduced easily. ∎

Let us now relate the recurrence sets of the partial inverse semigroup action
$\vartheta$ to the two types of recurrence sets of the associated prefix
expansion inverse semigroup action $\theta$.

######  Proposition 3.2.

Let $\theta:\mathcal{S}_{\mathcal{A}}\to{\rm PHomeo}(\Sigma)$ be the inverse
semigroup action attached to the partial action
$(\mathcal{A},\vartheta,\Sigma)$ and let $M,N\subset\Sigma$ . Then

$\iota\big{[}\mathcal{A}(\vartheta)_{M}^{N}\big{]}=\mathcal{S}_{\mathcal{A}}(\theta)^{N}_{M}\cap\iota(\mathcal{A})\,,$
(3.4)
$q\Big{(}\mathcal{S}_{\mathcal{A}}(\theta)^{N}_{M}\Big{)}=\mathcal{A}(\vartheta)_{M}^{N}\,.$
(3.5)

###### Proof.

One has
$s=[a]\in\mathcal{S}_{\mathcal{A}}(\theta)^{N}_{M}\cap\,\iota(\mathcal{A})$ if
and only if there is some $\sigma\in
M\cap\Sigma_{[a]^{\\#}}\\!=M\cap\Sigma_{a^{\sharp}}$ with
$\theta_{[a]}(\sigma)=\vartheta_{a}(\sigma)\in N$, and this means exactly that
$[a]=\iota(a)\in\iota\big{[}\mathcal{A}(\vartheta)_{M}^{N}\big{]}$ . Thus
(3.4) is proven.

The relation (3.5) follows by analyzing the definitions, or we may write

$\displaystyle q\Big{(}\mathcal{S}_{\mathcal{A}}(\theta)^{N}_{M}\Big{)}$
$\displaystyle=q\Big{(}\mathcal{S}_{\mathcal{A}}(\theta)^{N}_{M}\Big{)}\cap
q\big{(}\iota(\mathcal{A})\big{)}$
$\displaystyle=q\Big{(}\mathcal{S}_{\mathcal{A}}(\theta)^{N}_{M}\cap\iota(\mathcal{A})\Big{)}$
$\displaystyle=q\Big{(}\iota\big{[}\mathcal{A}(\vartheta)_{M}^{N}\big{]}\Big{)}$
$\displaystyle=\mathcal{A}(\vartheta)_{M}^{N}\,.$

We used the fact that $\iota$ is a section of $q$ , equation (3.4) and the
fact that $\iota(\mathcal{A})$ is saturated (to distribute $q$ to the
intersection). ∎

######  Theorem 3.3.

Let $\theta:\mathcal{S}_{\mathcal{A}}\to{\rm PHomeo}(\Sigma)$ be the inverse
semigroup action attached to the partial action
$(\mathcal{A},\vartheta,\Sigma)$ and let $\sigma\in\Sigma$ .

  1. (i)

The map $q:\mathcal{S}_{\mathcal{A}}\to\mathcal{A}$ restricts to a surjection
$q_{\sigma}\\!:\mathcal{S}_{\mathcal{A}}\\{\theta,\sigma\\}\to\mathcal{A}\\{\vartheta,\sigma\\}$
.

  2. (ii)

The map

$Q_{\sigma}:\mathscr{S}_{\mathcal{A}}\\{\theta,\sigma\\}\\!:=\mathcal{S}_{\mathcal{A}}\\{\theta,\sigma\\}/_{\\!\overset{\theta}{\underset{\sigma}{\leftrightarrow}}}\to\mathcal{A}\\{\vartheta,\sigma\\}\,,\quad
Q_{\sigma}\big{[}\pi^{\theta}_{\sigma}(s)\big{]}:=q_{\sigma}(s)$

is a well-defined bijection. One has bijectively

$Q_{\sigma}\big{[}\mathscr{S}_{\mathcal{A}}(\theta)_{\sigma}^{N}\big{]}=\mathcal{A}(\vartheta)_{\sigma}^{N},\quad\forall\,N\subset\Sigma\,.$
(3.6)

###### Proof.

$(i)$ As mentioned above, the domain of
$\theta_{\epsilon_{c_{1}}\dots\epsilon_{c_{n}}[a]}$ is
$\Sigma_{a^{\sharp}}\cap\,\vartheta_{a}^{-1}\big{(}\Sigma_{\epsilon_{c_{1}}}\cap\dots\cap\Sigma_{\epsilon_{c_{n}}}\big{)}$
, contained in $\Sigma_{a^{\sharp}}$ , the domain of $\vartheta_{a}$ , where
$a=q\big{(}\epsilon_{c_{1}}\dots\epsilon_{c_{n}}[a]\big{)}$ . This shows that
$q\big{(}\mathcal{S}_{\mathcal{A}}\\{\theta,\sigma\\}\big{)}\subset\mathcal{A}\\{\vartheta,\sigma\\}$
. Surjectivity is obvious: if $a\in\mathcal{A}\\{\vartheta,\sigma\\}$ then
$[a]\in\mathcal{S}_{\mathcal{A}}\\{\theta,\sigma\\}$ and $q([a])=a$ .

$(ii)$ The statement about $Q_{\sigma}$ relies on showing that
$q_{\sigma}(s)=q_{\sigma}(t)$ if and only if
$s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$ ; this proves both
the correctness of the definition and the injectvity. (Surjectivity follows
from the fact that $q_{\sigma}$ is onto.) Using normal forms

$s:=\epsilon_{c_{1}}\dots\epsilon_{c_{n}}[a]\,,\quad
t:=\epsilon_{d_{1}}\dots\epsilon_{d_{m}}[b]\,,$

the assumption $q_{\sigma}(s)=q_{\sigma}(t)$ reads $a=b$ . Then

$r\\!:=\epsilon_{c_{1}}\dots\epsilon_{c_{n}}\epsilon_{d_{1}}\dots\epsilon_{d_{m}}[a]\in\mathcal{S}_{\mathcal{A}}$

satisfies $r\leq s,t$ . From (3.3) one gets easily

$\sigma\in{\rm dom}\big{(}\theta_{r}\big{)}={\rm
dom}\big{(}\theta_{s}\big{)}\cap{\rm dom}\big{(}\theta_{t}\big{)}\,,$

so $r\in\mathcal{S}_{\mathcal{A}}\\{\theta,\sigma\\}$ and we get
$s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$ . For the inverse
implication, using (2.1) and normal forms, one sees that $r\leq s$ implies
$q(r)=q(s)$ . Then $s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$
implies $r\leq s,t$ for some $r$, therefore $q(t)=q(r)=q(s)$ . The relation
(3.6) follows from previous results and definitions:

$\mathcal{A}(\vartheta)_{\sigma}^{N}\overset{\eqref{babulic}}{=}q\Big{(}\mathcal{S}_{\mathcal{A}}(\theta)^{N}_{\sigma}\Big{)}=Q_{\sigma}\Big{[}\pi^{\theta}_{\sigma}\Big{(}\mathcal{S}_{\mathcal{A}}(\theta)^{N}_{\sigma}\Big{)}\Big{]}\overset{\eqref{reciurset}}{=}Q_{\sigma}\Big{[}\mathscr{S}_{\mathcal{A}}(\theta)^{N}_{\sigma}\Big{]}\,.$

∎

##  4 Quotients through idempotent pure congruences

Let us fix an idempotent pure congruence $\approx$ in $\mathcal{S}$. By
definition [10], this means that $\approx$ is an equivalence relation on
$\mathcal{S}$ such that

$s\approx t\,,\ u\approx v\ \Rightarrow\ su\approx tv,$ $s\approx e\,,\
e\in\mathcal{E}(\mathcal{S})\ \Rightarrow\ s\in\mathcal{E}(\mathcal{S})\,.$

We set $p(s)$ for the $\approx$-equivalence class of $s\in\mathcal{S}$ . Note
that $p:\mathcal{S}\to\mathcal{S}/_{\\!\approx}$ is an inverse semigroup
epimorphism, where the inversion in the quotient is
$p(s)^{\dagger}\\!:=p\big{(}s^{\sharp}\big{)}$ . It is convenient to set
$\mathcal{R}:=\mathcal{S}/_{\\!\approx}$ . A crucial fact ([9, Lemma 2.2]) is
that if $(\mathcal{S},\theta,\Sigma)$ is a partial action of the inverse
semigroup $\mathcal{S}$, there exists a unique partial action
$\big{(}\mathcal{R},\vartheta,\Sigma\big{)}$ such that

  * •

$\Sigma^{\vartheta}_{a}=\bigcup_{p(s)=a}\Sigma^{\theta}_{s}\,,\
\forall\,a\in\mathcal{R}$ ,

  * •

$\vartheta_{p(s)}(\sigma)=\theta_{s}(\sigma)\,,\
\forall\,\sigma\in\Sigma^{\theta}_{s^{\sharp}}\big{(}\\!\subset\Sigma^{\vartheta}_{p(s)^{\dagger}}\big{)}$
.

######  Theorem 4.1.

Let $(\mathcal{S},\theta,\Sigma)$ and
$\big{(}\mathcal{R},\vartheta,\Sigma\big{)}$ as above.

  1. (i)

The two partial actions have the same orbits and the same invariant sets.

  2. (ii)

For every $\sigma\in\Sigma$ the map $p:\mathcal{S}\to\mathcal{R}$ restricts to
a surjection
$p_{\sigma}\\!:\mathcal{S}\\{\theta,\sigma\\}\to\mathcal{R}\\{\vartheta,\sigma\\}$
. For $N\subset\Sigma$ one has
$p_{\sigma}\big{[}\mathcal{S}(\theta)_{\sigma}^{N}\big{]}=\mathcal{R}(\vartheta)_{\sigma}^{N}$.

  3. (iv)

The map

$P_{\sigma}:\mathscr{S}\\{\theta,\sigma\\}\\!:=\mathcal{S}\\{\theta,\sigma\\}/_{\\!\overset{\theta}{\underset{\sigma}{\leftrightarrow}}}\to\mathscr{R}\\{\vartheta,\sigma\\}:=\mathcal{R}\\{\vartheta,\sigma\\}/_{\\!\overset{\vartheta}{\underset{\sigma}{\leftrightarrow}}}\,,\quad
P_{\sigma}\big{[}\pi^{\theta}_{\sigma}(s)\big{]}:=\pi^{\vartheta}_{\sigma}\big{[}p_{\sigma}(s)\big{]}$

is a well-defined surjection. For $N\subset\Sigma$ one has

$P_{\sigma}\big{[}\mathscr{S}(\theta)_{\sigma}^{N}\big{]}=\mathscr{R}(\vartheta)_{\sigma}^{N}.$
(4.1)

###### Proof.

$(i)$ follow easily from the definitions and from the way $\vartheta$ has been
constructed out of $\theta$ .

For $(ii)$, we have

$a\in\mathcal{R}\\{\vartheta,\sigma\\}\,\Leftrightarrow\,\sigma\in\Sigma^{\vartheta}_{a^{\dagger}}\\!=\bigcup_{s\in
a}\Sigma^{\theta}_{s^{\sharp}}\,\Leftrightarrow\,\exists\,s\in
a\,,\sigma\in\,\Sigma^{\theta}_{s^{\sharp}}\,\Leftrightarrow\,\exists\,s\in\mathcal{S}\\{\theta,\sigma\\}\,,\,p(s)=a\,.$

Including the set $N$ in the arguments is trivial.

$(iii)$ It is clear that $P_{\sigma}$ is onto. One still has to check that if
$s,t\in\mathcal{S}\\{\theta,\sigma\\}$ and
$s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$ then
$p_{\sigma}(s)\overset{\vartheta}{\underset{\sigma}{\leftrightarrow}}p_{\sigma}(t)$
, ensuring that $P_{\sigma}$ is well-defined. This follows easily from the
definition of the equivalences, since a homomorphism of inverse semigroups
preserves the orderings. Then (4.1) is a consequence of the previous
definitions and results:

$\mathscr{R}(\vartheta)_{\sigma}^{N}=\pi^{\vartheta}_{\sigma}\big{[}\mathcal{R}(\vartheta)^{N}_{\sigma}\big{]}=\pi^{\vartheta}_{\sigma}\big{[}p_{\sigma}\big{(}\mathcal{S}(\theta)_{\sigma}^{N}\big{)}\big{]}=P_{\sigma}\big{[}\pi^{\theta}_{\sigma}\big{(}\mathcal{S}(\theta)_{\sigma}^{N}\big{)}\big{]}=P_{\sigma}\big{[}\mathscr{S}(\theta)_{\sigma}^{N}\big{]}\,.$

∎

######  Example 4.2.

Let us suppose now that $\mathcal{S}$ is a $E$-unitary inverse semigroup. It
is known then [10, Sect. 2.4] that $\leftrightarrow$ is an idempotent pure
congruence, that we use instead of $\approx$ in the above arguments. In
addition $\mathcal{R}:=\mathcal{S}/_{\\!\leftrightarrow}$ is a group (the
maximum group homomorphic image of $\mathcal{S}$). To the partial action
$\theta$ of $\mathcal{S}$ on $\Sigma$ we associate as above the partial action
$\vartheta$ of $\mathcal{R}$ in $\Sigma$ . Since now $\mathcal{R}$ is a group,
the equivalence relation
$\overset{\vartheta}{\underset{\sigma}{\leftrightarrow}}$ on
$\mathcal{R}\\{\vartheta,\sigma\\}$ is just the equality, so
$\mathscr{R}\\{\vartheta,\sigma\\}=\mathcal{R}\\{\vartheta,\sigma\\}$. By
Theorem 4.1 one has the well-defined surjection
$P_{\sigma}:\mathscr{S}\\{\theta,\sigma\\}\to\mathscr{R}\\{\vartheta,\sigma\\}$
. In fact, it is even a bijection. To show this, one has to check for
$s,t\in\mathcal{S}\\{\theta,\sigma\\}$ that $p_{\sigma}(s)=p_{\sigma}(t)$
implies that $\pi^{\theta}_{\sigma}(s)=\pi^{\theta}_{\sigma}(s)$ . Taking the
definitions into consideration, this amounts to $s\leftrightarrow
t\,\Rightarrow\,s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$ . This
is solved by Lemma 2.9. Consequently, in this case, the recurrence sets
$\mathscr{S}(\theta)_{\sigma}^{N}$ and $\mathscr{R}(\vartheta)_{\sigma}^{N}$
are in a one-to-one correspondence for every $N\subset\Sigma$ .

##  5 From partial actions of inverse semigroups to groupoids

Groupoids are small categories in which all the arrows are inverible. The
object part of such a category $\Xi$ is also called the unit space and denoted
by $\Xi^{(0)}\\!\equiv X$. The source and range maps, denoted by ${\rm
d,r}:\Xi\to\Xi^{(0)}$, define the family of composable pairs

$\Xi^{(2)}\\!=\\{(\xi,\eta)\\!\mid\\!{\rm d}(\xi)={\rm
r}(\eta)\\}\\!\subset\Xi\times\Xi\,.$

For $M,N\subset X$ one sets

$\Xi_{M}\\!:={\rm d}^{-1}(M)\,,\quad\Xi^{N}\\!:={\rm
r}^{-1}(M)\,,\quad\Xi_{M}^{N}:=\Xi_{M}\cap\Xi^{N},$ (5.1)

with the particular notations $\Xi_{x}\equiv\Xi_{\\{x\\}}$ and
$\Xi^{x}\equiv\Xi^{\\{x\\}}$. A topological groupoid is a groupoid $\Xi$ with
a topology such that the inversion $\xi\mapsto\xi^{-1}$ and multiplication
$(\xi,\eta)\mapsto\xi\eta$ are continuous.

Following [2], we now define and use the groupoid of germs associated to the
partial action $\theta$ of the inverse semigroup $\mathcal{S}$ on the
topological space $\Sigma$ . (For the case of genuine actions, see [6, 7, 12]
and references therein.) In this generality, this also covers the
transformation groupoids of partial group actions from [5] . First define

$\displaystyle\mathcal{S}\,\square\,\Sigma\\!:$
$\displaystyle=\big{\\{}(s,\sigma)\in\mathcal{S}\\!\times\\!\Sigma\,\big{|}\,\sigma\in\Sigma_{s^{\sharp}}\big{\\}}$
(5.2)
$\displaystyle=\big{\\{}(s,\sigma)\in\mathcal{S}\\!\times\\!\Sigma\,\big{|}\,s\in\mathcal{S}\\{\theta,\sigma\\}\big{\\}}\,.$

######  Definition 5.1.

We say that $(s,\sigma),(t,\tau)\in\mathcal{S}\,\square\,\Sigma$ are germ
equivalent, and write $(s,\sigma)\overset{\theta}{\rightleftharpoons}(t,\tau)$
, if $\sigma=\tau$ and
$s\overset{\theta}{\underset{\sigma}{\leftrightarrow}}t$ .

######  Remark 5.2.

Note that $\overset{\theta}{\leftrightharpoons}$ is an equivalence relation,
that depends on the partial action $\theta$. However, when $\mathcal{S}$ is
$E$-unitary, by Lemma 2.9, it becomes independent of $\theta$ . In the case of
partial group actions, $\leftrightharpoons$ is just the equality.

The quotient
$\mathcal{S}\,\square\,\Sigma/_{\\!\overset{\theta}{\leftrightharpoons}}$ is
also denoted as $\mathcal{S}\triangleright_{\theta}\\!\Sigma$ . Its generic
elements are denoted by $\langle
s,\sigma\rangle\\!:=[(s,\sigma)]_{\overset{\theta}{\leftrightharpoons}}$ .
Then $\mathcal{S}\triangleright_{\theta}\\!\Sigma$ is a groupoid with unit
space $\Sigma$ , with the well-defined algebraic structure

$\mathfrak{d}\big{(}\langle
s,\sigma\rangle\big{)}:=\sigma\,,\quad\mathfrak{r}\big{(}\langle
s,\sigma\rangle\big{)}:=\theta_{s}(\sigma)\,,$ (5.3)
$\big{\langle}t,\theta_{s}(\sigma)\big{\rangle}\langle
s,\sigma\rangle:=\langle ts,\sigma\rangle\,,\quad\langle
s,\sigma\rangle^{\star}\\!:=\big{\langle}s^{\sharp},\theta_{s}(\sigma)\big{\rangle}\,.$

Note that $\langle e,\sigma\rangle$ is independent of
$e\in\mathcal{E}(\mathcal{S})$ if $\sigma\in\Sigma_{e}$ , hence it will be
identified with $\sigma$.

One defines a topology on $\mathcal{S}\triangleright_{\theta}\\!\Sigma$ as
follows: Given $s\in\mathcal{S}$ and an open set
$U\subset\Sigma_{s^{\sharp}}$, one sets

$\langle s,U\rangle:=\big{\\{}\langle
s,\sigma\rangle\in\mathcal{S}\triangleright_{\theta}\\!\Sigma\,\big{|}\,\sigma\in
U\big{\\}}\,.$

The family of all these sets forms a basis for a topology that makes
$\mathcal{S}\triangleright_{\theta}\\!\Sigma$ an étale topological groupoid.

We are going to use systematically groupoid actions in the next section. One
only needs here a particular case (the canonical action, cf. Example 6.2). For
a topological groupoid $\Xi$ with unit space $\Sigma$ , we write $\xi\ast{\rm
d}(\xi):={\rm r}(\xi)$ for every $\xi\in\Xi$ . This may be reformulated as
$\xi\ast\sigma:=\xi\sigma\xi^{-1}$ if $\sigma={\rm d}(\xi)$ . Orbit
equivalence is denoted by $\sigma\overset{\Xi}{\sim}\tau$, and it means that
$\sigma={\rm d}(\xi)$ and $\tau={\rm r}(\xi)$ for some $\xi\in\Xi$ , i.e.
$\Xi_{\sigma}^{\tau}\neq\emptyset$ .

######  Proposition 5.3.

Use the shorthand $\,\Xi\equiv\mathcal{S}\triangleright_{\theta}\\!\Sigma$ and
consider its canonical action on its unit space. One has
$\sigma\overset{\Xi}{\sim}\tau$ if and only if
$\sigma\overset{\theta}{\sim}\tau$ . Hence the orbits in the unit space of the
germ groupoid coincide with the orbits of the initial inverse semigroup
partial action. Invariant subsets are the same.

###### Proof.

By (2.3), $\sigma\overset{\theta}{\sim}\tau$ if and only if
$\theta_{s}(\sigma)=\tau$ for some $s\in\mathcal{S}$ such that
$\sigma\in\Sigma_{s^{\sharp}}$. Equivalently, $\langle
s,\sigma\rangle\ast\sigma=\tau$. Indeed, the only not-straightforward piece is
the computation

$\langle s,\sigma\rangle\ast\sigma=\langle s,\sigma\rangle\langle
s^{\sharp}s,\sigma\rangle\langle s,\sigma\rangle^{-1}=\langle
ss^{\sharp}s,\sigma\rangle\langle s^{\sharp},\theta_{s}(\sigma)\rangle=\langle
ss^{\sharp},\theta_{s}(\sigma)\rangle=\theta_{s}(\sigma)=\tau$

and the proof is finished. The remaining assertions are direct consequences. ∎

Let us define

$\Theta:\mathcal{S}\,\square\,\Sigma\to\Sigma\times\Sigma\,,\quad{\Theta}(s,\sigma):=\big{(}\theta_{s}(\sigma),\sigma\big{)}\,.$

It is easy to see, by Remark 2.5, that the map

$\widehat{\Theta}:\mathcal{S}\triangleright_{\theta}\\!\Sigma\to\Sigma\times\Sigma\,,\quad\widehat{\Theta}\big{(}\langle
s,\sigma\rangle\big{)}:=\big{(}\theta_{s}(\sigma),\sigma\big{)}$

is well-defined. We get the diagram

$\begin{diagram}$ (5.4)

where $j$ is an obvious embedding, $p_{1}$ is the first projection,
${\mathfrak{p}}_{1}$ its restriction to $\mathcal{S}\,\square\,\Sigma$ and
$\pi$ is the quotient map.

######  Definition 5.4.

For $M,N\subset\Sigma$ , the corresponding groupoid recurrence set of the
action $\theta$ is $\widehat{\Theta}^{-1}(N\\!\times\\!M)$ .

The pure groupoid interpretation of this set is seen in the first part of the
following Proposition, while its connection with the naïve recurrence set of
the inverse semigroup partial actions is subject of the second.

######  Proposition 5.5.

For every $M,N\subset\Sigma$ , and using the notation (5.1), one has

$\big{(}\mathcal{S}\triangleright_{\theta}\\!\Sigma\big{)}^{N}_{M}=\widehat{\Theta}^{-1}(N\\!\times\\!M)\,.$
(5.5)

On the other hand,

$\mathcal{S}(\theta)_{M}^{N}=\mathfrak{p}_{1}\big{[}{\Theta}^{-1}(N\\!\times\\!M)\big{]}\,.$
(5.6)

###### Proof.

Taking into account (5.3), one has

$\displaystyle\big{(}\mathcal{S}\triangleright_{\theta}\\!\Sigma\big{)}^{N}_{M}$
$\displaystyle=\big{\\{}\langle
s,\sigma\rangle\in\mathcal{S}\triangleright_{\theta}\\!\Sigma\,\big{|}\,\mathfrak{d}\big{(}\langle
s,\sigma\rangle\big{)}=\sigma\in M,\,\mathfrak{r}\big{(}\langle
s,\sigma\rangle\big{)}=\theta_{s}(\sigma)\in N\big{\\}}$
$\displaystyle=\big{\\{}\langle
s,\sigma\rangle\in\mathcal{S}\triangleright_{\theta}\\!\Sigma\,\big{|}\,\big{(}\theta_{s}(\sigma),\sigma\big{)}\in
N\times M\big{\\}}$ $\displaystyle=\widehat{\Theta}^{-1}(N\times M)\,,$

and (5.5) is proven. For (5.6):

$\displaystyle\mathfrak{p}_{1}\big{[}{\Theta}^{-1}(N\\!\times\\!M)\big{]}$
$\displaystyle=\\{s\in\mathcal{S}\\!\mid\\!\exists\,(s,\sigma)\in\mathcal{S}\square\Sigma\,,\,{\Theta}(s,\sigma)\in
N\\!\times\\!M\\}$
$\displaystyle=\\{s\in\mathcal{S}\\!\mid\\!\exists\,(s,\sigma)\in\mathcal{S}\square\Sigma\,,\,\theta_{s}(\sigma)\in
N,\sigma\in M\\}$
$\displaystyle=\\{s\in\mathcal{S}\\!\mid\\!\exists\,\sigma\in\Sigma_{s^{\sharp}}\cap
M,\,\theta_{s}(\sigma)\in N\\}$ $\displaystyle=\mathcal{S}(\theta)_{M}^{N}\,.$

We used (5.2) and (2.4). ∎

######  Remark 5.6.

Let
$i:\mathcal{S}\triangleright_{\theta}\\!\Sigma\to\mathcal{S}\,\square\,\Sigma$
be a section of $\pi$ in diagram 5.4. Setting

$J:=\mathfrak{p}_{1}\circ i=p_{1}\circ j\circ
i:\mathcal{S}\triangleright_{\theta}\\!\Sigma\to\mathcal{S},$

it follows immediately that

$J\Big{[}\big{(}\mathcal{S}\triangleright_{\theta}\\!\Sigma\big{)}_{M}^{N}\Big{]}\subset\mathcal{S}(\theta)_{M}^{N}\,.$

The inclusion is usually strict, and $J$ has no remarkable properties.

For a one-point set $M=\\{\sigma\\}$ there is a direct way to relate germ
groupoid recurrence sets with the inverse semigroup recurrent sets (2.7).
Taking into account Definition 5.1, the map

$\gamma_{\sigma}:\big{(}\mathcal{S}\triangleright_{\theta}\Sigma\big{)}_{\sigma}\\!\to\mathscr{S}\\{\theta,\sigma\\}\,,\quad\gamma_{\sigma}\big{(}\langle
s,\sigma\rangle\big{)}:=\langle
s\rangle^{\theta}_{\sigma}=\pi^{\theta}_{\sigma}(s)$

is well-defined and bijective. From the definitions we deduce that, for every
$N\subset\Sigma$ , the map $\gamma_{\sigma}$ sends
$\big{(}\mathcal{S}\triangleright_{\theta}\Sigma\big{)}_{\sigma}^{N}$ to
$\mathscr{S}(\theta)_{\sigma}^{N}$ . We formulate for further reference

######  Proposition 5.7.

For every $N\subset\Sigma$ , the map
$\big{(}\mathcal{S}\triangleright_{\theta}\Sigma\big{)}_{\sigma}^{N}\ni\langle
s,\sigma\rangle\to\langle
s\rangle^{\theta}_{\sigma}\in\mathscr{S}\\{\theta,\sigma\\}$ restricts to a
bijection between
$\big{(}\mathcal{S}\triangleright_{\theta}\\!\Sigma\big{)}^{N}_{\sigma}$ and
$\mathscr{S}(\theta)_{\sigma}^{N}$ .

######  Remark 5.8.

Actually one can also introduce

$\gamma:\mathcal{S}\triangleright_{\theta}\Sigma\to\mathcal{S}/_{\\!\leftrightarrow}\,,\quad\gamma\big{(}\langle
s,\sigma\rangle\big{)}:=\langle s\rangle=\pi(s)\,,$

involving the maximal group homomorphic image from Definition 2.7. It is a
well-defined surjection, but it is generally not one-to-one and it seems less
convenient for our purposes.

We indicate now a result that complements Proposition 5.7. Recall that to a
partial action $(\mathcal{S},\theta,\Xi)$ one associates the germ groupoid
$S\triangleright_{\theta}\\!\Sigma$ .

######  Proposition 5.9.

Let us set $\,\Gamma:2^{\mathcal{S}}\to
2^{\mathcal{S}\triangleright_{\theta}\Sigma}$ , where

$\Gamma(R):=\big{\\{}\langle s,\sigma\rangle\,\big{|}\,s\in
R\,,\,\sigma\in\Sigma_{s^{\sharp}}\big{\\}}\,.$

For every $M,N\subset\Sigma$ one has

$\big{(}\mathcal{S}\triangleright_{\theta}\\!\Sigma\big{)}^{N}_{M}\subset\Gamma\big{[}\mathcal{S}(\theta)^{N}_{M}\big{]}\,.$

The inclusion may be strict. For a one-point set $M=\\{\sigma\\}$ , it is an
equality:

$\big{(}\mathcal{S}\triangleright_{\theta}\\!\Sigma\big{)}^{N}_{\sigma}\\!=\Gamma\big{[}\mathcal{S}(\theta)^{N}_{\sigma}\big{]}\,.$

###### Proof.

For $s\in\mathcal{S}$ and $\sigma\in\Sigma$ we have

$\displaystyle\langle
s,\tau\rangle\in\big{(}\mathcal{S}\triangleright_{\theta}\\!\Sigma\big{)}^{N}_{M}$
$\displaystyle\ \Leftrightarrow\ \tau\in\Sigma_{s^{\sharp}}\,,\
\mathfrak{d}\langle s,\tau\rangle\in M\,,\ \mathfrak{r}\langle
s,\tau\rangle\in N$ $\displaystyle\ \Leftrightarrow\
\tau\in\Sigma_{s^{\sharp}}\,,\ \tau\in M\,,\ \theta_{s}(\tau)\in N$
$\displaystyle\ \Rightarrow\ \tau\in\Sigma_{s^{\sharp}}\,,\ \theta_{s}(M)\cap
N\neq\emptyset$ $\displaystyle\ \Leftrightarrow\
\tau\in\Sigma_{s^{\sharp}}\,,\ s\in\mathcal{S}(\theta)^{N}_{M}$
$\displaystyle\ \Leftrightarrow\ \langle
s,\tau\rangle\in\Gamma\big{[}\mathcal{S}(\theta)^{N}_{M}\big{]}\,.$

At the third line, of course, there is just a one-sided implication: the
condition $\theta_{s}(M)\cap N\neq\emptyset$ has nothing to do with $\tau$.
However, if $M$ is specified to be a singleton, one clearly gets an equality.
∎

######  Remark 5.10.

We indicate briefly the connection between Propositions 5.9 and 5.7. First,
note that the map $\Gamma$ restricts to a surjection
$\Gamma_{\sigma}\\!:2^{\mathcal{S}\\{\theta,\sigma\\}}\\!\to
2^{(\mathcal{S}\triangleright_{\theta}\Sigma)_{\sigma}}$, which in its turn
gives rise to a well-defined bijection
$\dot{\Gamma}_{\sigma}\\!:2^{\mathscr{S}\\{\theta,\sigma\\}}\\!\to
2^{(\mathcal{S}\triangleright_{\theta}\Sigma)_{\sigma}}$ given by

$\dot{\Gamma}_{\sigma}\big{[}\pi^{\theta}_{\sigma}(R)\big{]}:=\Gamma_{\sigma}(R)=\gamma_{\sigma}^{-1}\big{[}\pi^{\theta}_{\sigma}(R)\big{]}\,,\quad\forall\,R\subset\mathcal{S}\\{\theta,\sigma\\}\,.$
(5.7)

In (5.7), the first equality is the definition. The second equality, involving
an inverse image, is the desired connection, easy to prove. To summarize, we
have $\dot{\Gamma}_{\sigma}=\gamma_{\sigma}^{-1}$, where the right hand side
should be seen as a set function.

##  6 From groupoid actions to inverse semigroup actions

######  Definition 6.1.

A continuous groupoid action $(\Xi,\rho,\kappa,\Sigma)$ consists of a
topological groupoid $\Xi$ , a topological space $\Sigma$ , a continuous
surjective map $\rho:\Sigma\rightarrow X$ (the anchor) and the continuous
action map

$\kappa:\Xi\\!\Join\\!\Sigma:=\\{(\xi,\sigma)\\!\mid\\!{\rm
d}(\xi)=\rho(\sigma)\\}\ni(\xi,\sigma)\mapsto{\kappa_{\xi}(\sigma)\equiv\xi\\!\bullet_{\kappa}\\!\sigma}\in\Sigma$

satisfying the axioms:

  1. 1.

$\rho(\sigma)\\!\bullet_{\kappa}\\!\sigma=\sigma,\,\forall\,\sigma\in\Sigma$ ,

  2. 2.

if $(\xi,\eta)\in\Xi^{(2)}$ and $(\eta,\sigma)\in\Xi\\!\Join\\!\Sigma$, then
$(\xi,\eta\\!\bullet_{\kappa}\\!\sigma)\in\Xi\\!\Join\\!\Sigma$ and
$(\xi\eta)\\!\bullet_{\kappa}\\!\sigma=\xi\\!\bullet_{\kappa}\\!(\eta\\!\bullet_{\kappa}\\!\sigma)$
.

If the action $\kappa$ is understood, we will write $\xi\bullet\sigma$ instead
of $\xi\bullet_{\kappa}\sigma$. The theory is developed in [13].

######  Example 6.2.

Each topological groupoid acts continuously in a canonical way on its unit
space: $\xi$ sends ${\rm d}(\xi)$ into ${\rm r}(\xi)$ . Here $\Sigma=X$ and
$\rho={\rm id}_{X}$, and then (special notation) $\xi\\!\ast\\!x:=\xi
x\xi^{-1}$ if ${\rm d}(\xi)=x$.

######  Example 6.3.

The topological groupoid $\Xi$ also acts on itself, with $\Sigma:=\Xi$ ,
$\rho:={\rm r}$ and $\xi\bullet\eta:=\xi\eta$ .

For $\xi\in\Xi\,,\,{\sf A},{\sf B}\subset\Xi\,,\,M\subset\Sigma$ we use the
notations

${\sf A}{\sf B}:=\big{\\{}\xi\eta\,\big{|}\,\xi\in{\sf A}\,,\,\eta\in{\sf
B}\,,\,{\rm d}(\xi)={\rm r}(\eta)\big{\\}}\,,$ (6.1) ${\sf A}\bullet
M:=\big{\\{}\xi\bullet\sigma\,\big{|}\,\xi\in{\sf A}\,,\sigma\in M\,,{\rm
d}(\xi)=\rho(\sigma)\big{\\}}=\bigcup_{\xi\in{\sf A}}\xi\bullet M\,.$

A subset $M\subset\Sigma$ is called invariant if $\xi\bullet M\subset M$, for
every $\xi\in\Xi$ . Particular cases are the orbits
$\mathfrak{O}_{\sigma}\\!:=\Xi_{\rho(\sigma)}\\!\bullet\sigma$ and the orbit
closures $\overline{\mathfrak{O}}_{\sigma}$ . The orbit equivalence relation
will be denoted by $\overset{\kappa}{\sim}$ or by $\overset{\bullet}{\sim}$ .

######  Definition 6.4.

For every $M,N\subset\Sigma$ one defines the recurrence set

$\widetilde{\Xi}_{M}^{N}=\\{\xi\in\Xi\\!\mid\\!(\xi\bullet M)\cap
N\neq\emptyset\\}\,.$

######  Remark 6.5.

Note that $\widetilde{\Xi}_{M}^{N}\subset\Xi_{\rho(M)}^{\rho(N)}$ . When
$\rho$ is also injective, one has
$\,\widetilde{\Xi}_{M}^{N}=\Xi_{\rho(M)}^{\rho(N)}$ . In Exemple 6.2 one has
$\widetilde{\Xi}_{M}^{N}=\Xi_{M}^{N}=\\{\xi\in\Xi\\!\mid\\!{\rm d}(\xi)\in
M,\,{\rm r}(\xi)\in N\\}$ , making the connection with the previous section.

A bisection of the topological groupoid $\Xi$ is an open subset of the
groupoid on which the restrictions of both ${\rm d}$ and ${\rm r}$ are
injective. We recall that the groupoid $\Xi$ is called étale if ${\rm
d}:\Xi\to\Xi$ is a local homeomorphism. If $\Xi$ is étale, $X\equiv\Xi^{(0)}$
is a clopen (closed and open) subset of $\Xi$ , all the fibres $\Xi^{x}$ and
$\Xi_{x}$ are discrete, and ${\rm d},{\rm r}$ and the multiplication are open
maps.

Let $\Xi$ be an étale groupoid over the unit space $X$ and let ${\rm
Bis}(\Xi)$ be the inverse semigroup associated to $\Xi$ , formed of bisections
(they form a basis of clopen sets for the topology of $\Xi$) . The
multiplication is given by (6.1) and the inverse is ${\sf A}^{\sharp}\\!:={\sf
A}^{-1}$, under which

${\sf A}^{-1}{\sf A}={\rm d}({\sf A})\,,\quad{\sf A}{\sf A}^{-1}={\rm r}({\sf
A})$

hold. The family of idempotents $\mathcal{E}[{\rm Bis}(\Xi)]={\rm Top}(X)$ is
formed exactly of the open subsets of the unit space, $X$ being the unit and
$\emptyset$ the zero element.

######  Definition 6.6.

Let $(\Xi,\rho,\kappa,\Sigma)$ be a groupoid action, with $\Xi$ étale. The
inverse semigroup ${\rm Bis}(\Xi)$ acts continuously on $\Sigma$ by

${\rm dom}(\theta_{\sf A})\equiv\Sigma_{{\sf A}^{-1}}\\!:=\rho^{-1}[{\rm
d}({\sf A})]\,,\quad{\rm im}(\theta_{\sf A})\equiv\Sigma_{{\sf
A}}\\!:=\rho^{-1}[{\rm r}({\sf A})]\,,\quad$ $\theta_{\sf A}(\sigma):=\xi^{\sf
A}_{\rho(\sigma)}\\!\bullet_{\kappa}\sigma\equiv\kappa_{\xi^{\sf
A}_{\rho(\sigma)}}\\!(\sigma)\,,\quad\forall\,{\sf A}\in{\rm Bis}(\Xi)\,,\
\sigma\in\rho^{-1}[{\rm d}({\sf A})]\,,$ (6.2)

where $\xi^{\sf A}_{x}={\rm d}|_{A}^{-1}(x)$ denotes the unique element
$\xi\in{\sf A}$ such that ${\rm d}(\xi)=x\in{\sf A}\subset X$. The action is
genuine.

######  Proposition 6.7.

The orbits and the invariant sets in $\Sigma$ of the action $\kappa$ of the
groupoid $\Xi$ coincide with those of the associated action $\theta$ of
$\,{\rm Bis}(\Xi)$ in $\Sigma$ .

###### Proof.

We will prove that $\sigma\overset{\kappa}{\sim}\tau\ \Leftrightarrow\
\sigma\overset{\theta}{\sim}\tau$. We have $\sigma\overset{\theta}{\sim}\tau$
if and only if $\exists\,{\sf A}\in{\rm Bis}(\Xi)$ with $\sigma\in\Sigma_{{\sf
A}^{-1}}$ and $\theta_{\sf A}(\sigma)=\tau$. By definition, this means that
$\rho(\sigma)\in{\rm d}({\sf A})$ and $\xi_{\rho(\sigma)}^{\sf
A}\bullet\sigma=\tau$. This is equivalent with
$\sigma\overset{\kappa}{\sim}\tau$, taking into account the defining property
of a bisection. The remaining assertion of the proposition follows from this.
∎

To simplify, we are going to use the notation $\mathcal{B}:={\rm Bis}(\Xi)$ .
Thus there are also notations as $\mathcal{B}\\{\theta,\sigma\\}$ ,
$\mathscr{B}\\{\theta,\sigma\\}$ , for instance. We recall the quotient map
(2.6), that in our concrete situation reads

$\pi^{\theta}_{\sigma}:\mathcal{B}\\{\theta,\sigma\\}\to\mathscr{B}\\{\theta,\sigma\\}:=\mathcal{B}\\{\theta,\sigma\\}/_{\\!\overset{\theta}{\underset{\sigma}{\leftrightarrow}}}\,.$

We are looking for connections between different types of recurrence sets,
attached to the actions, cf. Definitions 2.3, 2.6 and 6.4.

######  Theorem 6.8.

The map

$\dot{\delta}_{\sigma}:\mathscr{B}\\{\theta,\sigma\\}\to\widetilde{\Xi}_{\sigma}^{\Sigma}=\Xi_{\rho(\sigma)}\,,\quad\dot{\delta}_{\sigma}\big{[}\pi^{\theta}_{\sigma}({\sf
A})\big{]}:=\xi^{\sf A}_{\rho(\sigma)}$

is a well-defined bijection. For every $N\subset\Sigma$ one has

$\dot{\delta}_{\sigma}\big{[}\mathscr{B}(\theta)_{\sigma}^{N}\big{]}=\widetilde{\Xi}_{\sigma}^{N}.$
(6.3)

###### Proof.

First set

$\delta_{\sigma}:\mathcal{B}\\{\theta,\sigma\\}\to\Xi_{\rho(\sigma)}\,,\quad\delta_{\sigma}({\sf
A}):=\xi^{\sf A}_{\rho(\sigma)}\,,$

which is onto, since the bisections cover the groupoid. Let ${\sf A},{\sf
B}\in\mathcal{B}\\{\theta,\sigma\\}$ , which means that ${\sf A},{\sf B}$ are
bisections and $\rho(\sigma)\in{\rm d}({\sf A})\cap{\rm d}({\sf B})$ . The
first statement of the Proposition follows if we check that

$\delta_{\sigma}({\sf A})=\delta_{\sigma}({\sf B})\,\Leftrightarrow\ {\sf
A}\overset{\theta}{\underset{\sigma}{\leftrightarrow}}{\sf B}\,.$ (6.4)

It is easy to see for two bisections that ${\sf C}\leq{\sf D}$ if and only if
${\sf C}\subset{\sf D}$ . Using the definitions, we see that (6.4) reads in
detail

$\xi^{\sf A}_{\rho(\sigma)}\\!=\xi^{\sf B}_{\rho(\sigma)}\,\Leftrightarrow\
\exists\,{\sf C}\subset{\sf A}\cap{\sf B}\ {\rm with}\ \rho(\sigma)\in{\rm
d}({\sf C})\,.$ (6.5)

The implication $\Rightarrow$ in (6.5) is solved by taking ${\sf C}:={\sf
A}\cap{\sf B}\ni\xi^{\sf A}_{\rho(\sigma)}\\!=\xi^{\sf B}_{\rho(\sigma)}$ .
The other implication is trivial, by the unicity involved in the definition of
the bisection ${\sf C}$ . Then (6.3) is a consequence of (6.2) and of the
definitions of the recurrence sets $\mathscr{B}(\theta)_{\sigma}^{N}$ and
$\widetilde{\Xi}_{\sigma}^{N}$. ∎

The next result yields a second point of view upon the connection between
recurrence sets. We set

$\Delta:2^{\Xi}\to 2^{\mathcal{B}},\quad\Delta(E):=\\{{\sf
A}\in\mathcal{B}\\!\mid\\!{\sf A}\cap E\neq\emptyset\\}\,.$

######  Proposition 6.9.

Let $(\Xi,\rho,\kappa,\Sigma)$ be an étale groupoid action and denote by
$\big{(}\mathcal{B},\theta,\Sigma\big{)}$ the associated inverse semigroup
partial action, where $\mathcal{B}={\rm Bis}(\Xi)$ . Let $\sigma\in\Sigma$ and
$M,N\subset\Sigma$ .

  1. (i)

One has

$\mathcal{B}(\theta)^{N}_{M}=\Delta\big{(}\widetilde{\Xi}^{N}_{M}\big{)}\,.$
(6.6)

  2. (ii)

Let us denote by $\Delta_{\sigma}:2^{\Xi_{\rho(\sigma)}}\to
2^{\mathcal{B}\\{\theta,\sigma\\}}$ the restriction of $\Delta$ to
$2^{\Xi_{\rho(\sigma)}}$. Then $\Delta_{\sigma}=\delta_{\sigma}^{-1}$ (the
last one regarded as a set function).

###### Proof.

$(i)$ For ${\sf A}\in\mathcal{B}$ one may write

$\displaystyle{\sf A}\in\mathcal{B}(\theta)^{N}_{M}$ $\displaystyle\
\Leftrightarrow\ \theta_{A}(M)\cap N\neq\emptyset$ $\displaystyle\
\Leftrightarrow\ \exists\ \sigma\in M\cap\rho^{-1}[{\rm d}({\sf A})]\,,\
\theta_{\sf A}(\sigma)\in N$ $\displaystyle\ \Leftrightarrow\ \exists\
\sigma\in M\cap\rho^{-1}[{\rm d}({\sf A})]\,,\ \xi^{\sf
A}_{\rho(\sigma)}\\!\bullet_{\kappa}\\!\sigma\in N$ $\displaystyle\
\Leftrightarrow\ \exists\ \sigma\in M\cap\rho^{-1}[{\rm d}({\sf A})]\,,\
\exists\,\xi\in{\sf A}\cap\Xi_{\rho(\sigma)}\,,\
\xi\bullet_{\kappa}\\!\sigma\in N$ $\displaystyle\ \Leftrightarrow\
\exists\,\xi\in{\sf A}\,,\ \big{(}\xi\bullet_{\kappa}\\!M\big{)}\cap
N\neq\emptyset$ $\displaystyle\ \Leftrightarrow\ {\sf
A}\cap\widetilde{\Xi}^{N}_{M}\neq\emptyset$ $\displaystyle\ \Leftrightarrow\
{\sf A}\in\Delta\big{(}\widetilde{\Xi}^{N}_{M}\big{)}\,.$

The forth equivalence is due to the uniqueness of an element $\xi$ of the
bisection ${\sf A}$ such that ${\rm d}(\xi)=\rho(\sigma)$ .

$(ii)$ One gets as a particular case of (6.6) the relation

$\mathcal{B}\\{\theta,\sigma\\}=\mathcal{B}(\theta)_{\sigma}^{\Sigma}=\Delta\big{(}\,\widetilde{\Xi}_{\sigma}\big{)}=\Delta\big{(}\,\Xi_{\rho(\sigma)}\big{)}\,.$

This and the fact that $\Delta$ is increasing justifies the restriction
$\Delta_{\sigma}$ in the diagram. If $E\in 2^{\Xi_{\rho(\sigma)}}$ then for
every $\eta\in E$ one has ${\rm d}(\eta)=\rho(\sigma)$ , which justifies the
non-trivial part of the third step below:

$\displaystyle\delta_{\sigma}^{-1}(E)$ $\displaystyle=\big{\\{}{\sf
A}\in\mathcal{B}\\{\theta,\sigma\\}\,\big{|}\,\delta_{\sigma}({\sf A})\in
E\big{\\}}$ $\displaystyle=\big{\\{}{\sf
A}\in\mathcal{B}\\{\theta,\sigma\\}\,\big{|}\,\xi^{\sf A}_{\rho(\sigma)}\\!\in
E\big{\\}}$ $\displaystyle=\big{\\{}{\sf
A}\subset\mathcal{B}\\{\theta,\sigma\\}\,\big{|}\,{\sf A}\cap
E\neq\emptyset\big{\\}}$ $\displaystyle=\Delta_{\sigma}(E)\,.$

∎

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M. Măntoiu:

Departamento de Matemáticas, Universidad de Chile,

Las Palmeras 3425, Casilla 653, Santiago, Chile

E-mail<EMAIL_ADDRESS>