# Globalization of Partial Actions of Ordered Groupoids on Rings

Wesley G. Lautenschlaeger and Thaísa Tamusiunas  Instituto de Matemática,
Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 9500,
91509-900. Porto Alegre-RS, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We provide a necessary and sufficient condition to the existence of an ordered
globalization of a partial ordered action of an ordered groupoid on a ring and
we also present criteria to obtain uniqueness. Furthermore, we apply those
results to obtain a Morita context and to show that an inverse semigroup
partial action has a globalization (unique up to isomorphism) if, and only if,
it is unital.

2020 AMS Subject Classification: Primary. 20L05, 18B40, 20N02. Secondary.
16S35, 20M18

Keywords: ordered groupoid, partial action, ordered globalization, Morita
theory, inverse semigroup

##  1\. Introduction

Much has been developed in the theory of actions and partial actions of
groupoids. For instance, Galois theories were considered in [4], [5], [8],
[14], [24], [25], [26] and [27], actions in sets and duality theorems in [9],
generalizations of classic results of group theory such as isomorphism
theorems, Lagrange’s Theorem, Sylow theorems, classification of groupoids and
properties of cosets in [1], [2], [6] and [21], and partial actions of ordered
groupoids in [3], [15] and [22].

The concept of partial group action was given by Exel in [13] aiming to
classify a certain class of $C^{*}$-algebras. J. Kellendonk and M. Lawson [19]
investigated applications in several other areas, such as graph immersions and
inverse semigroups [19], $\mathbb{R}$-trees [28] and tilings of the Euclidean
space [18]. Much more can be found in the surveys written by Dokuchaev in [10]
and [11]. The definition of partial action of a groupoid was given by Bagio
and Paques in [4]. Among other results, they proved a condition for the
existence and uniqueness of a globalization. Partial ordered actions of
ordered groupoids on rings were introduced in [3], but no study on the
existence of globalization has been done so far.

Our main goal in this work is to analyze the existence and uniqueness of
ordered globalizations of partial ordered actions of ordered groupoids on
rings. In Section 2 we provide a background about partial ordered actions of
ordered groupoids on rings and fix some notations. In Section 3 we prove that
a partial ordered action of an ordered groupoid on a ring has a globalization
if, and only if, it is unital. In Section 4 we define strong partial ordered
actions, pseudoassociative groupoids and minimal globalizations, showing that
a unital strong partial ordered action of a pseudoassociative groupoid on a
ring always has a unique minimal globalization. Finally, in Section 5 we
present two applications: we construct a Morita context between the crossed
products of an ordered groupoid acting partially on a ring and its
globalization and we apply our results in the case of inverse semigroups
acting on rings via the Ehresmann-Schein-Nambooripad (ESN) Theorem [20,
Theorem 4.1.8], which states that the category of inverse semigroups and
premorphisms is isomorphic to the category of inductive groupoids and ordered
functors.

Throughout, rings and algebras are associative and unital unless otherwise
stated.

##  2\. Preliminaries

We recall that a _groupoid_ $\mathcal{G}$ is a small category in which every
morphism is an isomorphism. We denote by $\mathcal{G}_{0}$ the set of objects
of $\mathcal{G}$. Observe that
$\text{id}:\mathcal{G}_{0}\rightarrow\mathcal{G}$, given by
$\text{id}(x)=\text{id}_{x}$, is an injective map and whence we identify
$\mathcal{G}_{0}\subseteq\mathcal{G}$. Given $g\in\mathcal{G}$, the _domain_
and the _range_ of $g$ will be denoted by $d(g)$ and $r(g)$, respectively.
Hence, $d(g)=g^{-1}g$ and $r(g)=gg^{-1}$. For all $g,h\in\mathcal{G}$, we
write $\exists gh$ whenever the product $gh$ is defined. We denote
$\mathcal{G}_{2}=\\{(g,h)\in\mathcal{G}\times\mathcal{G}:\exists gh\\}$.

We say that $\mathcal{G}$ is _ordered_ if there is a partial order $\leq$ on
$\mathcal{G}$ such that:

  * (OG1)

If $g\leq h$ then $g^{-1}\leq h^{-1}$;

  * (OG2)

For all $g,h,k,\ell\in\mathcal{G}$ such that $g\leq h$, $k\leq\ell$, $\exists
gk$ and $\exists h\ell$, we have $gk\leq h\ell$;

  * (OG3)

Given $g\in\mathcal{G}$ and $e\in\mathcal{G}_{0}$ such that $e\leq d(g)$,
there is a unique element $(g|e)\in\mathcal{G}$ which satisfies $(g|e)\leq g$
and $d(g|e)=e$.

  * (OG3*)

Given $g\in\mathcal{G}$ and $e\in\mathcal{G}_{0}$ such that $e\leq r(g)$,
there is a unique element $(e|g)\in\mathcal{G}$ which satisfies $(e|g)\leq g$
and $r(e|g)=e$.

In a partially ordered set $P$, we say that $z\in P$ is the _meet_ of $x\in P$
and $y\in P$ if $z\leq x$, $z\leq y$ and if $w\in P$ is such that $w\leq x,y$
then $w\leq z$. We denote $z=x\wedge y$. A poset where every pair of elements
has a meet is called a _meet semilattice_. If $\mathcal{G}$ is such that
$\mathcal{G}_{0}$ is a meet semilattice with respect to $\leq$, we say that
$\mathcal{G}$ is an _inductive_ groupoid.

######  Definition 2.1.

Let $\mathcal{G}$ be an ordered groupoid and $A$ a ring. We say that
$\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ is a partial action of
$\mathcal{G}$ on $A$ if $A_{r(g)}$ is an ideal of $A$, $A_{g}$ is an ideal of
$A_{r(g)}$, $\alpha_{g}:A_{g^{-1}}\to A_{g}$ is an isomorphism of rings, for
all $g\in\mathcal{G}$, and

  1. (P1)

$\alpha_{e}=\text{Id}_{A_{e}}$, for all $e\in\mathcal{G}_{0}$, and
$A\subseteq\sum_{e\in\mathcal{G}_{0}}A_{e}$;

  2. (P2)

$\alpha_{h}^{-1}(A_{g^{-1}}\cap A_{h})\subset A_{(gh)^{-1}}$, for all
$(g,h)\in\mathcal{G}_{2}$;

  3. (P3)

$\alpha_{g}\circ\alpha_{h}(a)=\alpha_{gh}(a)$, for all
$(g,h)\in\mathcal{G}_{2},a\in\alpha_{h}^{-1}(A_{g^{-1}}\cap A_{h})$.

The condition $A\subseteq\sum_{e\in\mathcal{G}_{0}}A_{e}$ is required to
recover the concept of partial group actions on rings, once if $G$ is a group
with identity element $1_{G}$ acting partially on a ring $A$ via $\alpha$,
then $A_{1_{G}}=A$ and $\alpha_{1_{G}}=\text{Id}_{A}$. Similar conditions have
already appeared in some other papers, such as [9] (on the notion of a “split”
$\mathcal{G}$-set) and in (P4) of [23].

By [4, Lemma 1.1], it is also true that:

  1. (i)

$\alpha_{g}^{-1}=\alpha_{g^{-1}}$, for all $g\in\mathcal{G}$.

  2. (ii)

$\alpha_{g}(A_{g^{-1}}\cap A_{h})=A_{g}\cap A_{gh}$, for all
$(g,h)\in\mathcal{G}_{2}$.

We say that a partial action $\alpha$ is _unital_ if each $A_{g}$ is a unital
ring with central and idempotent identity $1_{g}$. Clearly,
$A_{g}=A1_{g}=1_{g}A$, for all $g\in\mathcal{G}$. We say that $\alpha$ is
_preunital_ if $A_{e}$ is a unital ring with central and idempotent identity
$1_{e}$, for all $e\in\mathcal{G}_{0}$.

Furthermore, $\alpha$ is said to be a _partial ordered action_ , or simply
_P.O. action_ , if

  1. (PO)

If $g\leq h$, then $A_{g}\subseteq A_{h}$ and
$\alpha_{g}=\alpha_{h}|_{A_{g^{-1}}}$.

Two P.O. actions $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ of
$\mathcal{G}$ on a ring $A$ and $\gamma=(C_{g},\gamma_{g})_{g\in\mathcal{G}}$
of $\mathcal{G}$ on a ring $C$ are _isomorphic_ (we denote by
$\alpha\simeq\gamma$) if there is a collection of ring isomorphisms
$\\{\varphi_{e}:A_{e}\to C_{e}:e\in\mathcal{G}_{0}\\}$ such that

  1. (i)

$\varphi_{r(g)}(A_{g})=C_{g}$, for all $g\in\mathcal{G}$,

  2. (ii)

$\varphi_{r(g)}\circ\alpha_{g}(a)=\gamma_{g}\circ\varphi_{d(g)}(a)$, for all
$a\in A_{g^{-1}}$.

A global action of $\mathcal{G}$ on $A$ is a partial action such that
$A_{g}=A_{r(g)}$, for all $g\in\mathcal{G}$. Ordered global actions and unital
global actions are defined analogously. Of course a global action is unital if
and only if it is preunital.

Let $\mathcal{G}$ be an ordered groupoid and
$\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ a P.O. action of $\mathcal{G}$
on a ring $A$. We define the _partial skew groupoid ring_
$A\ltimes_{\alpha}\mathcal{G}$ by
$A\ltimes_{\alpha}\mathcal{G}=\bigoplus_{g\in\mathcal{G}}A_{g}\delta_{g}$,
where each $\delta_{g}$ is a symbol. We consider the usual sum and the product
is given by

$\displaystyle(a_{g}\delta_{g})(b_{h}\delta_{h})=\begin{cases}\alpha_{g}(\alpha_{g^{-1}}(a_{g})b_{h})\delta_{gh},\text{
if }\exists gh,\\\ 0,\text{ otherwise,}\end{cases}$

linearly extended. We have that $A\ltimes_{\alpha}\mathcal{G}$ is a ring but
it is not necessarily associative nor unital. A condition to ensure the
associativity of $A\ltimes_{\alpha}\mathcal{G}$ is to ask $\alpha$ to be a
unital partial action (c.f. [3, Proposition 3.1] and [12, Corollary 3.2]).
Also, by [3, Propostion 3.3], if $\mathcal{G}_{0}$ is finite, then
$A\ltimes_{\alpha}\mathcal{G}$ is unital with identity
$1_{A\ltimes_{\alpha}\mathcal{G}}=\sum_{e\in\mathcal{G}_{0}}1_{e}$.

######  Definition 2.2.

Let $\mathcal{G}$ be an ordered groupoid and
$\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ a P.O. action of $\mathcal{G}$
on a ring $A$ such that $A\ltimes_{\alpha}\mathcal{G}$ is associative. Define
the ideal $N$ of $A\ltimes_{\alpha}\mathcal{G}$ as $N=\langle
a\delta_{g}-a\delta_{h}:g\leq h,a\in A_{g}\subseteq A_{h}\rangle$. We define
the _partial skew ordered groupoid ring_ $A\ltimes_{\alpha}^{o}\mathcal{G}$ as
$A\ltimes_{\alpha}^{o}\mathcal{G}=\frac{A\ltimes_{\alpha}\mathcal{G}}{N}$.

##  3\. Globalization

From now on, assume that all actions and partial actions are preunital, unless
stated otherwise.

Let $\beta=(B_{g},\beta_{g})_{g\in\mathcal{G}}$ be an ordered action of an
ordered groupoid $\mathcal{G}$ on a ring $B$. Let $A$ be a unital ideal of
$B$. We shall construct a P.O. action $\alpha$ of $\mathcal{G}$ on $A$ by
restricting $\beta$. Consider, for each $e\in\mathcal{G}_{0}$, $A_{e}:=A\cap
B_{e}$, which is an ideal of $A$. Now, for all $g\in\mathcal{G}$, define
$A_{g}:=A_{r(g)}\cap\beta_{g}(A_{d(g)})$, which is an ideal of $A_{r(g)}$ and
also define $\alpha_{g}=\beta_{g}|_{A_{g^{-1}}}$, which is a ring isomorphism.

Observe that if $e\leq f$ are elements of $\mathcal{G}_{0}$, then
$B_{e}\subseteq B_{f}$, which implies $A_{e}=A\cap B_{e}\subseteq A\cap
B_{f}=A_{f}$. Now, if $g\leq h$, then

$\displaystyle A_{g}$ $\displaystyle=A_{r(g)}\cap\beta_{g}(A_{d(g)})\subseteq
A_{r(h)}\cap\beta_{g}(A_{d(g)})$
$\displaystyle=A_{r(h)}\cap\beta_{h}(A_{d(g)})\subseteq
A_{r(h)}\cap\beta_{h}(A_{d(h)})=A_{h},$

and given arbitrary $a\in A_{g^{-1}}$,
$\alpha_{h}(a)=\beta_{h}(a)=\beta_{g}(a)=\alpha_{g}(a)$, so
$\alpha_{g}=\alpha_{h}|_{A_{g^{-1}}}$. Hence
$\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ is a P.O. action of
$\mathcal{G}$ on $A$. We say that $\alpha$ is the _standard restriction_ of
$\beta$ to the ideal $A$.

######  Example 3.1.

Let $\mathcal{G}=\\{s,s^{-1},r(s),d(s),e\\}$ with
$\mathcal{G}_{0}=\\{r(s),d(s),e\\}$, $e\leq s,$ $and\leq s^{-1}$, $e\leq
r(s)$, $e\leq d(s)$.

Let $R$ be a commutative ring and $B=Re_{1}\oplus Re_{2}\oplus Re_{3}$, where
$\\{e_{1},e_{2},e_{3}\\}$ is a set of pairwise orthogonal idempotents whose
sum is $1_{B}$. Define

$\displaystyle B_{d(s)}=B_{s^{-1}}=Re_{1}\oplus Re_{2},\quad
B_{r(s)}=B_{s}=Re_{2}\oplus Re_{3},\quad B_{e}=Re_{2}$

and $\beta_{g}=Id_{B_{g}}$, for all $g\in\mathcal{G}_{0}$,
$\beta_{s}(ae_{1}+be_{2})=be_{2}+ae_{3}$, and $\beta_{s^{-1}}=\beta_{s}^{-1}$.

Let $A=Re_{2}\oplus Re_{3}$. The standard restriction
$\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ of $\beta$ to $A$ is given by
$A_{r(s)}=Re_{2}\oplus Re_{3}$, $A_{d(s)}=Re_{2}=A_{e}$,
$A_{s}=Re_{2}=A_{s^{-1}}$, and
$\alpha_{g}=\beta_{g}|_{A_{g^{-1}}}=Id_{A_{g^{-1}}},\text{ for all
}g\in\mathcal{G}$.

######  Remark 3.2.

Every P.O. action $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ of
$\mathcal{G}$ on a ring $A$ obtained by a standard restriction of an ordered
action $\beta=(B_{g},\beta_{g})_{g\in\mathcal{G}}$ of $\mathcal{G}$ on a ring
$B$, where $A$ is an ideal of $B$, satisfies the following condition: if
$e\leq r(g)$, then $A_{(e|g)}=A_{e}\cap A_{g}$. In fact,

$\displaystyle A_{e}\cap A_{g}$ $\displaystyle=A_{e}\cap
A_{r(g)}\cap\beta_{g}(A_{d(g)})=A_{e}\cap\beta_{g}(A_{d(g)})$
$\displaystyle=\beta_{g}(\beta_{g^{-1}}(A_{e}))\cap\beta_{g}(A_{d(g)})=\beta_{g}(A_{d(g)}\cap\beta_{g^{-1}}(A_{e}))$
$\displaystyle=\beta_{g}(A_{d(g)}\cap\beta_{(g^{-1}|e)}(A_{e}))=\beta_{g}(A_{d(g)}\cap
B_{r(g^{-1}|e)}\cap\beta_{(g^{-1}|e)}(A_{e}))$
$\displaystyle=\beta_{g}(A_{d(g)}\cap
B_{d(e|g)}\cap\beta_{(g^{-1}|e)}(A_{e}))=\beta_{g}(A\cap B_{d(g)}\cap
B_{d(e|g)}\cap\beta_{(g^{-1}|e)}(A_{e}))$ $\displaystyle=\beta_{g}(A\cap
B_{d(e|g)}\cap\beta_{(g^{-1}|e)}(A_{e}))=\beta_{g}(A_{d(e|g)}\cap\beta_{(g^{-1}|e)}(A_{e}))$
$\displaystyle=\beta_{(e|g)}(A_{d(e|g)}\cap\beta_{(g^{-1}|e)}(A_{e}))=A_{e}\cap\beta_{(e|g)}(A_{d(e|g)})=A_{(e|g)}.$

Now we present a new kind of restriction of $\beta$, which generalizes the
standard one. Let $\beta=(B_{g},\beta_{g})_{g\in\mathcal{G}}$ be an ordered
global action of $\mathcal{G}$ on a ring $B$. Consider a unital ideal $A_{e}$
of $B_{e}$, for all $e\in\mathcal{G}_{0}$, such that $A_{e}\subseteq A_{f}$ if
$e\leq f$ in $\mathcal{G}_{0}$. Then $A_{g}=A_{r(g)}\cap\beta_{g}(A_{d(g)})$
is an ideal of $A_{r(g)}$, for all $g\in\mathcal{G}$. Define
$\alpha_{g}=\beta_{g}|_{A_{g^{-1}}}$, for all $g\in\mathcal{G}$. Consider $A$
the (unital) ideal generated by $\bigcup_{e\in\mathcal{G}_{0}}A_{e}$. Since
$A_{e}$ is an ideal of $B_{e}$, for all $e\in\mathcal{G}_{0}$, and $\beta$ is
unital, then $A_{e}$ is an ideal of $B$, for all $e\in\mathcal{G}_{0}$. Hence
$A=\sum_{e\in\mathcal{G}_{0}}A_{e}$ and
$\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ is a P.O. action of
$\mathcal{G}$ on $A$. We say that $\alpha$ is a _restriction_ of $\beta$ to
the ring $A$. Every standard restriction of $\beta$ is a restriction of
$\beta$.

Consider now the ideal $C_{g}$ of $B$ generated by $\bigcup_{r(h)\leq
r(g)}\beta_{h}(A_{d(h)})$. Since $A_{d(h)}$ is unital and $\beta_{h}$ is an
isomorphism, for all $h\in\mathcal{G}$, we have that $C_{g}=\sum_{r(h)\leq
r(g)}\beta_{h}(A_{d(h)})$. Notice that $C_{g}\subseteq B_{g}$. Defining
$\gamma_{g}=\beta_{g}|_{C_{g^{-1}}}$, we have that
$\gamma=(C_{g},\gamma_{g})_{g\in\mathcal{G}}$ is an ordered global action of
$\mathcal{G}$ on $C=\sum_{e\in\mathcal{G}_{0}}C_{e}$ such that the restriction
of $\gamma$ to the ring $A$ coincides with $\alpha$.

The next example illustrates that there are restrictions which are not
standard.

######  Example 3.3.

Let $\mathcal{G}=\\{m_{0},m_{1},n_{0},n_{1}\\}$, where
$\mathcal{G}_{0}=\\{m_{0},n_{0}\\}$, $m_{1}^{2}=m_{0}$, $n_{1}^{2}=n_{0}$,
$m_{0}\leq n_{0}$ and $m_{1}\leq n_{1}$. Let $R$ be a commutative ring and
$A=Re_{1}\oplus Re_{2}\oplus Re_{3}\oplus Re_{4}$, where the $e_{i}$’s are
central idempotents pairwise orthogonal whose sum is $1_{A}$. Define

$\displaystyle A_{m_{0}}=A_{n_{1}}=A_{n_{0}}=A,\qquad A_{m_{1}}=Re_{1}\oplus
Re_{3},$

and

$\displaystyle\alpha_{n_{1}}(ae_{1}+be_{2}+ce_{3}+de_{4})$
$\displaystyle=ce_{1}+de_{2}+ae_{3}+be_{4},$
$\displaystyle\alpha_{m_{1}}(ae_{1}+ce_{3})$ $\displaystyle=ce_{1}+ae_{3},$
$\displaystyle\alpha_{e}=I_{A_{e}},$ $\displaystyle\text{ for all
}e\in\mathcal{G}_{0},$

for all $a,b,c,d\in R$.

Then $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ is a unital P.O. action of
$\mathcal{G}$ on $A$. In Theorem 3.5 we will see that $\alpha$ is a
restriction of some ordered action $\beta$ of $\mathcal{G}$ on some ring $B$.
However, $\alpha$ is not a standard restriction, since
$A_{(m_{0}|n_{1})}=A_{m_{1}}\neq A_{n_{1}}=A_{m_{0}}\cap A_{n_{1}}$. Hence
there are restrictions that are not standard.

######  Definition 3.4.

Let $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ be a preunital P.O. action
of an ordered groupoid $\mathcal{G}$ on a ring $A$. A unital ordered global
action $\beta=(B_{g},\beta_{g})_{g\in\mathcal{G}}$ of $\mathcal{G}$ on a ring
$B$ is said to be a _globalization_ of $\alpha$ if for all
$e\in\mathcal{G}_{0}$ there is an injective homomorphism of rings
$\varphi_{e}:A_{e}\to B_{e}$ such that

  1. (i)

$\varphi_{e}(A_{e})$ is an ideal of $B_{e}$;

  2. (ii)

$\varphi_{r(g)}(A_{g})=\varphi_{r(g)}(A_{r(g)})\cap\beta_{g}(\varphi_{d(g)}(A_{d(g)}))$,
for all $g\in\mathcal{G}$;

  3. (iii)

$\beta_{g}\circ\varphi_{d(g)}(a)=\varphi_{r(g)}\circ\alpha_{g}(a)$, for all
$a\in A_{g^{-1}}$ and $g\in\mathcal{G}$;

  4. (iv)

$B_{g}=\sum_{r(h)\leq r(g)}\beta_{h}\left(\varphi_{d(h)}(A_{d(h)})\right)$.

Now we shall prove that a P.O. action has a globalization if, and only if, it
is unital.

######  Theorem 3.5.

Let $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ be a preunital P.O. action
of an ordered groupoid $\mathcal{G}$ on a ring $A$. Then $\alpha$ has a
globalization $\beta$ if and only if $\alpha$ is unital.

###### Proof.

($\Rightarrow$): Once
$\varphi_{r(g)}(A_{g})=\varphi_{r(g)}(A_{r(g)})\cap\beta_{g}(\varphi_{d(g)}(A_{d(g)}))$
and $A_{r(g)}$ and $A_{d(g)}$ are unital, the assertion clearly holds.

($\Leftarrow$): Assume that every $A_{g}$ is unital with identity $1_{g}$. Let
$\mathcal{F}:=\mathcal{F}(\mathcal{G},A)$ be the ring of all maps
$\mathcal{G}\to A$ and define $\mathcal{G}_{g}=\\{h\in\mathcal{G}:r(h)\leq
r(g)\\}$.

Take $F_{g}=\\{f\in\mathcal{F}:f(h)=0,\text{ for all
}h\notin\mathcal{G}_{g}\\}$. Since $\mathcal{G}_{g}=\mathcal{G}_{r(g)}$, it
follows that $F_{g}=F_{r(g)}$. If $h\leq g$, then $r(h)\leq r(g)$, so
$F_{h}\subseteq F_{g}$. Clearly $F_{g}$ is an ideal of $\mathcal{F}$, for all
$g\in\mathcal{G}$.

Now fix the notation $f(h):=f|_{h}$, for all $f\in\mathcal{F}$ and
$h\in\mathcal{G}$.

For $g\in\mathcal{G}$ and $f\in F_{g^{-1}}$, define $\gamma_{g}:F_{g^{-1}}\to
F_{g}$ by

$\displaystyle\gamma_{g}(f)|_{h}=\begin{cases}f((g^{-1}|r(h))h),\text{ if
}h\in\mathcal{G}_{g},\\\ 0,\text{ otherwise.}\end{cases}$

Notice that if $h\in\mathcal{G}_{g}$, then $r(h)\leq r(g)=d(g^{-1})$, and
hence the restriction $(g^{-1}|r(h))$ is defined, as well as the product
$(g^{-1}|r(h))h$. Moreover, $r((g^{-1}|r(h))h)=r(g^{-1}|r(h))\leq r(g^{-1})$,
from where it follows that $\gamma_{g}$ is well defined. It is easy to see
that $\gamma_{g}$ is a ring homomorphism.

For $g\in\mathcal{G}$, $f\in F_{g^{-1}}$ and $h\in\mathcal{G}_{g^{-1}}$, we
have that

$\displaystyle\gamma_{g^{-1}}\circ\gamma_{g}(f)|_{h}$
$\displaystyle=\gamma_{g}(f)|_{(g|r(h))h}=f((g^{-1}|r(g|r(h)))(g|r(h))h)$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}f((g^{-1}g|r(h))h)=f((d(g)|r(h))h)\stackrel{{\scriptstyle(**)}}{{=}}f(r(h)h)=f(h),$

where the equality ($*$) follows by [20, Proposition 4.1.3] and the equality
($**$) follows since $r(h)\leq d(g)$. Similarly
$\gamma_{g}\circ\gamma_{g^{-1}}(f)|_{h}=f(h)$. Thus we have that $\gamma_{g}$
is a ring isomorphism.

Observe that $\gamma_{e}=I_{F_{e}}$, for all $e\in\mathcal{G}_{0}$ and

$\displaystyle\gamma_{gh}(f)|_{k}$
$\displaystyle=f((h^{-1}g^{-1}|r(k))k)\stackrel{{\scriptstyle(*)}}{{=}}f((h^{-1}|r(g^{-1}|r(k)))(g^{-1}|r(k))k)=\gamma_{g}\circ\gamma_{h}(f)|_{k},$

for all $f\in F_{h^{-1}}$, $k\in\mathcal{G}_{g}$ and
$(g,h)\in\mathcal{G}_{2}$, where the equality ($*$) follows again by [20,
Proposition 4.1.3]. Then $\gamma=(F_{g},\gamma_{g})_{g\in\mathcal{G}}$ is a
global action of $\mathcal{G}$ on $\mathcal{F}$.

To see that $\gamma$ is ordered, observe that if $h\leq g$ in $\mathcal{G}$
and $f\in F_{h^{-1}}$, then

$\displaystyle\gamma_{g}(f)|_{k}$
$\displaystyle=f((g^{-1}|r(k))k)\stackrel{{\scriptstyle(*)}}{{=}}f((h^{-1}|r(k))k)=\gamma_{h}(f)|_{k},$

that is, $\gamma_{h}=\gamma_{g}|_{F_{h^{-1}}}$. The equality $(*)$ follows
from the uniqueness of restrictions, because $(h^{-1}|r(k))\leq h^{-1}\leq
g^{-1}$ and $d(h^{-1}|r(k))=r(k)$. Hence $\gamma$ is an ordered global action
of $\mathcal{G}$ on $\mathcal{F}$.

Now, for all $e\in\mathcal{G}_{0}$, define $\varphi_{e}:A_{e}\to F_{e}$ by

$\displaystyle\varphi_{e}(a)|_{h}=\begin{cases}\alpha_{h^{-1}}(a1_{h}),\text{
if }r(h)=e,\\\ 0,\text{ otherwise,}\end{cases}$

for all $a\in A_{e},h\in\mathcal{G}$. We have that
$\varphi_{e}(a)|_{e}=\alpha_{e}(a1_{e})=a$, thus $\varphi_{e}$ is a
monomorphism, for all $e\in\mathcal{G}_{0}$.

Let $B_{g}$ be the ideal generated by $\bigcup_{r(h)\leq
r(g)}\gamma_{h}(\varphi_{d(h)}(A_{d(h)}))$, for all $g\in\mathcal{G}$. Notice
that $B_{g}\subseteq F_{g}$. Consider
$B=\sum_{e\in\mathcal{G}_{0}}B_{e}=\left\langle\bigcup_{e\in\mathcal{G}_{0}}B_{e}\right\rangle$.
Define $\beta_{g}:=\gamma_{g}|_{B_{g^{-1}}}$, for all $g\in\mathcal{G}$. By
construction, $\beta=(B_{g},\beta_{g})_{g\in\mathcal{G}}$ is an ordered global
action of $\mathcal{G}$ on $B$. The next step is to show that $\beta$ is a
globalization of $\alpha$.

We will start by verifying (iii). Let $g\in\mathcal{G}$, $a\in A_{g^{-1}}$ and
$h\in\mathcal{G}$. If $r(h)=r(g)$, then

$\displaystyle\varphi_{r(g)}(\alpha_{g}(a))|_{h}=\alpha_{h^{-1}}(\alpha_{g}(a)1_{h})$

and

$\displaystyle\beta_{g}(\varphi_{d(g)}(a))|_{h}=\varphi_{d(g)}(a)|_{g^{-1}h}=\alpha_{h^{-1}g}(a1_{g^{-1}h}).$

Since $a\in A_{g^{-1}}$, $a1_{g^{-1}h}\in A_{g^{-1}}\cap A_{g^{-1}h}$. Thus

$\displaystyle\beta_{g}(\varphi_{d(g)}(a))$
$\displaystyle=\alpha_{h^{-1}g}(a1_{g^{-1}h})=\alpha_{h^{-1}}(\alpha_{g}(a1_{g^{-1}h}1_{g^{-1}}))$
$\displaystyle=\alpha_{h^{-1}}(\alpha_{g}(a1_{g^{-1}})\alpha_{g}(1_{g^{-1}h}1_{g^{-1}}))=\alpha_{h^{-1}}(\alpha_{g}(a1_{g^{-1}})1_{g}1_{h})$
$\displaystyle=\alpha_{h^{-1}}(\alpha_{g}(a1_{g^{-1}})1_{h}),$

because $\alpha_{g}(A_{g^{-1}}\cap A_{g^{-1}h})=A_{g}\cap A_{h}$ implies
$\alpha_{g}(1_{g^{-1}h}1_{g^{-1}})=1_{g}1_{h}$.

On the one hand, if $r(g)\neq r(h)$, we have that
$\varphi_{r(g)}(\alpha_{g}(a))|_{h}=0$. On the other hand, if $r(h)<r(g)$,

$\displaystyle\beta_{g}(\varphi_{d(g)}(a))|_{h}=\varphi_{d(g)}(a)|_{(g^{-1}|r(h))h}=\begin{cases}\alpha_{h^{-1}(r(h)|g)}(a1_{(g^{-1}|r(h))h}),\text{
if }r(g^{-1}|r(h))=d(g),\\\ 0,\text{ otherwise.}\end{cases}$

Notice that if $r(g^{-1}|r(h))=d(g)$, then since $g^{-1}\leq g^{-1}$ and
$r(g^{-1})=d(g)$, from the uniqueness of corestrictions, we have that
$g^{-1}=(d(g)|g^{-1})=(d(g)|(g^{-1}|r(h)))=(g^{-1}|r(h))$. But this implies
that $r(g)=d(g^{-1})=d(g^{-1}|r(h))=r(h)$, which is a contradiction. Therefore
$d(g^{-1}|r(h))<d(g^{-1})$. Thus $\beta_{g}(\varphi_{d(g)}(a))|_{h}=0$. If
$r(g)<r(h)$ or $r(g)$ and $r(h)$ are incomparable, this equality also holds,
showing (iii).

Now we proceed to (ii). Let $g\in\mathcal{G}$ and
$c\in\varphi_{r(g)}(A_{r(g)})\cap\beta_{g}(\varphi_{d(g)}(A_{d(g)}))$. Then
there are $a\in A_{r(g)}$, $b\in A_{d(g)}$ such that
$c=\varphi_{r(g)}(a)=\beta_{g}(\varphi_{d(g)}(b))$. Therefore,

$\displaystyle
a=\alpha_{r(g)^{-1}}(a1_{r(g)})=\varphi_{r(g)}(a)|_{r(g)}=\beta_{g}(\varphi_{d(g)}(b))|_{r(g)}=\varphi_{d(g)}(b)|_{g^{-1}}=\alpha_{g}(b1_{g^{-1}})\in
A_{g},$

from where $c=\varphi_{r(g)}(a)\in\varphi_{r(g)}(A_{g})$.

Conversely, if $c\in\varphi_{r(g)}(A_{g})$, then $c=\varphi_{r(g)}(a)$, for
some $a\in A_{g}$. Taking $b=\alpha_{g^{-1}}(a)\in A_{g^{-1}}$, we have by
(iii) that
$\beta_{g}(\varphi_{d(g)}(b))=\varphi_{r(g)}(\alpha_{g}(b))=\varphi_{r(g)}(a)$.
Hence
$c\in\varphi_{r(g)}(A_{g})\cap\beta_{g}(\varphi_{d(g)}(A_{g^{-1}}))\subseteq\varphi_{r(g)}(A_{r(g)})\cap\beta_{g}(\varphi_{d(g)}(A_{d(g)}))$.

It only remains for us to show that $\varphi_{e}(A_{e})$ is an ideal of
$B_{e}$, for all $e\in\mathcal{G}_{0}$. For this, it is sufficient to show
that
$\varphi_{e}(b)\beta_{h}(\varphi_{d(h)}(a)),\beta_{h}(\varphi_{d(h)}(a))\varphi_{e}(b)\in\varphi_{e}(A_{e})$,
for all $h\in\mathcal{G}_{e},a\in A_{d(h)}$ and $b\in A_{e}$. Given
$k\in\mathcal{G}$, we have two possibilities:

Case (1): $r(k)\leq r(h)\leq e$.

Observe that

$\displaystyle\beta_{h}(\varphi_{d(h)}(a))\varphi_{e}(b)|_{k}=\beta_{h}(\varphi_{d(h)}(a))|_{k}\varphi_{e}(b)|_{k}=\varphi_{d(h)}(a)|_{(h^{-1}|r(k))k}\varphi_{e}(b)|_{k}.$

We have that

$\displaystyle\varphi_{e}(b)|_{k}=\begin{cases}\alpha_{k^{-1}}(b1_{k}),\text{
if }r(k)=e,\\\ 0,\text{ otherwise.}\end{cases}$

Since $r(k)=e$ and $r(k)\leq r(h)\leq e$ imply $r(k)=r(h)=e$, we obtain

$\displaystyle\beta_{h}(\varphi_{d(h)}(a))\varphi_{e}(b)|_{k}$
$\displaystyle=\begin{cases}\alpha_{k^{-1}h}(a1_{h^{-1}k})\alpha_{k^{-1}}(b1_{k}),\text{
if }r(k)=r(h)=e,\\\ 0,\text{ otherwise}\end{cases}$

$\displaystyle=\begin{cases}\alpha_{k^{-1}h}(a1_{h^{-1}k})1_{k^{-1}h}1_{k^{-1}}\alpha_{k^{-1}}(b1_{k}),\text{
if }r(k)=r(h)=e,\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}\begin{cases}\alpha_{k^{-1}h}(a1_{h^{-1}k})\alpha_{k^{-1}h}(1_{h^{-1}k}1_{h^{-1}})\alpha_{k^{-1}}(b1_{k}),\text{
if }r(k)=r(h)=e,\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle=\begin{cases}\alpha_{k^{-1}h}(a1_{h^{-1}}1_{h^{-1}k})\alpha_{k^{-1}}(b1_{k}),\text{
if }r(k)=r(h)=e,\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle=\begin{cases}\alpha_{k^{-1}}(\alpha_{h}(a1_{h^{-1}})b1_{k}),\text{
if }r(k)=r(h)=e,\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle=\varphi_{e}(\alpha_{h}(a1_{h^{-1}})b)|_{k},$

where $(*)$ holds because Proposition (i)(ii) implies that
$\alpha_{k^{-1}h}(A_{h^{-1}k}\cap A_{h^{-1}})=A_{k^{-1}h}\cap A_{k^{-1}}$.

Case (2): $r(h)\leq r(k)\leq e$ or $r(k)$ and $r(h)$ incomparable.

In this case
$\beta_{h}(\varphi_{d(h)}(a))\varphi_{e}(b)|_{k}=\beta_{h}(\varphi_{d(h)}(a))|_{k}\varphi_{e}(b)|_{k}=0\cdot\varphi_{e}(b)|_{k}=0$
directly from the definition of $\beta$.

The verification that
$\varphi_{e}(b)\beta_{h}(\varphi_{d(h)}(a))\in\varphi_{e}(A_{e})$ is similar,
and then $\varphi_{e}(A_{e})$ is an ideal of $B_{e}$. ∎

######  Example 3.6.

Let $\mathcal{G}$, $R$, $B$ and $A$ be as in Example 3.1. We will construct a
globalization for $\alpha$. Consider $\mathcal{F}=\mathcal{F}(\mathcal{G},A)$.
We have that $\mathcal{F}\simeq A^{5}$ via
$f\in\mathcal{F}\mapsto(f(s),f(s^{-1}),f(r(s)),f(d(s)),$ $f(e))$.

Since $\mathcal{G}_{s}=\mathcal{G}_{r(s)}=\\{s,r(s),e\\}$,
$\mathcal{G}_{s^{-1}}=\mathcal{G}_{d(s)}=\\{s^{-1},d(s),e\\}$ and
$\mathcal{G}_{e}=\\{e\\}$, we have that

$\displaystyle F_{s}$ $\displaystyle=\\{f\in\mathcal{F}:f(g)=0,\text{ for all
}g\notin\mathcal{G}_{s}\\}\simeq A\times 0\times A\times 0\times A,$
$\displaystyle F_{s^{-1}}$ $\displaystyle=\\{f\in\mathcal{F}:f(g)=0,\text{ for
all }g\notin\mathcal{G}_{s^{-1}}\\}\simeq 0\times A\times 0\times A\times A,$
$\displaystyle F_{e}$ $\displaystyle=\\{f\in\mathcal{F}:f(g)=0,\text{ for all
}g\notin\mathcal{G}_{e}\\}\simeq 0\times 0\times 0\times 0\times A.$

Thus, $\gamma_{s}(0,a,0,b,c)=(b,0,a,0,c)$,
$\gamma_{s^{-1}}(a,0,b,0,c)=(0,b,0,a,c)$ and $\gamma_{g}=Id_{F_{g}}$, for all
$g\in\mathcal{G}_{0}$. Furthermore, notice that
$\varphi_{r(s)}(ae_{2}+be_{3})=(ae_{2},0,ae_{2}+be_{3},0,0)$,
$\varphi_{d(s)}(ae_{2})=(0,ae_{2},0,ae_{2},0)$ and
$\varphi_{e}(ae_{2})=(0,0,0,0,ae_{2})$.

Given $ae_{2}+be_{3}\in A_{r(s)}$, $ce_{2}\in A_{d(s)}$, $de_{2}\in A_{e}$, we
obtain

$\displaystyle\gamma_{r(s)}(\varphi_{r(s)}(ae_{2}+be_{3}))+\gamma_{s}(\varphi_{d(s)}(ce_{2}))+\gamma_{e}(\varphi_{e}(de_{2}))$
$\displaystyle=\;$
$\displaystyle\gamma_{r(s)}(ae_{2},0,ae_{2}+be_{3},0,0)+\gamma_{s}(0,ce_{2},0,ce_{2},0)+\gamma_{e}(0,0,0,0,de_{2})$
$\displaystyle=\;$
$\displaystyle(ae_{2},0,ae_{2}+be_{3},0,0)+(ce_{2},0,ce_{2},0,0)+(0,0,0,0,de_{2})$
$\displaystyle=\;$ $\displaystyle((a+c)e_{2},0,(a+c)e_{2}+be_{3},0,de_{2}),$

from where it follows that
$B_{s}^{\prime}=\\{(ae_{2},0,ae_{2}+be_{3},0,ce_{2}):a,b,c\in R\\}$.

Similarly,
$B_{s^{-1}}^{\prime}=\\{(0,ae_{2}+be_{3},0,ae_{2},0,ce_{2}):a,b,c\in R\\}$ and
$B_{e}^{\prime}=Re_{2}(0,0,0,0,1)$, thus
$B^{\prime}=B_{s}^{\prime}+B_{s^{-1}}^{\prime}+B_{e}^{\prime}=\\{(ae_{2},be_{2}+ce_{3},ae_{2}+de_{3},be_{2},fe_{3}):a,b,c,d,f\in
R\\}$.

Then $\beta^{\prime}=(B_{g}^{\prime},\beta_{g}^{\prime})_{g\in\mathcal{G}}$ is
a globalization of $\alpha$. Observe that $\beta$ is also a globalization of
$\alpha$, taking $\varphi_{g}=\iota_{g}:A_{g}\to B_{g}$ as the usual inclusion
$A_{g}\mapsto A\cap B_{g}$, for all $g\in\mathcal{G}_{0}$. However
$\beta^{\prime}\not\simeq\beta$, since $B_{s}\simeq R^{2}\not\simeq
R^{3}\simeq B_{s}^{\prime}$.

##  4\. Uniqueness

In this section we will give a sufficient condition for a P.O. action to have
a unique globalization. In order to achieve this goal, we introduce the notion
of a minimal globalization.

######  Definition 4.1.

Let $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ be a preunital P.O. action
of $\mathcal{G}$ on a ring $A$ and $\beta=(B_{g},\beta_{g})_{g\in\mathcal{G}}$
be a globalization of $\alpha$ on a ring $B$. We say that $\beta$ is a
_minimal globalization_ if

  1. (iv’)

$B_{g}=\sum_{r(h)=r(g)}\beta_{h}(\varphi_{d(h)}(A_{d(h)}))$, for all
$g\in\mathcal{G}$,

holds.

Moreover, we introduce a specific type of P.O. action. Given an ordered
groupoid $\mathcal{G}$, we define the _pseudoproduct_ $*$ as

$\displaystyle g*h=\begin{cases}(g|d(g)\wedge r(h))(d(g)\wedge r(h)|h),\text{
if }d(g)\wedge r(h)\text{ is defined},\\\ \text{undefined,
otherwise.}\end{cases}$

######  Definition 4.2.

Let $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ be a P.O. action of
$\mathcal{G}$ on a ring $A$. We say that $\alpha$ is a _strong_ P.O. action
[16, Definition 3.7] if

  1. (PS)

$\alpha_{g}\circ\alpha_{h}=\alpha_{g*h}\circ\text{Id}_{A_{h^{-1}}}$, for all
$g,h\in\mathcal{G}$ such that $\exists g*h$.

######  Remark 4.3.

Since (PS) holds, notice that
$\text{dom}(\alpha_{g}\circ\alpha_{h})=\text{dom}(\alpha_{g*h}\circ\text{Id}_{A_{h^{-1}}})$,
for all $g,h\in\mathcal{G}$ such that $\exists g*h$. But this is the same as
saying that $\alpha_{g}(A_{g^{-1}}\cap A_{h})=A_{g}\cap A_{g*h}$.

In the case where the action is global, we have that strong global ordered
actions are precisely global ordered actions. In the case where $\mathcal{G}$
is trivially ordered with the equality, the concepts of strong P.O. action and
partial action coincide.

######  Lemma 4.4.

Let $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ be a strong P.O. action of
$\mathcal{G}$ on a ring $A$. The following statements are true:

  1. (i)

If $e\leq r(g)$, $A_{(e|g)}=A_{e}\cap A_{g}$.

  2. (ii)

If $e\leq d(g)$, then $A_{(g|e)}=A_{g}\cap A_{r(g|e)}$.

  3. (iii)

If $e,f\in\mathcal{G}_{0}$ are such that $e\wedge f$ is defined, then
$A_{e\wedge f}=A_{e}\cap A_{f}$.

###### Proof.

(i): If $e\leq r(g)$, then
$\alpha_{e}\circ\alpha_{g}=\alpha_{e*g}\circ\text{Id}_{A_{g^{-1}}}=\alpha_{(e|g)}\circ\text{Id}_{A_{g^{-1}}}=\alpha_{(e|g)}$.

Now
$A_{(e|g)}=\text{Im}(\alpha_{(e|g)})=\text{Im}(\alpha_{e}\circ\alpha_{g})=\alpha_{e}(A_{e}\cap
A_{g})=A_{e}\cap A_{g}$.

(ii): Notice that $(g|e)=(r(g|e)|g)$ by the uniqueness of restrictions. Hence
$A_{(g|e)}=A_{(r(g|e)|g)}=A_{g}\cap A_{r(g|e)}$.

(iii): From (PS) it follows that
$\alpha_{e}\circ\alpha_{f}=\alpha_{e*f}\circ\alpha_{f}=\alpha_{e\wedge
f}\circ\alpha_{f}=\alpha_{e\wedge f}$. Therefore $A_{e\wedge
f}=\text{Im}(\alpha_{e\wedge
f})=\text{Im}(\alpha_{e}\circ\alpha_{f})=\alpha_{e}(A_{e}\cap A_{f})=A_{e}\cap
A_{f}.$ ∎

######  Definition 4.5.

Let $\mathcal{G}$ be an ordered groupoid. We say that $\mathcal{G}$ is
_pseudoassociative_ if $\exists(g*h)*k\iff\exists g*(h*k)$, for all
$g,h,k\in\mathcal{G}$.

######  Remark 4.6.

In the case that $\exists(g*h)*k$ and $\exists g*(h*k)$, we have by [20, Lemma
4.1.6] that these pseudoproducts coincide. Inductive groupoids are always
pseudoassociative.

######  Theorem 4.7.

Let $\alpha=(A_{g},\alpha_{g})_{g\in\mathcal{G}}$ be a strong P.O. action of a
pseudoassociative groupoid $\mathcal{G}$ on a ring $A$. Then $\alpha$ admits a
minimal globalization if, and only if, $\alpha$ is unital. Furthermore, the
minimal globalization is unique up to isomorphisms.

###### Proof.

Consider, as in the proof of Theorem 3.5,
$\mathcal{F}=\mathcal{F}(\mathcal{G},A)$ the ring of the maps of $\mathcal{G}$
on $A$. Given $g\in\mathcal{G}$, define
$\mathcal{E}_{g}=\\{h\in\mathcal{G}:\exists g^{-1}*h\\}$. Define
$F_{g}=\\{f\in\mathcal{F}:f(h)=0,\text{ for all }h\notin\mathcal{E}_{g}\\}$.

Clearly $F_{g}=F_{r(g)}$, for all $g\in\mathcal{G}$. Consider
$\gamma_{g}:F_{g^{-1}}\to F_{g}$, where

$\displaystyle\gamma_{g}(f)|_{h}=\begin{cases}f(g^{-1}*h),\text{ if
}h\in\mathcal{E}_{g},\\\ 0,\text{ otherwise.}\end{cases}$

It is easy to see that $\gamma_{g}$ is a well defined ring homomorphism, for
all $g\in\mathcal{G}$. If $(g,h)\in\mathcal{G}_{2}$, $f\in F_{h^{-1}}$ and
$k\in\mathcal{E}_{g}$ then

$\displaystyle\gamma_{gh}(f)|_{k}$
$\displaystyle=f((gh)^{-1}*k)=f((h^{-1}g^{-1})*k)$
$\displaystyle=f((h^{-1}*g^{-1})*k)=f(h^{-1}*(g^{-1}*k))$
$\displaystyle=\gamma_{h}(f)|_{g^{-1}*k}=\gamma_{g}\circ\gamma_{h}(f)|_{k}.$

If $k\notin\mathcal{E}_{g}$, then
$\gamma_{gh}(f)|_{k}=0=\gamma_{g}\circ\gamma_{h}(f)|_{k}$, so
$\gamma_{gh}=\gamma_{g}\circ\gamma_{h}$.

For all $e\in\mathcal{G}_{0}$, define $\psi_{e}:A_{e}\to F_{e}$ by

$\displaystyle\psi_{e}(a)|_{h}=\begin{cases}\alpha_{h^{-1}}(a1_{h}),\text{ if
}h\in\mathcal{E}_{e},\\\ 0,\text{ otherwise},\end{cases}$

for all $a\in A_{e},h\in\mathcal{G}$. We have that
$\psi_{e}(a)|_{e}=\alpha_{e}(a1_{e})=a$, for all $a\in A_{e}$. Hence
$\psi_{e}$ is a monomorphism, for all $e\in\mathcal{G}_{0}$.

Let $B_{g}$ be the ideal generated by
$\bigcup_{r(h)=r(g)}\gamma_{h}(\psi_{d(h)}(A_{d(h)}))$, for all
$g\in\mathcal{G}$. Notice that $B_{g}\subseteq F_{g}$. Consider
$B=\sum_{e\in\mathcal{G}_{0}}B_{e}$. Define
$\beta_{g}:=\gamma_{g}|_{B_{g^{-1}}}$, for all $g\in\mathcal{G}$.

Given $e\in\mathcal{G}_{0}$, $g\in\mathcal{G}$ with $r(g)=e$ and $a\in
A_{d(g)}$, we have that

$\displaystyle\beta_{e}\circ\beta_{g}(\psi_{d(g)}(a))=\gamma_{e}\circ\gamma_{g}(\psi_{d(g)}(a))=\gamma_{eg}(\psi_{d(g)}(a))=\gamma_{g}(\psi_{d(g)}(a))=\beta_{g}(\psi_{d(g)}(a)),$

from where it follows that $\beta_{e}=\text{Id}_{B_{e}}$. Since
$\gamma_{g}\circ\gamma_{h}=\gamma_{gh}$, for all $(g,h)\in\mathcal{G}_{2}$,
then $\beta_{g}\circ\beta_{h}=\beta_{gh}$, for all $(g,h)\in\mathcal{G}_{2}$.

Now we prove that $\beta$ satisfies (PS). Indeed, given
$g,h,k,\ell\in\mathcal{G}$ such that $h\in\mathcal{E}_{g}$, $r(k)=d(h)$ and
$a\in A_{d(k)}$, we have that

$\displaystyle\beta_{g}\circ\beta_{h}(\beta_{k}(\psi_{d(k)}(a)))|_{\ell}$
$\displaystyle=\begin{cases}\beta_{h}(\beta_{k}(\psi_{d(k)}(a)))|_{g^{-1}*\ell},\text{
if }\ell\in\mathcal{E}_{g},\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle=\begin{cases}\beta_{k}(\psi_{d(k)}(a))|_{h^{-1}*(g^{-1}*\ell)},\text{
if }\ell\in\mathcal{E}_{g},\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle=\begin{cases}\beta_{k}(\psi_{d(k)}(a))|_{(h^{-1}*g^{-1})*\ell},\text{
if }\ell\in\mathcal{E}_{g},\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle=\begin{cases}\beta_{k}(\psi_{d(k)}(a))|_{((d(h)*h^{-1})*g^{-1})*\ell},\text{
if }\ell\in\mathcal{E}_{g},\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle=\begin{cases}\beta_{k}(\psi_{d(k)}(a))|_{d(h)*(h^{-1}*g^{-1})*\ell},\text{
if }\ell\in\mathcal{E}_{g},\\\ 0,\text{ otherwise}\end{cases}$

$\displaystyle=\begin{cases}\beta_{d(h)}(\beta_{k}(\psi_{d(k)}(a)))|_{(h^{-1}*g^{-1})*\ell},\text{
if }\ell\in\mathcal{E}_{g},\\\ 0,\text{ otherwise}\end{cases}$
$\displaystyle=\beta_{g*h}\circ\beta_{d(h)}(\beta_{k}(\psi_{d(k)}(a)))|_{\ell}.$

To finish the proof, we shall show that $\beta$ satisfies (i)-(iii) from the
definition of minimal globalization.

(iii): Let $g,h\in\mathcal{G}$ e $a\in A_{g^{-1}}$. Then

$\displaystyle\psi_{r(g)}\circ\alpha_{g}(a)|_{h}$
$\displaystyle=\begin{cases}\alpha_{h^{-1}}(\alpha_{g}(a)1_{h}),\text{ if
}h\in\mathcal{E}_{g},\\\ 0,\text{ otherwise}\end{cases}$

and

$\displaystyle\beta_{g}\circ\psi_{d(g)}(a)|_{h}$
$\displaystyle=\begin{cases}\psi_{d(g)}(a)|_{g^{-1}*h},\text{ if
}h\in\mathcal{E}_{g},\\\ 0,\text{
otherwise}\end{cases}=\begin{cases}\alpha_{h^{-1}*g}(a1_{g^{-1}*h}),\text{ if
}h\in\mathcal{E}_{g},\\\ 0,\text{ otherwise.}\end{cases}$

Then we can work only with the case in which the meet $e:=r(g)\wedge r(h)$ is
defined. As $a\in A_{g^{-1}}$ we have that $a1_{g^{-1}*h}\in A_{g^{-1}}\cap
A_{g^{-1}*h}=A_{(e|g^{-1})}\cap A_{g^{-1}*h}$. Hence

$\displaystyle\beta_{g}(\psi_{d(g)}(a))|_{h}$
$\displaystyle=\alpha_{h^{-1}*g}(a1_{g^{-1}*h})=\alpha_{(h^{-1}|e)}\circ\alpha_{(e|g)}(a1_{g^{-1}*h})$
$\displaystyle=\alpha_{(h^{-1}|e)}(\alpha_{(e|g)}(a1_{(g^{-1}|e)})1_{(e|h)})=\alpha_{h^{-1}}(\alpha_{g}(a1_{(g^{-1}|e)})1_{(e|h)})$
$\displaystyle=\alpha_{h^{-1}}(\alpha_{g}(a)\alpha_{g}(1_{g^{-1}}1_{(g^{-1}|e)})1_{(e|h)})=\alpha_{h^{-1}}(\alpha_{g}(a)\alpha_{(e|g)}(1_{(g^{-1}|e)})1_{(e|h)})$
$\displaystyle=\alpha_{h^{-1}}(\alpha_{g}(a)1_{(e|g)}1_{(e|h)})=\alpha_{h^{-1}}(\alpha_{g}(a)1_{e}1_{g}1_{h})$
$\displaystyle=\alpha_{h^{-1}}(\alpha_{g}(a)1_{r(g)}1_{r(h)}1_{g}1_{h})=\alpha_{h^{-1}}(\alpha_{g}(a)1_{g}1_{h})=\alpha_{h^{-1}}(\alpha_{g}(a)1_{h}).$

Therefore $\beta_{g}\circ\psi_{d(g)}(a)=\psi_{r(g)}\circ\alpha_{g}(a)$.

(ii): Let $g\in\mathcal{G}$ and
$c\in\psi_{r(g)}(A_{r(g)})\cap\beta_{g}(\psi_{d(g)}(A_{d(g)}))$. Then there
are $a\in A_{r(g)},b\in A_{d(g)}$ such that
$c=\psi_{r(g)}(a)=\beta_{g}(\psi_{d(g)}(b))$. Thus,

$\displaystyle
a=\alpha_{r(g)^{-1}}(a1_{r(g)})=\psi_{r(g)}(a)|_{r(g)}=\beta_{g}(\psi_{d(g)}(b))|_{r(g)}=\psi_{d(g)}(b)|_{g^{-1}}=\alpha_{g}(b1_{g^{-1}})\in
A_{g},$

from where it follows that $c=\psi_{r(g)}(a)\in\psi_{r(g)}(A_{g})$.

Conversely, if $c\in\psi_{r(g)}(A_{g})$, then $c=\psi_{r(g)}(a)$, for some
$a\in A_{g}$. Taking $b=\alpha_{g^{-1}}(a)\in A_{g^{-1}}$, we have from (iii)
that $\beta_{g}(\psi_{d(g)}(b))=\psi_{r(g)}(\alpha_{g}(b))=\psi_{r(g)}(a)=c$.

Thus
$c\in\psi_{r(g)}(A_{g})\cap\beta_{g}(\psi_{d(g)}(A_{g^{-1}}))\subseteq\psi_{r(g)}(A_{r(g)})\cap\beta_{g}(\psi_{d(g)}(A_{d(g)}))$.

(i): Let $e\in\mathcal{G}_{0}$, $h\in\mathcal{G}$ be such that $r(h)=e$, $a\in
A_{d(h)}$ and $b\in A_{e}$. It is enough to prove that
$\psi_{e}(b)\beta_{h}(\psi_{d(h)}(a)),\beta_{h}(\psi_{d(h)}(a))\psi_{e}(b)\in\psi_{e}(A_{e})$.
For that, consider $k\in\mathcal{G}$. We have that
$\beta_{h}(\psi_{d(h)}(a))\psi_{e}(b)|_{k}=\beta_{h}(\psi_{d(h)}(a))|_{k}\psi_{e}(b)|_{k}$.
On the one hand,

$\displaystyle\psi_{e}(b)|_{k}=\begin{cases}\alpha_{k^{-1}}(b1_{k}),\text{ if
}k\in\mathcal{E}_{e},\\\ 0,\text{ otherwise.}\end{cases}$

On the other hand,

$\displaystyle\beta_{h}(\psi_{d(h)}(a))|_{k}$
$\displaystyle=\begin{cases}\psi_{d(h)}(a)|_{h^{-1}*k},\text{ if
}k\in\mathcal{E}_{h}=\mathcal{E}_{e},\\\ 0,\text{
otherwise}\end{cases}=\begin{cases}\alpha_{k^{-1}*h}(a1_{h^{-1}*k}),\text{ if
}k\in\mathcal{E}_{e},\\\ 0,\text{ otherwise.}\end{cases}$

Thus we can work only with the case in which $k\in\mathcal{E}_{e}$. Then,
denoting by $f=r(k)\wedge r(h)$,

$\displaystyle\beta_{h}(\psi_{d(h)}(a))\psi_{e}(b)|_{k}$
$\displaystyle=\alpha_{k^{-1}*h}(a1_{h^{-1}*k})\alpha_{k^{-1}}(b1_{k})=\alpha_{(k^{-1}|f)(f|h)}(a1_{(h^{-1}|f)(f|k)})\alpha_{k^{-1}}(b1_{k})$
$\displaystyle=\alpha_{(k^{-1}|f)(f|h)}(a1_{(h^{-1}|f)(f|k)})1_{(k^{-1}|f)(f|h)}1_{k^{-1}}\alpha_{k^{-1}}(b1_{k})$
$\displaystyle=\alpha_{(k^{-1}|f)(f|h)}(a1_{(h^{-1}|f)(f|k)})1_{(k^{-1}|f)(f|h)}1_{r(k^{-1}|f)}1_{k^{-1}}\alpha_{k^{-1}}(b1_{k}).$

By Lemma 4.4(ii), we have that $A_{(k^{-1}|f)}=A_{k^{-1}}\cap
A_{r(k^{-1}|f)}$, from where it follows that
$1_{(k^{-1}|f)}=1_{k^{-1}}1_{r(k^{-1}|f)}$. Therefore,

$\displaystyle\beta_{h}(\psi_{d(h)}(a))\psi_{e}(b)|_{k}$
$\displaystyle=\alpha_{(k^{-1}|f)(f|h)}(a1_{(h^{-1}|f)(f|k)})1_{(k^{-1}|f)(f|h)}1_{(k^{-1}|f)}\alpha_{k^{-1}}(b1_{k}).$

Now, by Proposition (i)(ii), we have that

$A_{(k^{-1}|f)(f|h)}\cap
A_{(k^{-1}|f)}=\alpha_{(k^{-1}|f)(f|h)}(A_{(h^{-1}|f)(f|k)}\cap
A_{(h^{-1}|f)}),$

so
$1_{(k^{-1}|f)(f|h)}1_{(k^{-1}|f)}=\alpha_{(k^{-1}|f)(f|h)}(1_{(h^{-1}|f)(f|k)}1_{(h^{-1}|f)})$.
Thus,

$\displaystyle\beta_{h}(\psi_{d(h)}(a))\psi_{e}(b)|_{k}$
$\displaystyle=\alpha_{(k^{-1}|f)(f|h)}(a1_{(h^{-1}|f)(f|k)})\alpha_{(k^{-1}|f)(f|h)}(1_{(h^{-1}|f)(f|k)}1_{(h^{-1}|f)})\alpha_{k^{-1}}(b1_{k})$
$\displaystyle=\alpha_{(k^{-1}|f)(f|h)}(a1_{(h^{-1}|f)(f|k)}1_{(h^{-1}|f)})\alpha_{k^{-1}}(b1_{k})$
$\displaystyle=\alpha_{(k^{-1}|f)}(\alpha_{(f|h)}(a1_{(h^{-1}|f)})1_{(f|k)})\alpha_{k^{-1}}(b1_{k})$
$\displaystyle=\alpha_{k^{-1}}(\alpha_{(f|h)}(a1_{(h^{-1}|f)})1_{(f|k)})\alpha_{k^{-1}}(b1_{k})$
$\displaystyle=\alpha_{k^{-1}}(\alpha_{(f|h)}(a1_{(h^{-1}|f)})1_{(f|k)}b1_{k})=\alpha_{k^{-1}}(\alpha_{(f|h)}(a1_{(h^{-1}|f)})1_{f}1_{k}b1_{k})$
$\displaystyle=\alpha_{k^{-1}}(\alpha_{(f|h)}(a1_{(h^{-1}|f)})b1_{k})=\psi_{e}(\alpha_{(f|h)}(a1_{(h^{-1}|f)})b)|_{k}.$

Notice that $b\in A_{e}$ implies $\alpha_{(f|h)}(a1_{(h^{-1}|f)})b\in A_{e}$,
since $A_{e}$ is an ideal of $A$. Analogously we have that
$\psi_{e}(b)\beta_{h}(\psi_{d(h)})\in\psi_{e}(A_{e})$.

The uniqueness now follows by [4, Theorem 2.1]. ∎

######  Example 4.8.

Resuming Example 3.6 and applying the construction of Theorem 4.7, we obtain
$\mathcal{E}_{g}=\mathcal{G},\text{ for all }g\in\mathcal{G}$, from where
$F_{g}=\mathcal{F},\text{ for all }g\in\mathcal{G}$.

Thus, $\gamma_{s}(a,b,c,d,f)=(d,c,b,a,f)=\gamma_{s^{-1}}(a,b,c,d,f)$ and
$\gamma_{g}=Id_{\mathcal{F}}$, for all $f\in\mathcal{G}_{0}$. Furthermore,
notice that
$\psi_{r(s)}(ae_{2}+be_{3})=(ae_{2},ae_{2},ae_{2}+be_{3},ae_{2},ae_{2})$ and
$\psi_{d(s)}(ae_{2})=\psi_{e}(ae_{2})=(ae_{2},ae_{2},ae_{2},ae_{2},ae_{2})$.

Therefore
$B_{s}^{\prime}=\\{(ae_{2},ae_{2},ae_{2}+be_{3},ae_{2},ae_{2}):a,b\in R\\}$,
$B_{s^{-1}}^{\prime}=\\{(ae_{2},ae_{2}+be_{3},ae_{2},$ $ae_{2},ae_{2}):a,b\in
R\\}$ and $B_{e}^{\prime}=Re_{2}(1,1,1,1,1)$. Hence
$B^{\prime}=B_{s}^{\prime}+B_{s^{-1}}^{\prime}+B_{e}^{\prime}=\\{(ae_{2},ae_{2}+be_{3},ae_{2}+ce_{3},ae_{2},ae_{2}):a,b,c\in
R\\}$ and then $B^{\prime}\simeq B$ and $B_{g}^{\prime}\simeq B_{g}$, for all
$g\in\mathcal{G}$. So $\beta^{\prime}\simeq\beta$.

##  5\. Applications

We conclude by presenting two applications of the results of the previous
sections. We first show that the partial skew ordered groupoid ring is Morita
equivalent to the skew ordered groupoid ring relative to its globalization.
Afterwards, as a consequence of the ESN Theorem, we provide a necessary and
sufficient condition for a partial action of an inverse semigroup to have a
(unique) minimal globalization.

###  5.1. A Morita Context

To simplify the notation, in this subsection we will assume that
$\varphi_{e}=\iota_{e}:A_{e}\to B_{e}$ is the usual inclusion, so that $A_{e}$
is an ideal of $B_{e}$, for all $e\in\mathcal{G}_{0}$.

######  Proposition 5.1.

Let $\alpha=(A_{g},\delta_{g})_{g\in\mathcal{G}}$ be a unital P.O. action of
the ordered groupoid $\mathcal{G}$ on a ring $A$ and
$\beta=(B_{g},\beta_{g})_{g\in\mathcal{G}}$ be a globalization of $\alpha$ on
a ring $B$. Let $R=A\ltimes_{\alpha}^{o}\mathcal{G}$ and
$T=B\ltimes_{\beta}^{o}\mathcal{G}$. If $\mathcal{G}_{0}$ is finite,

  1. (i)

$T1_{R}=\sum_{g\in\mathcal{G}}\overline{\beta_{g}(A_{d(g)})\delta_{g}}$;

  2. (ii)

$1_{R}T=\sum_{g\in\mathcal{G}}\overline{A_{r(g)}\delta_{g}}$;

  3. (iii)

$1_{R}T1_{R}=R$;

  4. (iv)

$T1_{R}T=T$.

###### Proof.

First notice that the crossed products $A\ltimes_{\alpha}\mathcal{G}$ and
$B\ltimes_{\beta}\mathcal{G}$ are associative, so $R$ and $T$ are defined. The
proof of (i)-(iii) are similar to the nonordered case [4, Proposition 3.1]. In
order to prove (iv), it suffices to show that $T\subseteq 1_{R}T1_{R}$. Since
for all $h\in\mathcal{G}$ we have that $B_{h}=\sum_{r(g)\leq
r(h)}\beta_{g}(A_{d(g)})$, the result follows from

$\displaystyle\overline{\beta_{g}(a)\delta_{h}}=\overline{\beta_{g}(a)\delta_{(r(g)|h)}}=\overline{\beta_{g}(a)\delta_{g}}\cdot\overline{1_{r(g^{-1}(r(g)|h))}\delta_{g^{-1}(r(g)|h)}}\in(T1_{R})(1_{R}T)=T1_{R}T,$

for all $a\in A_{d(g)}$. ∎

A _Morita context_ is a sextuple
$(R,R^{\prime},M,M^{\prime},\varphi,\varphi^{\prime})$, where $R,R^{\prime}$
are unital rings, $M$ is a $(R,R^{\prime})$-bimodule, $M^{\prime}$ is a
$(R^{\prime},R)$-bimodule and $\varphi:M\otimes_{R^{\prime}}M^{\prime}\to R$
and $\varphi^{\prime}:M^{\prime}\otimes_{R}M\to R^{\prime}$ are bimodule
morphisms such that:

  1. (i)

$\varphi(x\otimes x^{\prime})y=x\varphi^{\prime}(x^{\prime}\otimes y)$, for
all $x,y\in M$ and $x^{\prime}\in M^{\prime}$,

  2. (ii)

$\varphi^{\prime}(x^{\prime}\otimes x)y^{\prime}=x^{\prime}\varphi(x\otimes
y^{\prime})$, for all $x^{\prime},y^{\prime}\in M^{\prime}$ and $x\in M$.

According to [29, Theorems 4.1.4 and 4.1.17], if $\varphi$ and
$\varphi^{\prime}$ are surjective, then the categories of left $R$-modules and
of left $R^{\prime}$-modules are equivalent and the rings $R$ and $R^{\prime}$
are said to be _Morita equivalent_. In this case, we also say that the Morita
context $(R,R^{\prime},M,M^{\prime},\varphi,\varphi^{\prime})$ is _strict_.

######  Theorem 5.2.

The rings $R$ and $T$ as above are Morita equivalent.

###### Proof.

Consider $M=1_{R}T$ and $N=T1_{R}$. Clearly $M$ is a $(R,T)$-bimodule and $N$
is a $(T,R)$-bimodule. Defining the maps $\varphi:M\otimes_{T}N\to R$ and
$\varphi^{\prime}:N\otimes_{R}M\to T$ by $\varphi(m\otimes n)=mn$ and
$\varphi^{\prime}(n\otimes m)=nm$, we have that
$(R,T,M,N,\varphi,\varphi^{\prime})$ is a strict Morita context by Proposition
5.1. ∎

###  5.2. Partial Actions of Inverse Semigroups

Let $S$ be an inverse semigroup. According to [7], a partial action of an
inverse semigroup $S$ on a ring $A$ is a collection
$\alpha=(A_{s},\alpha_{s})_{s\in S}$ of ideals $A_{ss^{-1}}$ of $A$, ideals
$A_{s}$ of $A_{ss^{-1}}$ and ring isomorphisms $\alpha_{s}:A_{s^{-1}}\to
A_{s}$ such that

  1. (P1)

$A=\sum_{e\in E(S)}A_{e}$.

  2. (P2)

$\alpha_{s}(A_{s^{-1}}\cap A_{t})=A_{s}\cap A_{st}$, for all $s,t\in S$.

  3. (P3)

$\alpha_{s}(\alpha_{t}(a))=\alpha_{st}(a)$, for all $a\in A_{t^{-1}}\cap
A_{(st)^{-1}}$, $s,t\in S$.

By [17, Theorem 6.1, Lemma 6.2], we have that the partial actions of an
inverse semigroup $S$ on a ring $A$ are in one-to-one correspondence with the
strong P.O. actions of an associated inductive groupoid $\mathbb{G}(S)$ on
$A$. The ESN Theorem [20, Theorem 4.1.8] also ensures that the global actions
of $S$ are in one-to-one correspondence with the global actions of
$\mathbb{G}(S)$.

These remarks enable us to translate the theory that we have developed so far
to the case of inverse semigroups. More precisely, we have the following
result.

######  Theorem 5.3.

Let $\alpha$ be a partial inverse semigroup action of $S$ on a ring $A$ such
that $A_{e}$ is unital for every $e\in E(S)$. Then $\alpha$ has a minimal
globalization $\beta$ if and only if $\alpha$ is unital. Moreover, $\beta$ is
unique up to isomorphism.

Consider the definition of partial skew inverse semigroup ring given in [7],
that is, $A\ltimes_{\alpha}S=\mathcal{L}/\mathcal{N}$, where $\alpha$ is a
unital partial action of $S$ on a ring $A$, $\mathcal{L}=\bigoplus_{s\in
S}A_{s}\delta_{s}$ with component-wise addition and product given by
$(a_{s}\delta_{s})(b_{t}\delta_{t})=\alpha_{s}(\alpha_{s^{-1}}(a_{s})b_{t})\delta_{st}$
and $\mathcal{N}=\langle a_{s}\delta_{s}-a_{s}\delta_{t}:s\leq t\text{ and
}a_{s}\in A_{s}\rangle$ is an ideal of $\mathcal{L}$. Then the same argument
used in Theorem 5.2 also proves the following result.

######  Theorem 5.4.

Let $\alpha$ be a unital partial inverse semigroup action of $S$ on a ring
$A$. Assume that $E(S)$ is finite. Let $\beta$ be the minimal globalization of
$\alpha$, global action of $S$ on a ring $B$. Then $A\ltimes_{\alpha}S$ and
$B\ltimes_{\beta}S$ are Morita equivalent.

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# Circular orbits and chaos bound in slow-rotating curved acoustic black holes

Balbeer Singh<EMAIL_ADDRESS>Nibedita Padhi
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Kharagpur,
Kharagpur-721302, India  Rashmi R. Nayak<EMAIL_ADDRESS>Centre for
Ocean, River, Atmosphere and Land Sciences,
Indian Institute of Technology Kharagpur,
Kharagpur-721302, India

###### Abstract

In recent years, the incorporation of acoustic black holes into Schwarzschild
spacetime has enabled the simultaneous existence of event and acoustic
horizons, as derived from the Gross-Pitaevskii theory. This paper investigates
the dynamics of a particle-like vortex in free fall as it approaches the
horizon of a curved slowly-rotating acoustic black hole, specifically within
the context of the Lens-Thirring spacetime. We analyze the circular orbits of
vortices around the acoustic black hole and delve into the chaotic motion of
the vortex at the unstable equilibrium in the vicinity of the horizon.
Furthermore, we explore whether a universal bound exists for the Lyapunov
exponent characterising this chaotic motion, drawing parallels to the known
bounds for the particle motion in general relativity. We anticipate that
future experiments will be able to corroborate these findings.

††preprint: APS/123-QED

##  I Introduction

Understanding the propagation of acoustic disturbance in a non-homogeneous
fluid flow is difficult but it can be tractable by invoking the language of
Lorentzian differential geometry. Even if the underlying fluid dynamics is
Newtonian and the phenomena take place in flat space plus time, the
fluctuations (sound waves) are governed by an effective Lorentzian space-time
geometry. This in fact gives us the basis to draw an analogy between the black
holes, Einstein gravity and supersonic flow. The study of analogies has indeed
shed some light in the understanding of black holes and this analogy is
fascinating enough to offer a tangible laboratory model [1, 2, 3, 4] for
exploring concepts from curved-space quantum field theory. By considering
supersonic fluid flow, researchers have been able to create an acoustic
analogue of a black hole known as a ”dumb hole.” This analogy has been
extended to demonstrate the presence of phononic Hawking radiation from the
acoustic horizon, mirroring the phenomenon predicted by Stephen Hawking for
black holes. Motivated by these facts we are interested to consider an
analogue acoustic background of a slowly rotating curve Lense-Thirring black
hole geometry [5, 6]. The importance of Lens-Thirring metric lies in the fact
that handling the full Kerr black hole metric is somewhat difficult and also
in case of an actual rotating planet or star, the vacuum solution outside its
surface is not precisely Kerr; it is only asymptotically Kerr, so the only
region one can consider is the asymptotic region where it reduces to the Lens-
Thirring metric which shows the astrophysical relevence of the black hole.
Therefore, we are interested in the associated Lense-Thirring acoustic black
hole (LTABH) which has recently been obtained using the Gross- Pitaevskii and
Yang-Mills theory in [7].

Our aim is to understand how a particle motion or its geodesic will be
influenced by the presence of the horizon of the acoustic black hole. As we
know there is a difference how a test particle motion is treated in Newtonian
mechanics and general theory of relativity but the general principle for the
motion of free test particle in curved space time is the same as that for flat
space time. Considering the motion of the vortex as the test particle near the
horizon of the slowly rotating curved acoustic black hole certainly will give
us a better understanding of the black hole as a practically observed model.
Recently, Maldacena, Shenker and Stanford [8] have proposed a bound given by

$\lambda\leq 2\pi T,$ (1)

where T is the Hawking temperature i.e Lyapunov exponent $\lambda$ is bounded
above by and proportional to temperature T. The bound was originally proved
using the shock waves near the horizon of a black hole. Plethra of research
have been conducted on the Lyapunov exponent and the chaos bound including
black hole systems with a probe particle [9, 10, 11, 12, 13] and of the
AdS/CFT correspondence [14, 15, 16, 17, 18, 19]. Although originally proposed
for holographic system, the bound is more robust and found to be satisfied by
various systems such as [20, 21, 22, 23], with some noticeable violations [24,
25, 26, 27]. The chaos bound discussed in eqn.(1) can be examined by analyzing
the geodesic motion of a single particle. Several interesting studies have
been conducted to test the chaos bound [21],[24],[28],[29],[30]. In their
study, Hashimoto et. al. have examined a particle influenced by external
forces near the horizon of a spherically symmetric static black hole,
demonstrating that the Lyapunov exponent of the static equilibrium reaches the
bound near the horizon [21]. Simultaneously, several studies have also been
carried out with the geodesic movement of particles in different models
showcasing the violations of the chaos bound. For instance, Lei et. al.
examined the circular motion, both null and timelike, of charged particles
near the horizon of a charged black hole [23]. Their findings indicate that
locally, for circular motions of RN(AdS) black hole, the chaos bound may be
exceeded. Similarly, the authors of [31] have analysed the chaos bound
violation in both extremal and non extremal black holes considering the
geodesic motion. For more details of investigation see [31],[32],[33],[34].
Moreover, in the context of the acoustic black hole, Ge et. al. [35] have
recently proved that the bound saturates for vortex motion near the acoustic
black hole in the flat and curved spacetime. To our current knowledge, there
has been no prior research conducted in the context of acoustic black holes
investigating the bound using the geodesic circular orbit stability analysis.
Therefore with this motivation, our aim is to initially derive the Lyapunov
exponent and subsequently examine whether the chaos bound holds for the slowly
rotating acoustic black hole using the approach mentioned in works [36, 37].

The structure of the paper is as follows. In section II we give a brief
overview of the Gross-Pitaevskii theory and review of the derivation of slow-
rotating acoustic black hole from the related Lens-Thirring metric. In section
III we obtain the effective potential of the particle-like vortex in the
vicinity of the acoustic black hole and then further find out the innermost
stable circular orbit. Then in section IV we first develop the necessary
Jacobi matrix to find the Lyapunov exponent and related stability of circular
orbits followed by the analysis of the chaos bound using numerical methods. In
the last section we conclude our results with some comments.

##  II Slow-rotating acoustic black holes from Gross-Pitaevskii equation

Acoustic black holes can be realized through various approaches, extending
beyond laboratory experiments within condensed matter systems to phenomena
within high energy physics, astronomy and cosmology [38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50]. In recent works [51, 35, 52, 53], a set of
acoustic black hole solutions was derived through the application of
relativistic Gross-Pitaevskii and Yang-Mills theories.
In this section, we briefly provide an overview of the development of the
acoustic metric derived from the Gross-Pitaevskii (GP) theory, for more
details see [51, 7, 54]. The Gross-Pitaevskii theory can model the vortex
dynamics within fluid systems for both the flat and curved spacetimes [55].
The action of the GP theory is given by

$\displaystyle\mathcal{S}=\int
d^{4}x\sqrt{-g}\left(|\partial_{\mu}\Phi|^{2}+m^{2}|\Phi|^{2}-\frac{b}{2}|\Phi|^{4}\right),$
(2)

where $\Phi$ is the complex scalar field, b is a constant, and m is a
parameter which depends on the Hawking temperature T as $m^{2}\sim(T-T_{c})$
where $T_{c}$ is the critical temperature of the GP theory describing phase
transitions. Thus at critical temperature $T=T_{c}$, $m^{2}=0$. The complex
scalar field $\Phi$ serves as the order parameter in the phase transition of
the fluid system, propagating within a fixed background spacetime of the form

$\displaystyle
ds^{2}_{background}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$
$\displaystyle+2g_{t\phi}dtd\phi,$

Taking the field $\Phi=\sqrt{\rho(x^{\mu})}e^{i\Theta(x^{\mu})}$ around this
spacetime and then the perturbations of the fields $\Theta,\rho$

$\displaystyle\rho=\rho_{0}+\rho_{1},$ (4)
$\displaystyle\Theta=\Theta_{0}+\Theta_{1},$ (5)

where $(\rho_{0},\Theta_{0})$ are related to the fixed spacetime and
$(\rho_{1},\Theta_{1})$ are related to the fluctuations. The propagation of
the phase fluctuations $\Theta(x^{\mu})$ up to the subleading order is given
by a wave equation in cured spacetime similar to the massless Klein-Gordon
equation

$\displaystyle\frac{1}{\sqrt{-G}}\partial_{\mu}\left(\sqrt{-G}G^{\mu\nu}\partial_{\nu}\Theta\right)=0,$
(6)

with the effective metric spacetime tensor $G_{\mu\nu}$ given by

$\displaystyle ds^{2}_{ac}$ $\displaystyle=$ $\displaystyle
G_{\mu\nu}dx^{\mu}dx^{\nu}$ (8) $\displaystyle=$
$\displaystyle\frac{c_{s}}{\sqrt{c_{s}^{2}-v_{\mu}v^{\mu}}}\Bigg{[}(c_{s}^{2}-v_{r}v^{r})g_{tt}dt^{2}+$
$\displaystyle
c_{s}^{2}\frac{c_{s}^{2}-v_{\mu}v^{\mu}}{c_{s}^{2}-v_{r}v^{r}}g_{rr}dr^{2}+(c_{s}^{2}-v_{\mu}v^{\mu})g_{\theta\theta}d\theta^{2}+$
$\displaystyle(c_{s}^{2}-v_{\mu}v^{\mu})g_{\phi\phi}d\phi^{2}+2(c_{s}^{2}-v_{t}v^{t})g_{t\phi}dtd\phi\Bigg{]},$

where $v_{\mu}\equiv\partial_{\mu}\Theta$ is the velocity in the fluid system.
The $G_{\mu\nu}$ represents the propagation of phase fluctuations in fluid
systems.
Next, we focus on the Lens-Thirring acoustic black hole obtained in [6, 7, 5]
using eq (8) by assuming a simple background such that $g_{tt}(r)g_{rr}(r)=-1$
and taking the background phase $\Theta_{0}$ to be independent of the
coordinates $\theta,\phi$ which leads to $v_{\theta}=0=v_{\phi}$. Further, we
shall only deal with the critical temperature of the GP theory so that
$v_{\mu}v^{\mu}=-2c_{s}^{2}$ with the coordinate transformations

$\displaystyle
dt\xrightarrow{}dt+\frac{v_{r}v_{t}}{g_{tt}\left(c_{s}^{2}-v_{i}v^{i}dr\right)},$
(9) $\displaystyle
d\phi\xrightarrow{}d\phi-\frac{g_{t\phi}\left(c_{s}^{2}-v_{t}v^{t}\right)v_{t}v_{r}dr}{g_{\phi\phi}\left(c_{s}^{2}-v_{\mu}v^{\mu}\right)g_{tt}\left(c_{s}^{2}-v_{r}v^{r}\right)},$
(10)

Then the acoustic metric obtained in eq (8) reduces to the following

$\displaystyle ds^{2}$
$\displaystyle=\sqrt{3}c_{s}^{2}\Bigg{[}-\mathcal{F}(r)dt^{2}+\frac{1}{\mathcal{F}(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}$ $\displaystyle-\frac{4Ma\sin^{2}\theta}{r}dtd\phi\Bigg{]},$ (11)

where,

$\mathcal{F}(r)=f(r)\Bigg{[}1-f(r)\frac{2M\xi}{r}\Bigg{]},$ (12)

where, $a$ is the rotation parameter with the limit $a^{2}\approx 0$, the
function $f(r)=1-\frac{2M}{r}$, and $\xi$ is a tuning parameter. Note that as
mentioned in [7] after the essential rescaling of the four-velocity in the
critical temperature limit, the radial velocity and the temporal velocity are
considered as

$\displaystyle
v_{r}=\sqrt{\frac{2M\xi}{r}},~{}~{}v_{t}=\sqrt{f(r)+\frac{2M\xi}{r}f^{2}(r)},$
(13)

For a spatially two-dimensional model, we choose $\theta=\frac{\pi}{2}$,
therefore the metric in eq II reduces to

$ds^{2}=\sqrt{3}c_{s}^{2}\Bigg{[}-\mathcal{F}(r)dt^{2}+\frac{1}{\mathcal{F}(r)}dr^{2}+r^{2}d\phi^{2}-\frac{4Ma}{r}dtd\phi\Bigg{]},$
(14)

The metric in the matrix form can be rewritten as

$g_{\mu\nu}=\left(\begin{array}[]{ccc}-\mathcal{F}(r)&0&-\frac{2aM}{r}\\\
0&\frac{1}{\mathcal{F}(r)}&0\\\ -\frac{2aM}{r}&0&r^{2}\\\ \end{array}\right),$
(15) $\mathcal{F}=0=(r-r_{s})(r-r_{1})(r-r_{2}),$ (16)

On equating, $\mathcal{F}=0$, we obtain the outer and inner acoustic horizons
denoted by, respectively,
$r_{1}=M(\xi+\sqrt{\xi^{2}-4\xi}),~{}~{}r_{2}=M(\xi-\sqrt{\xi^{2}-4\xi})$
whereas the optical event horizon is given by $r_{s}=2M$. For the acoustic
horizons to exist, we must have $\xi\geq 4$, for the region of parameters
falling in $0\leq\xi<4$ only the optical horizon exists and in this case both
the interior and exterior acoustic event horizons vanish. Further when the
tuning parameter $\xi=4$, the inner and outer horizons coincide and we obtain
the extreme LTABH (Fig 1). But, taking the parameter $\xi>4$, there exist
three regions: In the region where $r<r_{s}$, both light rays (photons) and
sound waves (phonons) are unable to escape from the acoustic black hole.For
$r_{s}<r<r_{1}$, light rays can escape from the acoustic black hole, while
sound waves cannot.Beyond $r>r_{1}$, both light rays and sound waves have the
potential to escape from the acoustic black hole [52]. Moreover, when $\xi=0$,
the metric reduces to the original Schwarzchild metric (blue curve in Fig 1).
As $\xi\xrightarrow{}\infty$ the entire spacetime resides within the acoustic
black hole. We will henceforth regard the outer acoustic horizon as the
acoustic horizon.
The surface gravity of the slowly rotating acoustic black hole and the angular
velocity corresponding to the outer horizon respectively can be expressed as
[7]

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\frac{1}{r_{1}^{2}}\left.\mathcal{F}^{\prime}(r)\right|_{r=r_{1}}=\frac{(r_{1}-r_{2})(r_{1}-r_{s})}{2r_{1}^{2}},$
(17) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\frac{2Ma}{r_{1}^{2}},$ (18)

and therefore the Hawking temperature is given by

$\displaystyle T_{H}=\frac{\kappa}{2\pi k_{B}},$ (19)

where $k_{B}$ is the Boltzmann constant.

Figure 1: (Left panel) Plot for $r$ vs $\xi$ at fixed parameters $M=1$. The
blue line represents the $r_{s}$. The (right panel) $\mathcal{F}(r)$ vs $r$ at
same fixed parameters with $\xi=0,~{}6$(blue, red). The green dotted curve
depicts outermost acoustic horizon, $r_{1}=9.4641.$

##  III Circular orbits and ISCO

In this section, we turn our attention to the motion of the test particle
(vortex) of unit mass around the slowly-rotating acoustic black hole. It was
suggested that vortices could act like relativistic particles with their
motion dictated by the acoustic metric such that they do not exceed the speed
of sound [56, 57, 58]. Therefore we shall consider a particle-like vortex
(i.e. irrotational) of unit mass in the presence of the acoustic black hole.
We first find the effective potential and then the presence of the stable (and
unstable) circular orbits. Towards the end, we also obtain the innermost
stable circular orbit (ISCO).

###  III.1 Effective Potential and Circular Orbits

The metric in eq 14 is independent of the coodinates $t$ and $\phi$, therefore
there are conserved quantities along these directions. If the Killing vectors
for our metric are $\xi^{\alpha}=(1,0,0)$ and $\eta^{\alpha}=(0,0,1)$ and
$u^{\alpha}$ is the three-velocity of the vortex [59, 60], then, energy per
unit mass yields

$\displaystyle\epsilon=$ $\displaystyle-\xi^{\alpha}g_{\alpha\beta}u^{\beta}$
$\displaystyle=$
$\displaystyle\sqrt{3}c_{s}^{2}\mathcal{F}(r)\frac{dt}{d\tau}+\sqrt{3}c_{s}^{2}\left(\frac{2aM}{r}\right)\frac{d\phi}{d\tau},$
(20)

and the angular momentum per unit mass is given by

$\displaystyle l=$ $\displaystyle\eta^{\alpha}g_{\alpha\beta}u^{\beta}$
$\displaystyle=$
$\displaystyle-\sqrt{3}c_{s}^{2}\left(\frac{2aM}{r}\right)\frac{dt}{d\tau}+\sqrt{3}c_{s}^{2}r^{2}\frac{d\phi}{d\tau},$
(21)

Using the normalization condition $g_{\alpha\beta}u^{\alpha}u^{\beta}=-1$, we
obtain

$\displaystyle-\mathcal{F}(r)(u^{t})^{2}-\frac{4aM}{r}u^{t}u^{\phi}+\frac{1}{\mathcal{F}(r)}(u^{r})^{2}+r^{2}(u^{\phi})^{2}$
$\displaystyle=-\frac{1}{\sqrt{3}c_{s}^{2}},$

or,

$\displaystyle(u^{r})^{2}$
$\displaystyle=\mathcal{F}(r)\Bigg{[}-\frac{1}{\sqrt{3}c_{s}^{2}}+\mathcal{F}(r)(u^{t})^{2}+\frac{4aM}{r}u^{t}u^{\phi}-r^{2}(u^{\phi})^{2}\Bigg{]},$

On solving the eq (III.1) and eq (III.1), we obtain the following relation

$\displaystyle
u^{t}=\frac{-2alM+r^{3}\epsilon}{\sqrt{3}c_{s}^{2}r^{3}\mathcal{F}(r)},$ (24)

and,

$\displaystyle
u^{\phi}=\frac{2aM\epsilon+lr\mathcal{F}(r)}{\sqrt{3}c_{s}^{2}r^{3}\mathcal{F}(r)},$
(25)

Then putting these values in the eq (III.1), we obtain

$\displaystyle(u^{r})^{2}=$
$\displaystyle\frac{\epsilon\left(-4alM+r^{3}\epsilon\right)-r\mathcal{F}(r)\left(\sqrt{3}c_{s}^{2}r^{2}+l^{2}\right)}{3c_{s}^{4}r^{3}}$
(26) $\displaystyle=$
$\displaystyle\frac{\epsilon\left(-4alM+r^{3}\epsilon\right)}{r^{3}}+$
$\displaystyle\frac{\left(l^{2}+r^{2}\right)(2M-r)\left(r^{2}-2M\xi(r-2M)\right)}{r^{5}},$

where, in the last equation we have set $c_{s}=1/\sqrt{3}$.
Thus the effective potential is obtained as

$\displaystyle V_{eff}=$ $\displaystyle\epsilon^{2}-(u^{r})^{2}$ (28)
$\displaystyle=$
$\displaystyle\frac{1}{r^{5}}\Big{(}r^{2}\left(4alM\epsilon+\left(l^{2}+r^{2}\right)(r-2M)\right)$
$\displaystyle-2M\xi\left(l^{2}+r^{2}\right)(r-2M)^{2}\Big{)},$

which in the limit of $a\xrightarrow{}0$, and $\xi\xrightarrow{}0$ reduces to
that of the Schwarzchild [6].

Figure 2: Plot for Effective potential vs $r$ for $M=1$, and for various
parameter values of $a,~{}\xi,~{}\epsilon$ and $l$.

Figure 3: $V_{\text{eff}}$ at $M=0.1$, $a=0.1$, $\xi=4$, and different values
of $\epsilon$ and $l$.

Comparison of Effective Potential $V_{\text{eff}}$ for different parameters,
left panel for non-extremal and right panel for extremal case.

Let us now analyse the circular orbits in the acoustic black hole scenario for
which $u^{r}=0$. For a particle to describe the circular orbits we should have
$\left.\frac{dV_{eff}}{dr}\right|_{r=r_{*}}=0$. Further, the effective
potential for a stable circular orbit should have a local minimum ensured by
$\left.\frac{d^{2}V_{eff}}{dr^{2}}\right|_{r=r_{*}}>0$, whereas the unstable
circular orbit corresponds to the local maximum given by the condition
$\left.\frac{d^{2}V_{eff}}{dr^{2}}\right|_{r=r_{*}}<0$. As shown in Fig 3, the
effective potential attains a minimum value as well as a maximum, therefore
there exists stable and unstable circular orbits for the acoustic black hole.
However, for the extremal case, we only see the unstable circular orbit (Fig
3). The effective potential acts as a barrier for radial coordinate
approaching the near horizon, however, the effective potential quickly becomes
zero at the horizon. Further, on analysing for fixed $\xi,~{}M,~{}a$ and
$\epsilon$, since the horizon does not depend on $l$, so the effective
potential attains the zero at the same position and is independent of the
angular number $l$ which is quite evident from the Fig 4(left panel).
Moreover, the potential barrier increases on increasing $l$ similar to the
Schwarzchild scenario.

Figure 4: (Left panel)Effective potential with $M=0.1,~{}\xi=6,~{}a=0.01$ and
$\epsilon=10$ at different $l=5,10,12$ (black, red, blue). The (right panel)
plot showing $l^{2}-r$ plot at parameters $M=1,~{}\epsilon=2,$ for pairs
$~{}~{}a=0.1,0.3,0.08,~{}~{}\xi=6,8,4.5$, (blue,red,green).

###  III.2 Innermost Stable Circular Orbit

Further on analysing the equation $\frac{dV_{eff}}{dr}=0$ which gives

$\displaystyle 2r^{2}\left(-6alM\epsilon+l^{2}(3M-r)+Mr^{2}\right)+$
$\displaystyle 2M\xi(r-2M)\left(l^{2}(3r-10M)+r^{2}(r-6M)\right)=0$

and solving for the angular momentum $l^{2}$, we get

$\displaystyle l^{2}$ $\displaystyle=\frac{1}{\left(-20M^{3}\xi+16M^{2}\xi
r-3M(\xi+1)r^{2}+r^{3}\right)^{2}}$
$\displaystyle\quad\times\left(\left(\sqrt{Mr^{2}\left(9a^{2}Mr^{2}\epsilon^{2}-\left(20M^{3}\xi-16M^{2}\xi
r+3M(\xi+1)r^{2}-r^{3}\right)\left(12M^{2}\xi+r(-8M\xi+\xi
r+r)\right)\right)}+3aMr^{2}\epsilon\right)^{2}\right),$ (31)

due to the excessively challenging expression of $l$ we obtain the plots of
the angular momentum $l^{2}$ with the help of the numerics as shown in Fig
4(right panel) for different rotation parameter values. Since the minimum of
the angular momentum ensures the presence of the inner stable circular orbit
(ISCO) of a test particle, therefore from Fig 4 (right panel), it is evident
that there exists an ISCO [61, 62]. We further plot the $l^{2}$ vs $r$ plot at
different parameters as shown in Fig 4 (right panel). As $a$ decreases the
curve shifts towards the left to smaller $r$.
The radius of the ISCO in an acoustic rotating black hole depends on the
fluid’s angular velocity profile which we have notice from our analysis and
the speed of sound in the fluid. Similar to the Kerr black hole, where the
ISCO radius depends on the spin parameter a, in the acoustic case, it will
depend on the fluid’s rotational properties.

##  IV Lyapunov exponent and bound analysis

In the literature, there exist various techniques to determine the Lyapunov
exponent, for instance [63, 28, 23, 64, 21] and others such as [30, 24]. In
particular, with the help of the Lagrangian framework, the Lyapunov exponents
can be established for particle motion [63]. The equations of motion of the
particle can be written as

$\displaystyle\frac{dy^{i}}{dt}=F_{i}(x^{j}).$ (32)

On linearising the equations about a certain orbit

$\displaystyle\frac{d\delta y^{i}(t)}{dt}=K_{ij}\delta y^{j}(t),$ (33)

where $K_{ij}$ is the Jacobian matrix defined as

$\displaystyle K_{ij}=\frac{\partial F_{i}}{\partial y^{j}},$ (34)

about the solution

$\displaystyle\delta y^{i}(t)=L_{ij}\delta y^{i}(0),$ (35)

where $L_{ij}(t)$ is the evolution matrix satisfying
$\dot{L}_{ij}(t)=K_{il}L_{lj}$ and $L_{ij}(0)=\delta_{ij}$. The Lyapunov
exponent characterises the average exponential rate of divergence between two
nearby trajectories in a dynamical system. It signifies the typical rate of
contraction or expansion of nearby orbits in the phase space. The eigenvalues
of the matrix $L_{ij}$ determine the principal Lyapunov exponent

$\displaystyle\lambda=\lim_{t\xrightarrow{}\infty}\frac{1}{t}log\left(\frac{L_{ij}(t)}{L_{ij}(0)}\right).$
(36)

Then typically the eigenvalues of the Jacobi matrix give the Lyapunov
exponent. The positive Lyapunov exponent indicates the presence of chaos in
the system.

###  IV.1 Jacobi matrix method

In this section, we will employ the Jacobi matrix method to calculate the
Lyapunov exponent [36, 64, 65]. We consider a vortex(test particle) of unit
mass in the equatorial plane $(\theta=\frac{\pi}{2})$ of the slow-rotating
black hole whose Lagrangian is given by

$\displaystyle
2\mathcal{L}=\sqrt{3}c_{s}^{2}\Bigg{[}-\mathcal{F}(r)\dot{t}^{2}+\frac{1}{\mathcal{F}(r)}\dot{r}^{2}+r^{2}\dot{\phi}^{2}-\frac{4Ma}{r}\dot{t}\dot{\phi}\Bigg{]},$

where
$\dot{x}=\Big{(}\frac{dt}{d\tau},\frac{dr}{d\tau},\frac{d\phi}{d\tau}\Big{)}$.
On setting, $\sqrt{3}c_{s}^{2}=1$. then the generalised momenta are given by

$\displaystyle-p_{t}$
$\displaystyle=\mathcal{F}(r)\frac{dt}{d\tau}+\left(\frac{2aM}{r}\right)\frac{d\phi}{d\tau},$
(38) $\displaystyle p_{r}$ $\displaystyle=\frac{\dot{r}}{\mathcal{F}(r)},$
(39) $\displaystyle p_{\phi}$
$\displaystyle=-\left(\frac{2aM}{r}\right)\frac{dt}{d\tau}+r^{2}\frac{d\phi}{d\tau},$
(40)

Therefore, the Hamiltonian of the particle-like vortex is

$\displaystyle\mathcal{H}=-\frac{2aMp_{t}p_{\phi}}{r^{3}\mathcal{F}(r)}-\frac{p_{t}^{2}}{2\mathcal{F}(r)}+\frac{1}{2}\mathcal{F}(r)p_{r}(\tau){}^{2}+\frac{p_{\phi}^{2}}{2r^{2}},,$
(41)

Using the Hamiltonian equation of motions we obtain

$\displaystyle\dot{t}$
$\displaystyle=-\frac{2aMp_{\phi}+r^{3}p_{t}}{r^{3}\mathcal{F}(r)},$ (42)
$\displaystyle\dot{p}_{t}$ $\displaystyle=0,$ (43) $\displaystyle\dot{r}$
$\displaystyle=p_{r}\mathcal{F}(r),$ (44) $\displaystyle\dot{p}_{r}$
$\displaystyle=\frac{1}{2r^{4}\mathcal{F}(r)^{2}}\Bigg{(}2\mathcal{F}(r)p_{\phi}\left(r\mathcal{F}(r)p_{\phi}-6aMp_{t}\right)-$
$\displaystyle
r\mathcal{F}^{\prime}(r)\left(4aMp_{t}p_{\phi}+r^{3}\left(\mathcal{F}(r)^{2}p_{r}^{2}+p_{t}^{2}\right)\right)\Bigg{)},$
(45) $\displaystyle\dot{\phi}$
$\displaystyle=\frac{rp_{\phi}-\frac{2aMp_{t}}{\mathcal{F}(r)}}{r^{3}},$ (46)
$\displaystyle\dot{p}_{\phi}$ $\displaystyle=0,$ (47)

where $\prime$ denotes the derivative wrt coordinate r.
We further define

$\displaystyle F_{1}\equiv\frac{\dot{r}}{\dot{t}}$ $\displaystyle=$
$\displaystyle-\frac{r^{3}\mathcal{F}(r)^{2}p_{r}}{2aMp_{\phi}+r^{3}p_{t}},$
(48) $\displaystyle F_{2}\equiv\frac{\dot{p}_{r}}{\dot{t}}$ $\displaystyle=$
$\displaystyle-\frac{2\mathcal{F}(r)p_{\phi}\left(r\mathcal{F}(r)p_{\phi}-6aMp_{t}\right)-r\mathcal{F}^{\prime}(r)\left(4aMp_{t}p_{\phi}+r^{3}\left(\mathcal{F}(r)^{2}p_{r}^{2}+p_{t}^{2}\right)\right)}{2r\mathcal{F}(r)\left(2aMp_{\phi}+r^{3}p_{t}\right)},$

The normalization condition of the ”three-velocity” of the massive particle
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$ provides the following constraint

$\displaystyle
r^{4}\mathcal{F}(r)\left(r^{2}\mathcal{F}(r)p_{r}^{2}+p_{\phi}^{2}+r^{2}\right)=r^{3}p_{t}\left(4aMp_{\phi}+r^{3}p_{t}\right)$

then employing this constraint in the eq (48) and eq (LABEL:F2) to eliminate
the $p_{t}$ gives

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle-\frac{r^{3}\mathcal{F}(r)^{2}p_{r}}{\sqrt{r^{4}\mathcal{F}(r)\left(r^{2}\mathcal{F}(r)p_{r}^{2}+p_{\phi}^{2}+r^{2}\right)}},$
(51) $\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{12aMp_{\phi}}{r^{4}}+\frac{p_{\phi}^{2}\left(r\mathcal{F}^{\prime}(r)-2\mathcal{F}(r)\right)}{\sqrt{r^{4}\mathcal{F}(r)\left(r^{2}\mathcal{F}(r)p_{r}^{2}+p_{\phi}^{2}+r^{2}\right)}}+\frac{r^{3}\left(2\mathcal{F}(r)p_{r}^{2}+1\right)\mathcal{F}^{\prime}(r)}{\sqrt{r^{4}\mathcal{F}(r)\left(r^{2}\mathcal{F}(r)p_{r}^{2}+p_{\phi}^{2}+r^{2}\right)}}\right),$

Using the constraint $p_{r}=\frac{dp_{r}}{dt}=0$ for the equilibrium orbit of
the particle yields

$\displaystyle K_{11}$ $\displaystyle=0,$ (53) $\displaystyle K_{22}$
$\displaystyle=0,$ (54) $\displaystyle K_{12}$
$\displaystyle=-\frac{\mathcal{F}(r)\sqrt{r^{4}\mathcal{F}(r)\left(p_{\phi}^{2}+r^{2}\right)}}{r\left(p_{\phi}^{2}+r^{2}\right)},$
(55) $\displaystyle K_{21}$
$\displaystyle=\frac{1}{4r\left(r^{4}\mathcal{F}(r)\left(p_{\phi}^{2}+r^{2}\right)\right){}^{3/2}}\Bigg{[}r^{6}\mathcal{F}^{\prime}(r)^{2}\left(-\left(p_{\phi}^{2}+r^{2}\right){}^{2}\right)+2r^{2}\mathcal{F}(r)^{2}\left(6r^{4}p_{\phi}^{2}+4r^{2}p_{\phi}^{4}\right)$
$\displaystyle\quad+2\mathcal{F}(r)\left(p_{\phi}^{2}+r^{2}\right)\left(-48aMp_{\phi}\sqrt{r^{4}\mathcal{F}(r)\left(p_{\phi}^{2}+r^{2}\right)}+r^{5}p_{\phi}^{2}\left(r\mathcal{F}^{\prime\prime}(r)-2\mathcal{F}^{\prime}(r)\right)+r^{8}\mathcal{F}^{\prime\prime}(r)\right)\Bigg{]},$
(56)

and then finally evaluating the eigenvalues of the matrix

$\displaystyle\begin{pmatrix}0&K_{12}\\\ K_{21}&0\end{pmatrix},$ (57)

gives the following (squared of) Lyapunov exponent for the timelike circular
motion

$\displaystyle\lambda^{2}$
$\displaystyle=-\frac{\mathcal{F}(r)\left(-48alM\sqrt{r^{4}\mathcal{F}(r)\left(l^{2}+r^{2}\right)}+r^{6}\left(l^{2}+r^{2}\right)\mathcal{F}^{\prime\prime}(r)-2l^{2}r^{5}\mathcal{F}^{\prime}(r)\right)}{2r^{6}\left(l^{2}+r^{2}\right)}+\frac{1}{4}\mathcal{F}^{\prime}(r)^{2}-\frac{\mathcal{F}(r)^{2}\left(2l^{4}+3l^{2}r^{2}\right)}{r^{2}\left(l^{2}+r^{2}\right)^{2}},$
(58)

where we have set the $p_{\phi}\equiv l$, the angular momentum along $\phi$
direction. Notice that the presence of parameters $a,~{}l$ and $\xi$ is
significant since they make a nontrivial contribution to the Lyapunov
exponent.

The stability of the equilibrium circular orbits can be determined by
$\lambda^{2}$. It can be established that for the circular motion to be
unstable, it should correspond to $\lambda^{2}>0$ [23], which thus indicates
the presence of chaos. Our goal is to examine and if possible to establish the
chaos bound $\lambda<\kappa$. On rewriting the bound eq 1 as

$\displaystyle\lambda^{2}-\kappa^{2}\leq 0.$ (59)

Thus the sign of $\kappa^{2}-\lambda^{2}$ would decide whether the bound is
satisfied. Due to the complex expression of the $\lambda$, we resort to the
numerical calculation and checking of the bound presented in the next section.

###  IV.2 Numerical Analysis

Here, we study how different parameters affect the chaos bound, identifying
the spatial regions where the bound is respected or violated if any. The
region we are concerned about is not limited to the near-horizon region, but
also at a certain distance from the horizon i.e. in the vicinity of the
horizon.

To analyse the bound $\kappa^{2}-\lambda^{2}$, we first find the equilibrium
orbit $r_{0}$ using $p_{r}=F_{2}=0$ numerically at certain fixed parameter
values, and then obtain the corresponding $\kappa^{2}-\lambda^{2}$ value by
numerical calculation using eq (58), eq (12) and eq (17). To this end, without
loss of generality, we set

$\displaystyle M=0.1.~{}~{}$ (60)

The presence of unstable equilibrium points would depend on the parameter
values of $\xi,~{}~{}a,$ and $~{}l$. In Table 1-4 we have shown the position
$r_{0}$ of the circular orbits at different parameter values. Then in the
following analysis, we examine how the angular momentum, rotation parameter
and $\xi$ are influencing the exponent and the chaos bound for extreme LTABH
and the non-extrermal cases.

####  IV.2.1 Extremal cases

Here, since at extremality, the temperature of the black hole is zero
therefore $\kappa=0$ and the bound

$\displaystyle\kappa^{2}-\lambda^{2}\geq 0,$ (61)

reduces to

$\displaystyle-\lambda^{2}\geq 0.$ (62)

However, we observe that $\lambda^{2}>0$ for the equilibrium point (more
accurately the unstable equilibrium point) in the vicinity of the horizon and
therefore the bound is violated (see Fig 5). Moreover, from the table 1, it is
observed that as we increase the angular momentum the position of the orbits
slowly decreases.

Table 1:  Positions of the circular orbits of the particle around extreme LTABH at different values of $l$ when $M=a=0.1$ and $\xi\xrightarrow{}4$. The event horizon is located at $r_{1}=0.4$.  $l$ | 5 | 7 | 10 | 13 | 16 | 20 |
---|---|---|---|---|---|---|---
$r_{0}$ | 0.962649 | 0.953677 | 0.949045 | 0.947253 | 0.946375 | 0.945763 |
Figure 5: The plot for $\kappa^{2}-\lambda^{2}$ for extremal LTABH at fixed
parameters $a=M=0.1~{}~{}\xi\xrightarrow{}4$ for different values of
$l=5,7,10,13,16,20$.

####  IV.2.2 Non-extremal cases

For non-extremal scenarios of the acoustic black hole, the various observed
behaviour are summed up as follows

  * •

When the rotation parameter $a$ and parameter $\xi$ are kept fixed, different
values of angular momentum $l$ give rise to different values of the Lyapunov
exponent, see Fig 6. Further, the larger the value of $l$, the greater the
value of $\lambda$ which implies stronger chaos in the system. However, the
value of $\lambda$ does not exceed the bound.

  * •

With fixed (say) $l=12$, different values of the parameter $\xi$ produce the
different Lyapunov exponent corresponding to fixed rotation parameter $a$.
Moreover, for larger as well as smaller $\xi$ at fixed $a$, there is still no
violation of the bound. See Fig. 7 (top panel). From the figure, we also see
that the values of the $\lambda$ are close to each other even for varying $a$,
and therefore $\kappa^{2}-\lambda^{2}$ values are quite overlapping.

  * •

Finally, in Fig 7 (bottom panel) with fixed rotation parameter, different
angular momentum corresponds to different Lyapunov exponent values. Even if
$l$ increases sufficiently, we observe no bound violation.

  * •

Again similar to the extremal case, for the equilibrium point in vicinity of
the horizon we observe the positivity of the $\lambda^{2}$. However, the bound
is not violated.

Figure 6: The figure shows the plot of $\kappa^{2}-\lambda^{2}$ for different
values of $l$ for $M=0.1$ at $~{}\xi=6$ and fixed $a$.

In non-extremal cases, we see $\lambda^{2}>0$ for equilibrium points in the
vicinity of the horizon, however far away from the horizon we observe
$\lambda^{2}<0$. Further there exist points $r_{0}^{\ast}$ ($<r_{1}$) such
that the positivity of $\lambda^{2}$ is seen to be in region $(r_{1}$,
$r_{0}^{\ast})$. Also, we observe that as we move very far away (i.e.
asymptotic limit) from point $r_{0}^{\ast}$, the $\lambda^{2}$ approaches
zero. Further, the bound is satisfied at all unstable equilibrium points. Note
that similar behaviour was also observed for counter-rotating waves (negative
$l$).

Thus, we observe no bound violation throughout the scan of the parameter
space. Similar observations related to the non-rotating acoustic black hole
have also been reported earlier in the literature [35] where they have
observed the saturation of the bound i.e. $\lambda=2\pi T$. However, in the
slow-rotating ABH the chaos bound is not saturated even at the horizon $r_{1}$
which can be seen from using eq (58) and eq (17).

Table 2:  Positions of the circular orbits of the particle around non-extreme LTABH at different values of $l$ and $a$ at $\xi=6$. The event horizon is located at $r_{1}=0.94641$. | l | 5 | 10 | 20 | 30
---|---|---|---|---|---
| a=0.02 | 1.59189 | 1.54205 | 1.53060 | 1.52851
$r_{0}$ | a=0.04 | 1.59894 | 1.54851 | 1.53692 | 1.53481
| a=0.06 | 1.60598 | 1.55494 | 1.54322 | 1.54108
| a= 0.08 | 1.61298 | 1.56135 | 1.54949 | 1.54733

Table 3:  Positions of the circular orbits of the particle around non-extreme LTABH at different values of $\xi$ and $a$. The corresponding event horizon is located at $r_{1}$ for $l=12$. | $\xi$ | 6 | 10 | 12
---|---|---|---|---
$r_{1}$ |  | 0.94641 | 1.77460 | 2.17980
| a=0.02 | 1.53734 | 2.80220 | 3.45145
$r_{0}$ | a=0.04 | 1.54374 | 2.80537 | 3.45405
| a= 0.08 | 1.55647 | 2.81169 | 3.45924

Table 4:  Positions of the circular orbits of the particle around non-extreme LTABH at different values of $\xi$ and $l$. The corresponding event horizon is located at $r_{1}$ for $a=0.02$. | $\xi$ | 6 | 10 | 12 |
---|---|---|---|---|---
$r_{1}$ |  | 0.94641 | 1.77460 | 2.17980 |
| $l=5$ | 1.59188 | 3.15920 | 4.24152 |
$r_{0}$ | $l=10$ | 1.54205 | 2.82836 | 3.50026 |
| $l=20$ | 1.53059 | 2.76589 | 3.38527 |

Figure 7: The top panel shows the plot of $\kappa^{2}-\lambda^{2}$ for
$M=0.1,~{}~{}l=12$ for different values of $\xi$ with fixed $a$. The bottom
panel shows plot of $\kappa^{2}-\lambda^{2}$ for different values of $\xi$ and
$M=0.1$, $a=0.02$ at fixed $l$.

##  V Conclusions

We investigated the dynamics of the vortex in the vicinity of the horizon of
the slow-rotating acoustic black hole. We determined the effective potential
and identified the presence of an inner stable circular orbit. Further, we
discussed the chaotic dynamics of the vortex. We calculate the Lyapunov
exponent using the Jacobian matrix for the equilibrium orbits. It is found
that the Lyapunov exponent $\lambda$ tends to $\kappa$ for unstable
equilibrium points close to the horizon. We have observed explicit instances
where unstable equilibria are present for extremal black hole with $\kappa=0$.
Moreover, we observe that near-extreme slow-rotating ABH, there is a violation
of the bound. The violation of the chaos bound in the near-extremal region is
expected to be influenced by certain thermal properties of the acoustic black
hole, and therefore exploring this connection further would be valuable for
our understanding and demands further investigation.
Acoustic rotating black holes can be created in laboratory settings using
fluids such as Bose-Einstein condensates or other superfluid systems. By
controlling the rotation and properties of the fluid, researchers can simulate
the ISCO for sound waves, which can allow researchers to visualize and
experiment with concepts from general relativity and black hole physics in
accessible ways and which can have broader applications in physics and
engineering. An alternative way of establishing the results would be through
the use of effective potential. In our future work, we aim to extend this
study by establishing the results through rigorous analytical investigations.
Meanwhile, to the best of our knowledge, a similar study has not been
conducted on the real Lens-Thirring black hole, particularly focusing on
chaotic dynamics. One intriguing feature of rotating black holes in general
relativity is the possibility of extracting energy from them via the Penrose
process. We anticipate observing the analogous Penrose process in this
acoustic spacetime as well. Therefore, investigating these aspects in the
future would provide more insights into the relationship between real and
acoustic black holes specifically related to the generic rotating black holes.

###### Acknowledgements.

We would like to thank Kamal L. Panigrahi for valuable comments and
suggestions during the preparations of the manuscript.

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