# Globalization of partial inverse semigroupoids actions on sets

Paulinho Demeneghi111Universidade Federal de Santa Catarina., Felipe Augusto
Tasca222Instituto Federal do Paraná.

###### Abstract

We prove that every partial action of an inverse semigroupoid on a set admits
a universal globalization. Moreover, we show that our construction gives a
reflector from the category of partial actions on the full subcategory of
global actions. Finally, we investigate if the mediating function given by the
universal property of our construction is injective.

##  1 Introduction

The notion of partial action has appeared for the first time in the literature
at [12], in a work developed by Exel in the context of C*-algebras where the
concept of crossed product of a C*-algebra by a partial action of the infinite
cyclic group was introduced. The work of Exel was generalized afterward by
McClanahan in [21] where a formal definition of crossed product of a
C*-algebra by a partial action of a discrete group was presented. Since then,
partial actions of groups were extensively explored and studied in many other
areas.

The notion of partial action of groups on sets was introduced by Exel in [13]
motivated by the great interest in the concept of partial actions of groups on
C*-algebras. Since then, the concept of group partial actions have been
appeared in many contexts.

Global actions naturally induce partial actions via a restriction process and
it is an important problem to know when a partial action can be obtained
through the restriction of a global one. Explicitly, given a partial action of
a group $G$ on a set $X$, one can obtain a new action by restricting the
action of $G$ to a subset $Y\subseteq X$. The resultant action is a partial
action of $G$ on $Y$ which is not, in general, a global action.

On the other side, given a partial action of $G$ on a set $Y$, one may ask if
there exists a global action of $G$ on a set $X$ containing $Y$ such that the
original partial action of $G$ on $Y$ can be obtained by the restriction of
the global action of $G$ on $X$. This question motivates the notion of
globalization of partial actions. A globalization of a partial action $\theta$
of a group $G$ on a set $Y$ is (isomorphic to) a global action
$\tilde{\theta}$ of $G$ on a set $X\supseteq Y$ such that $\theta$ may be
obtained by the restriction of $\tilde{\theta}$ to $Y$.

The problem of the existence of globalizations was initially studied by Abadie
at [1, 2] and later by Kellendonk and Lawson in [17] in an independent way. At
[1], Abadie proved that every partial action of a group on a set admits a
universal globalization. In this same paper, Abadie has also studied
enveloping actions of groups on topological spaces and C*-algebras. At [17],
in addition to the context of sets and topological spaces, Kellendonk and
Lawson also studied partial action on semi-lattices.

Motivated by the problem of the existence of globalizations for partial
actions, many works were developed in many other contexts exploring: partial
actions of groups on objects like topological spaces, rings, C*-algebras;
partial actions of monoids, semigroups, inverse semigroups, Hopf algebras,
ordered groupoids, groupoids and small categories; and even contexts involving
twisted actions. We may cite [3, 4, 5, 6, 10, 11, 15, 18, 19, 22, 23, 24] as
example. Furthermore, we indicate Dokuchaev extensive survey on the subject
[9].

In this paper we are interested in studying the problem of globalization of
inverse semigroupoid partial actions on sets. The notion of semigroupoids
includes both the notions of categories and semigroups. A semigroupoid may be
thought as a set equipped with a partially defined associative operation.
Actually, there are two notions of semigroupoids in the literature and both
notions are obtained by weakening category axioms. The first notion of
semigroupoid is presented by Tilson in [25] in order to treat category ideals
independently of their parent categories. Roughly speaking, a Tilson
semigroupoid is a small category which may fail to have identity arrows. The
second notion is presented by Exel in [14] with the goal of providing a
unified treatment for C*-algebras associated to combinatorial objects. As a
matter of fact, Exel’s notion is more general than the Tilson’s one, since
Tilson semigroupoids comes with an underlying graph structure, peculiar from
categories, which completely characterizes which pairs of morphisms can be
composed, meanwhile Exel semigroupoids comes equipped with a set of composable
pairs which may not admit a compatible underlying graph structure. Roughly
speaking, in Exel semigroupoids it may not be possible to compose morphisms
even though the codomain of the first agrees with the domain of the second.

The inverse semigroupoids framework provides a unified treatment for inverse
semigroups and groupoids and, it turns out that, despite the difference
between the two existing notions of semigroupoids, when the unique pseudo-
inverse axiom is added, every Exel inverse semigroupoid can be realized as a
Tilson inverse semigroupoid in a unique way, as proved by Cordeiro in [8]. So
that, for a better fit in our approach, we have opted by Tilson’s definition.

Partial actions of inverse semigroupoids were introduced by Cordeiro in [7,
8], where an extensive study on semigroupoids is developed. Based on groupoid
partial actions, in this work we present an adaptation of this definition,
which is equivalent to Cordeiro’s definition in this context.

We prove that, just as group partial actions on sets, every inverse
semigroupoid partial action on a set admits a universal globalization. This
result generalizes simultaneously Theorem 4.23 of [1] for group partial
actions on sets, Theorem 6.10 of [16] for inverse semigroups partial actions
on sets and Theorem 4 (Remark 29) of [23] for groupoids partial actions on
sets. Moreover, we prove that our constructions gives a reflector from the
category of partial actions on the full subcategory of global actions.

The paper is structured as follows: in the second section we first present the
definitions of semigroupoid and inverse semigroupoid adopted, as well as
basics facts about these structures that we are going to need. Then we proceed
to define inverse semigroupoid partial actions on sets and introduce the
necessary machinery for the development of this work. In the third section we
fix a partial action $\theta$ of an inverse semigroupoid
$\operatorname{\mathcal{S}}$ on a set $X$ and then construct a globalization
of $\theta$ which we further prove that is a reflector of $\theta$ in the full
subcategory of global actions (Theorem 3.13) and, hence, is universal among
all globalizations of $\theta$ (Theorem 3.14). Finally, we explore some
examples, one of them, showing that the mediating function given by the
universal property of our construction may not be injective. However, we show
that it is injective on some special subsets (Proposition 3.17).

##  2 Inverse Semigroupoids and Partial Actions

In this section we present the necessary machinery involving inverse
semigroupoids and inverse semigroupoid partial actions on sets. For further
references, we recommend [8, 20].

We first present the definition of semigroupoid we shall adopt.

######  Definition 2.1.

A _semigroupoid_ is a quintuple
$(\operatorname{\mathcal{S}},\operatorname{\mathcal{S}}^{(0)},d,c,\mu)$, in
which

  * •

$\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{S}}^{(0)}$ are sets
called _arrow_ and _object_ sets, respectively;

  * •

$d,c:\operatorname{\mathcal{S}}\to\operatorname{\mathcal{S}}^{(0)}$ are maps
called _domain_ and _codomain_ maps, respectively;

  * •

$\mu:\operatorname{\mathcal{S}}^{(2)}\to\operatorname{\mathcal{S}}$ is a map
called _multiplication_ map (denoted by juxtaposition $\mu(s,t)=st$) defined
in the set
$\operatorname{\mathcal{S}}^{(2)}\subseteq\operatorname{\mathcal{S}}\times\operatorname{\mathcal{S}}$
of _composable pairs_ consisting of all pairs
$(s,t)\in\operatorname{\mathcal{S}}\times\operatorname{\mathcal{S}}$ such that
$d(s)=c(t)$.

satisfying

  1. (i)

$d(st)=d(t)$ and $c(st)=c(s)$ for any composable pair
$(s,t)\in\operatorname{\mathcal{S}}^{(2)}$;

  2. (ii)

$(ps)t=p(st)$ whenever $(p,s),(s,t)\in\operatorname{\mathcal{S}}^{(2)}$.

Notice that, if $p,s,t\in\operatorname{\mathcal{S}}$ are elements such that
$(p,s)$ and $(s,t)$ lies in $\operatorname{\mathcal{S}}^{(2)}$, then by axiom
(i) above $(ps,t)$ and $(p,st)$ also lies in
$\operatorname{\mathcal{S}}^{(2)}$, and hence axiom (ii) does make sense.

By an abuse of notation, whenever there is no chance of confusion, we shall
use just $\operatorname{\mathcal{S}}$ to refer to semigroupoid
$(\operatorname{\mathcal{S}},\operatorname{\mathcal{S}}^{(0)},d,c,\mu)$.

Notice that the class of semigroupoids includes semigroups and small
categories. Semigroups are precisely the semigroupoids with a single object
(and not necessarily an identity element) and small categories are precisely
the semigroupoids containing one identity for each object.

To be precise, by an identity element we mean an element
$e\in\operatorname{\mathcal{S}}$ in a semigroupoid
$\operatorname{\mathcal{S}}$ satisfying $es=s$ and $te=t$ for every
$s,t\in\operatorname{\mathcal{S}}$ such that
$(e,s),(t,e)\in\operatorname{\mathcal{S}}^{(2)}$. If necessary, we may say
that $e\in\operatorname{\mathcal{S}}$ is an identity over an object
$u\in\operatorname{\mathcal{S}}^{(0)}$ to emphasize that $e$ is an identity
element such that $d(e)=c(e)=u$.

Another sort of elements which play an essential role in this work are
idempotent elements. By an idempotent element in a semigroupoid
$\operatorname{\mathcal{S}}$, we mean an element
$e\in\operatorname{\mathcal{S}}$ such that
$(e,e)\in\operatorname{\mathcal{S}}^{(2)}$ and $ee=e$. Again, we may emphasize
that $d(e)=c(e)=u$ by saying that $e$ is an idempotent over
$u\in\operatorname{\mathcal{S}}^{(0)}$. We denote by
$E(\operatorname{\mathcal{S}})$ the set of idempotent elements of
$\operatorname{\mathcal{S}}$, which evidently includes the identity elements.

As mentioned before, we are mainly interested in inverse semigroupoids, as
follows.

######  Definition 2.2.

A semigroupoid $\operatorname{\mathcal{S}}$ is an _inverse semigroupoid_ if
for every $s\in\operatorname{\mathcal{S}}$ there exists a unique
$s^{*}\in\operatorname{\mathcal{S}}$, called the _inverse_ of $s$, such that
$(s,s^{*}),(s^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$ and

$ss^{*}s=s\quad\text{ and }\quad s^{*}ss^{*}=s^{*}.$

Just like the class of semigroupoids includes small categories and semigroups,
the class of inverse semigroupoids includes groupoids and inverse semigroups.

We shall see now that many inverse semigroup properties keep working for this
more general setting. For example, usual rules for inverses are preserved:
$(s^{*})^{*}=s$ for every $s\in\operatorname{\mathcal{S}}$ and if
$(s,t)\in\operatorname{\mathcal{S}}^{(2)}$ then
$(t^{*},s^{*})\in\operatorname{\mathcal{S}}^{(2)}$ and $(st)^{*}=t^{*}s^{*}$.
For every element $s\in\operatorname{\mathcal{S}}$ we have $s^{*}s,ss^{*}\in
E(\operatorname{\mathcal{S}})$ and for every $e\in
E(\operatorname{\mathcal{S}})$ we have $e^{*}=e$. Moreover, the idempotent
elements commute in the sense that, if $e,f\in E(\operatorname{\mathcal{S}})$
and $(e,f)\in\operatorname{\mathcal{S}}^{(2)}$, then
$(f,e)\in\operatorname{\mathcal{S}}^{(2)}$ and $ef=fe$. This implies that for
every $s\in\operatorname{\mathcal{S}}$ and $e\in
E(\operatorname{\mathcal{S}})$ such that
$(s,e)\in\operatorname{\mathcal{S}}^{(2)}$ we have $ses^{*}\in
E(\operatorname{\mathcal{S}})$.

The commutativity of $E(\operatorname{\mathcal{S}})$ enables us to define a
canonical order relation on any inverse semigroupoid
$\operatorname{\mathcal{S}}$ just like in the inverse semigroup case. More
precisely, if $s,t\in\operatorname{\mathcal{S}}$, we say that $s\leq t$ if
$d(s)=d(t)$, $c(s)=c(t)$ and one of the following equivalent statements holds:

  1. (i)

$s=ts^{*}s$;

  2. (ii)

$s=te$ for some $e\in E(\mathcal{S})$ such that
$(t,e)\in\operatorname{\mathcal{S}}^{(2)}$;

  3. (iii)

$s=ss^{*}t$;

  4. (iv)

$s=ft$ for some $f\in E(\mathcal{S})$ such that
$(f,t)\in\operatorname{\mathcal{S}}^{(2)}$.

Notice that, if $s,t\in\operatorname{\mathcal{S}}$, then $s\leq t$ if and only
if $s^{*}\leq t^{*}$ and that, if
$s_{1},s_{2},t_{1},t_{2}\in\operatorname{\mathcal{S}}$ are such that
$(s_{1},s_{2})\in\operatorname{\mathcal{S}}^{(2)}$, $s_{1}\leq t_{1}$ and
$s_{2}\leq t_{2}$, then $(t_{1},t_{2})\in\operatorname{\mathcal{S}}^{(2)}$ and
$s_{1}s_{2}\leq t_{1}t_{2}$. Moreover, it is immediate from the definition
that, if $e,f\in E(\operatorname{\mathcal{S}})$ and
$(e,f)\in\operatorname{\mathcal{S}}^{(2)}$, than $ef\leq e,f$. As a
consequence of the latter, $ses^{*}\leq ss^{*}$ whenever
$s\in\operatorname{\mathcal{S}}$, $e\in E(\operatorname{\mathcal{S}})$ and
$(s,e)\in\operatorname{\mathcal{S}}^{(2)}$.

Before we proceed to the definition of partial action, we present a inverse
semigroupoid example which is not an inverse semigroup, nor a groupoid (not
even a category).

######  Example 2.3.

As a set, $\operatorname{\mathcal{S}}$ is given by
$\operatorname{\mathcal{S}}=\\{a,a^{*},b,b^{*},a^{*}a,aa^{*},b^{*}b,bb^{*}\\}$.
The object set is a two element set and the graph structure may be viewed in
the following picture.

$a$$a^{*}$$a^{*}a$$b$$b^{*}$$b^{*}b$$bb^{*}$$aa^{*}$

Finally, the multiplication table is given by:

$\cdot$ |  $a$ |  $a^{*}$ |  $b$ |  $b^{*}$ |  $a^{*}a$ |  $aa^{*}$ |  $b^{*}b$ |  $bb^{*}$
---|---|---|---|---|---|---|---|---
$a$ |  $-$ |  $aa^{*}$ |  $a$ |  $a$ |  $a$ |  $-$ |  $a$ |  $a$
$a^{*}$ |  $a^{*}a$ |  $-$ |  $-$ |  $-$ |  $-$ |  $a^{*}$ |  $-$ |  $-$
$b$ |  $-$ |  $a^{*}$ |  $a^{*}a$ |  $bb^{*}$ |  $a^{*}a$ |  $-$ |  $b$ |  $a^{*}a$
$b^{*}$ |  $-$ |  $a^{*}$ |  $b^{*}b$ |  $a^{*}a$ |  $a^{*}a$ |  $-$ |  $a^{*}a$ |  $b^{*}$
$a^{*}a$ |  $-$ |  $a^{*}$ |  $a^{*}a$ |  $a^{*}a$ |  $a^{*}a$ |  $-$ |  $a^{*}a$ |  $a^{*}a$
$aa^{*}$ |  $a$ |  $-$ |  $-$ |  $-$ |  $-$ |  $aa^{*}$ |  $-$ |  $-$
$b^{*}b$ |  $-$ |  $a^{*}$ |  $a^{*}a$ |  $b^{*}$ |  $a^{*}a$ |  $-$ |  $b^{*}b$ |  $a^{*}a$
$bb^{*}$ |  $-$ |  $a^{*}$ |  $b$ |  $a^{*}a$ |  $a^{*}a$ |  $-$ |  $a^{*}a$ |  $bb^{*}$

in which the symbol “$-$” means that the product is undefined.

Notice that, the idempotent set is given by
$E(\operatorname{\mathcal{S}})=\\{a^{*}a,aa^{*},b^{*}b,bb^{*}\\}$.

We now proceed to present the definition of a partial action of an inverse
semigroupoid on a set.

######  Definition 2.4.

A _partial action_ $\theta$ of an inverse semigroupoid $\mathcal{S}$ on a set
$X$ is a pair

$\theta=\big{(}\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}}\big{)}$

consisting of a collection $\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}}$ of
subsets of $X$ and a collection
$\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}}$ of maps

$\theta_{s}\colon X_{s^{*}}\to X_{s}$

such that

  1. (P1)

$\theta_{e}=\operatorname{id}_{X_{e}}$ for all $e\in
E(\operatorname{\mathcal{S}})$. Moreover, for all $x\in X$, there exists $e\in
E(\operatorname{\mathcal{S}})$ such that $x\in X_{e}$;

  2. (P2)

$X_{s}\subseteq X_{ss^{*}}$ for every $s\in\operatorname{\mathcal{S}}$;

  3. (P3)

$\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})=X_{(st)^{*}}\cap X_{t^{*}}$ for all
$(s,t)\in\mathcal{S}^{(2)}$ and, moreover,
$\theta_{s}\big{(}\theta_{t}(x)\big{)}=\theta_{st}(x)$ for all $x\in
X_{(st)^{*}}\cap X_{t^{*}}$.

We say that $\theta$ is a _global action_ if moreover it satisfies:

  1. (P4)

$X_{s}=X_{ss^{*}}$ for every $s\in\operatorname{\mathcal{S}}$ (i.e., equality
holds in (P2)).

By an $\operatorname{\mathcal{S}}$-_set_ we shall mean a pair $(X,\theta)$ in
which $X$ is a set, $\operatorname{\mathcal{S}}$ is an inverse semigroupoid
and $\theta$ is a partial action of $\operatorname{\mathcal{S}}$ on $X$. We
shall, moreover, say that $(X,\theta)$ is a _global_
$\operatorname{\mathcal{S}}$-_set_ if $\theta$ is a global action.

We should also point out that, for an $\operatorname{\mathcal{S}}$-_set_
$(X,\theta)$, unless explicitly said otherwise, we will use the same symbols
$X$ and $\theta$, indexed by elements of $\operatorname{\mathcal{S}}$, to
denote the members of the family of subsets of $X$ and the members of the
family of maps that constitute $\theta$, respectively. That is, we will
implicitly assume that $\theta$ is given by
$(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$.

Before we proceed, let us point some observations about the last definition.

######  Remark 2.5.

First, notice that $\bigcup_{e\in E(\operatorname{\mathcal{S}})}X_{e}=X$ by
(P1). This is not a very restrictive requirement since, in its absence, we may
replace $X$ by the union of the domains $X_{e}$, leading to an action of
$\operatorname{\mathcal{S}}$ on $\bigcup_{e\in
E(\operatorname{\mathcal{S}})}X_{e}$ satisfying all the desired properties.
Actually, some authors refer to partial actions with this extra requirement as
_non-degenerate_ partial actions. We have chosen to not follow this approach
and ask for it directly in the definition.

Replacing $s$ by $s^{*}$ in (P2), we deduce that the domain $X_{s^{*}}$ of
$\theta_{s}$ is contained in the domain $X_{s^{*}s}$ of $\theta_{s^{*}s}$ and,
moreover, by (P4), $\theta_{s}$ and $\theta_{s^{*}s}$ share domains if the
action is global.

Given $s,t\in\operatorname{\mathcal{S}}$, we may not be able to compose
$\theta_{s}$ and $\theta_{t}$ in the traditional way, since the image of
$\theta_{t}$ may not be contained in the domain of $\theta_{s}$. However, we
will use the notation $\theta_{s}\circ\theta_{t}$ to refer to the map defined
on the largest domain in which the expression $\theta_{s}(\theta_{t}(x))$ does
make sense. Explicitly, the domain and codomain of $\theta_{s}\circ\theta_{t}$
are considered to be $\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})$ and
$\theta_{s}(X_{s^{*}}\cap X_{t})$, respectively.

Hence, under the convention explained in the last paragraph, we can deduce
from (P3) that, whenever $(s,t)\in\operatorname{\mathcal{S}}^{(2)}$, the
domain $\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})$ of $\theta_{s}\circ\theta_{t}$
is contained in the domain $X_{(st)^{*}}$ of $\theta_{st}$ and
$\theta_{s}\circ\theta_{t}$ coincides with $\theta_{st}$ in the domain of
$\theta_{s}\circ\theta_{t}$. In other words, from (P3) we deduce that
$\theta_{st}$ is an extension333We shall use the term extension even though
the codomain of the functions involved may not agree with each other. of
$\theta_{s}\circ\theta_{t}$ for $(s,t)\in\operatorname{\mathcal{S}}^{(2)}$,
which we shall denote by $\theta_{s}\circ\theta_{t}\subseteq\theta_{st}$. That
is, we shall use the symbol “$\subseteq$” to express that the function on the
right-hand side is an extension of the one in the left-hand side.

######  Proposition 2.6.

Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and $(X,\theta)$
an $\operatorname{\mathcal{S}}$-set. For every
$s\in\operatorname{\mathcal{S}}$, $\theta_{s}$ is a bijection from $X_{s^{*}}$
onto $X_{s}$ and, moreover, $\theta_{s}^{-1}=\theta_{s^{*}}$.

###### Proof.

Let $s\in\operatorname{\mathcal{S}}$. By (P3) we deduce that $\theta_{s^{*}s}$
is an extension of $\theta_{s^{*}}\circ\theta_{s}$. By (P1), it follows that
$\theta_{s^{*}s}$ is the identity map on $X_{s^{*}s}$. Hence,
$\theta_{s^{*}}\circ\theta_{s}$ is the identity map on its domain, which is
clearly $X_{s^{*}}$. Similarly, $\theta_{s}\circ\theta_{s^{*}}$ is the
identity map on $X_{s}$. ∎

With this result in hands we can rewrite the domain $\theta_{t}^{-1}(X_{t}\cap
X_{s^{*}})$ of $\theta_{s}\circ\theta_{t}$ as $\theta_{t^{*}}(X_{t}\cap
X_{s^{*}})$. This enables us to obtain a characterization for the codomain
$\theta_{s}(X_{s^{*}}\cap X_{t})$ of $\theta_{s}\circ\theta_{t}$ in the same
way (P3) provides one for the domain of $\theta_{s}\circ\theta_{t}$.

######  Proposition 2.7.

Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and $(X,\theta)$
an $\operatorname{\mathcal{S}}$-set. If
$(s,t)\in\operatorname{\mathcal{S}}^{(2)}$, then

$\theta_{s}(X_{s^{*}}\cap X_{t})=X_{s}\cap X_{st}.$

###### Proof.

Let $(s,t)\in\operatorname{\mathcal{S}}^{(2)}$. Hence,
$(t^{*},s^{*})\in\operatorname{\mathcal{S}}^{(2)}$ and, by (P3),
$\theta_{s^{*}}^{-1}(X_{s^{*}}\cap X_{t})=X_{st}\cap X_{s}$. The result now
follows from the comment immediately before the statement. ∎

We now compare $\theta_{s}$ and $\theta_{t}$ when $s\leq t$.

######  Proposition 2.8.

Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and $(X,\theta)$
an $\operatorname{\mathcal{S}}$-set. If $s\leq t$, then $X_{s}\subseteq X_{t}$
and $\theta_{s}\subseteq\theta_{t}$.

###### Proof.

Let $s,t\in\operatorname{\mathcal{S}}$ such that $s\leq t$. Since $s\leq t$,
we have that $s=ss^{*}t$. Let $x\in X_{s}$. First, we shall prove
$X_{s}\subseteq X_{t}$. By (P2), $x\in X_{ss^{*}}$. Hence, by 2.7 and (P1), we
have

$x\in X_{ss^{*}}\cap X_{s}=X_{ss^{*}}\cap
X_{ss^{*}t}\overset{\ref{ran_composition}}{=}\theta_{ss^{*}}(X_{ss^{*}}\cap
X_{t})\overset{\ref{p1}}{=}X_{ss^{*}}\cap X_{t}\subseteq X_{t}.$

For the extension, we also have $s^{*}\leq t^{*}$ and, analogously to the
previous argument, we have $X_{s^{*}}\subseteq X_{t^{*}}$. Then, for every
$x\in X_{s^{*}}$, by (P1) and (P3), we have that $x$ lies in the domain
$X_{s^{*}}\cap X_{t^{*}}$ of $\theta_{ss^{*}}\circ\theta_{t}$ and

$\theta_{s}(x)\overset{\ref{p3}}{=}\theta_{ss^{*}}\big{(}\theta_{t}(x)\big{)}\overset{\ref{p1}}{=}\theta_{t}(x).$

Hence, $\theta_{t}$ is an extension of $\theta_{s}$ as stated. ∎

If $e,f$ are composable idempotent elements, the next result shows that
$\theta_{e}\circ\theta_{f}$ and $\theta_{ef}$ share domains and, hence,
coincide.

######  Proposition 2.9.

Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and $(X,\theta)$
an $\operatorname{\mathcal{S}}$-set. If $e,f\in E(\operatorname{\mathcal{S}})$
and $(e,f)\in\operatorname{\mathcal{S}}^{(2)}$, then $X_{ef}=X_{e}\cap X_{f}$
and $\theta_{ef}=\theta_{e}\circ\theta_{f}$.

###### Proof.

Let $e,f\in E(\operatorname{\mathcal{S}})$ such that $(e,f)\in S^{(2)}$.
Since, $ef\leq e$, by 2.8, we have $X_{ef}\subseteq X_{e}$. Hence, by (P1) and
2.7, we have

$X_{ef}=X_{e}\cap X_{ef}\overset{\ref{ran_composition}}{=}\theta_{e}(X_{e}\cap
X_{f})\overset{\ref{p1}}{=}X_{e}\cap X_{f}.$

Since the domains (and codomains) of $\theta_{ef}$ and
$\theta_{e}\circ\theta_{f}$ coincide, we must have
$\theta_{ef}=\theta_{e}\circ\theta_{f}$, as desired. ∎

We now present one alternative for the definition of partial action. The
reader is invited to compare the next proposition with Definition 1.7 [7].

######  Proposition 2.10.

Let $X$ be a set and $\operatorname{\mathcal{S}}$ an inverse semigroupoid. Let
$\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}}$ be a collection of subsets of
$X$ and $\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}}$ a collection of
maps $\theta_{s}\colon X_{s^{*}}\to X_{s}$. Then, the pair
$\theta=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
is a partial action of $\operatorname{\mathcal{S}}$ on $X$ if, and only if, it
satisfies

  1. (E1)

For every $s\in\mathcal{S}$, $\theta_{s}$ is a bijection and
$\theta_{s}^{-1}=\theta_{s^{*}}$. Moreover,
$X=\cup_{s\in\operatorname{\mathcal{S}}}X_{s}$;

  2. (E2)

$\theta_{s}\circ\theta_{t}\subseteq\theta_{st}$ for every
$s,t\in\operatorname{\mathcal{S}}$ such that $(s,t)\in\mathcal{S}^{(2)}$;

  3. (E3)

$X_{s}\subseteq X_{t}$ for every $s,t\in\operatorname{\mathcal{S}}$ such that
$s\leq t$.

Furthermore, if $\theta$ is a partial action of $\operatorname{\mathcal{S}}$
on $X$, then it is a _global action_ if, and only if

  1. (E4)

$\theta_{st}=\theta_{s}\circ\theta_{t}$ for every
$s,t\in\operatorname{\mathcal{S}}$ such that $(s,t)\in\mathcal{S}^{(2)}$ (i.e.
equality holds in (E2)).

###### Proof.

If $\theta$ is a partial action, the first part of (E1) follows from
Proposition 2.6 and the second part follows immediately from the second part
of (P1). From the comments after Definition 2.4 we deduce (E2), and (E3) is a
direct consequence of Proposition 2.8.

Assume now that $\theta$ satisfies (E1), (E2) and (E3). Our goal is to prove
that $\theta$ is a partial action of $\operatorname{\mathcal{S}}$ on $X$.

By (E1) and (E2) we deduce (P2) since

$\operatorname{id}_{X_{s}}\overset{\ref{e1}}{=}\theta_{s}\circ\theta_{s^{*}}\overset{\ref{e2}}{\subseteq}\theta_{ss^{*}},$

and, hence, the domain $X_{s}$ of $\operatorname{id}_{X_{s}}$ is contained in
the domain $X_{ss^{*}}$ of $\theta_{ss^{*}}$. This also enables us to infer
the second part of (P1) since, by (E1), for any $x\in X$, there exists
$s\in\operatorname{\mathcal{S}}$ such that $x\in X_{s}$. Hence, $x\in
X_{s}\subseteq X_{ss^{*}}$ and $ss^{*}$ is an idempotent.

For the first part of (P1) we use again (E1) and (E2). In one hand we have

$\operatorname{id}_{X_{e}}\overset{\ref{e1}}{=}\theta_{e}\circ\theta_{e}\overset{\ref{e2}}{\subseteq}\theta_{e}$

and, since $\operatorname{id}_{X_{e}}$ and $\theta_{e}$ share domains, we must
have $\theta_{e}=\operatorname{id}_{X_{e}}$.

It remains to argue (P3). For that task, let
$(s,t)\in\operatorname{\mathcal{S}}^{(2)}$. By (E2),
$\theta_{s}\circ\theta_{t}\subseteq\theta_{st}$ and, hence, the domain
$\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})$ of $\theta_{s}\circ\theta_{t}$ is
contained in the domain $X_{(st)^{*}}$ of $\theta_{st}$. Combining this with
the fact that the domain of $\theta_{t}$ is $X_{t^{*}}$ we obtain

$\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})\subseteq X_{(st)^{*}}\cap X_{t^{*}}.$
(1)

To deduce the reverse inclusion, notice that we must also have
$(st,t^{*})\in\operatorname{\mathcal{S}}^{(2)}$ and using (E2) and a similar
argument we obtain

$\theta_{t^{*}}^{-1}(X_{t^{*}}\cap X_{(st)^{*}})\subseteq
X_{(stt^{*})^{*}}\cap X_{t}\overset{\ref{e3}}{\subseteq}X_{s^{*}}\cap X_{t},$
(2)

in which the last inclusion follows from (E3), since
$(stt^{*})^{*}=tt^{*}s^{*}\leq s^{*}$. Combining the fact that $X_{s^{*}}\cap
X_{t}$ is contained in the domain of $\theta_{t^{*}}$ and (E1), we obtain from
(2) that

$X_{t^{*}}\cap X_{(st)^{*}}\subseteq\theta_{t}^{-1}(X_{s^{*}}\cap X_{t}).$ (3)

Joining (1) and (3) we obtain the desired equality. Finally, if $x\in
X_{(st)^{*}}\cap X_{t^{*}}$, then $x$ lies in the domain
$\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})$ of $\theta_{s}\circ\theta_{t}$ and,
hence, by (E2), we have that

$\theta_{s}\big{(}\theta_{t}(x)\big{)}=\theta_{st}(x),$

completing the verification of (P3).

Suppose now that $\theta$ is a global action of $\operatorname{\mathcal{S}}$
on $X$. Hence, by (P4) and Proposition 2.9, we have

$X_{(st)^{*}}\cap X_{t^{*}}\overset{\ref{pglob}}{=}X_{(st)^{*}(st)}\cap
X_{t^{*}t}\overset{\ref{ef_domain}}{=}X_{(st)^{*}(st)t^{*}t}=X_{(st)^{*}(st)}=X_{(st)^{*}},$

which means that the domain of $\theta_{s}\circ\theta_{t}$ coincides with the
domain of $\theta_{st}$ and, hence, concluding (E4).

Conversely, if $\theta$ is partial action that satisfies (E4), then for any
$s\in\operatorname{\mathcal{S}}$ we would have
$\theta_{ss^{*}}=\theta_{s}\circ\theta_{s^{*}}=\operatorname{id}_{X_{s}}$ and,
consequently, $X_{ss^{*}}=X_{s}$ concluding (P4). ∎

We end this section with two examples of partial actions that we shall explore
again in the next section.

######  Example 2.11.

Let $X=\\{1,2,3,4\\}$ and let
$\operatorname{\mathcal{S}}=\\{a,a^{*},b,b^{*},a^{*}a,aa^{*},b^{*}b,bb^{*}\\}$
be the inverse semigroupoid presented in Example 2.3. Consider the following
family of subsets of $X$:

$X_{b^{*}}=\\{1,2\\}$,

$X_{b^{*}b}=\\{1,2\\}$,

$X_{b}=\\{1,4\\}$,

$X_{bb^{*}}=\\{1,4\\}$,

$X_{a^{*}}=\\{1\\}$

$X_{a^{*}a}=\\{1\\}$,

$X_{a}=\\{3\\}$,

$X_{aa^{*}}=\\{3,4\\}$,

and the following family of bijections:

$\begin{array}[]{cccc}\theta_{b}\colon&X_{b^{*}}&\rightarrow&X_{b}\\\
&1&\mapsto&1\\\
&2&\mapsto&4\end{array},\qquad\begin{array}[]{cccc}\theta_{a}\colon&X_{a^{*}}&\rightarrow&X_{a}\\\
&1&\mapsto&4\\\ &&&\end{array},$

$\theta_{a^{*}}=\theta_{a}^{-1}$, $\theta_{b^{*}}=\theta_{b}^{-1}$ and
$\theta_{e}=\operatorname{id}_{X_{e}}$ for $e\in
E(\operatorname{\mathcal{S}})=\\{b^{*}b,bb^{*},a^{*}a,aa^{*}\\}$. It is
routine to check that
$\theta=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
is, indeed, a partial action of $\operatorname{\mathcal{S}}$ on $X$.

######  Example 2.12.

Let $Y=\\{1,2,3\\}$ and let
$\operatorname{\mathcal{S}}=\\{a,a^{*},b,b^{*},a^{*}a,aa^{*},b^{*}b,bb^{*}\\}$
be the inverse semigroupoid presented in Example 2.3. Consider the following
family of bijections:

$\begin{array}[]{cccc}\theta^{Y}_{a}\colon&Y&\rightarrow&Y\\\ &1&\mapsto&2\\\
&2&\mapsto&3\\\ &3&\mapsto&1\end{array},$

$\theta^{Y}_{a^{*}}={\theta^{Y}_{a}}^{-1}$ and
$\theta^{Y}_{s}=\operatorname{id}_{Y}$ if $s\neq a,a^{*}$. Setting $Y_{s}=Y$
for every $s\in\operatorname{\mathcal{S}}$, it is routine to check that
$\theta^{Y}=(\\{Y_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta^{Y}_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
is a global action of $\operatorname{\mathcal{S}}$ on $Y$.

##  3 Restriction and Globalization of partial actions

It is well known that we can produce interesting examples of partial actions
via a process of restriction of global actions. Let us explain this process in
the present context: suppose
$\theta=(\\{Y_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
is a partial action of an inverse semigroupoid $\operatorname{\mathcal{S}}$ on
a set $Y$. If $X$ is a subset of $Y$, we may try to restrict the action of
$\operatorname{\mathcal{S}}$ to $X$ by restricting the domain of each
$\theta_{s}$ to $X\cap Y_{s^{*}}$. The main problem in this process is that,
even though $x\in X\cap Y_{s^{*}}$, we may have $\theta_{s}(x)\notin X$.

However, we can turn around this problem by restricting each $\theta_{s}$ to
the set formed by all $x\in X\cap Y_{s^{*}}$ such that $\theta_{s}(x)$ also
lies in $X$. Explicitly, setting $X_{s}=\theta_{s}(X\cap Y_{s^{*}})\cap X$ and
noticing that $\theta_{s}(X_{s^{*}})=X_{s}$, we may let

${(\left.\theta\right|_{X})}_{s}\colon X_{s^{*}}\to X_{s}$

to be the restriction of $\theta_{s}$ to $X_{s^{*}}$ onto $X_{s}$. It is then
easy to verify that

$\left.\theta\right|_{X}=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{{(\left.\theta\right|_{X})}_{s}\\}_{s\in\operatorname{\mathcal{S}}})$

satisfies the axioms of Definition 2.4 and, hence, is a partial action of
$\operatorname{\mathcal{S}}$ on $X$ which, from now on, we shall call the
_restriction_ of the action $\theta$ to $X$.

Let us point out that, even if the original action $\theta$ of
$\operatorname{\mathcal{S}}$ on $Y$ is global, there is no guarantee that the
restriction $\left.\theta\right|_{X}$ is a global action. Indeed, for any
$x\in X\cap Y_{s^{*}}$ such that $\theta_{s}(x)$ does not lie in $X$, we would
have $x\in X_{s^{*}s}$, but $x\notin X_{s^{*}}$.

The observation in the last paragraph raises an interesting problem: if we
start with a partial action $\theta$ of an inverse semigroupoid
$\operatorname{\mathcal{S}}$ on a set $X$, does there exist a set $Y$
containing $X$ and a global action $\eta$ of $\operatorname{\mathcal{S}}$ on
$Y$ such that $\theta$ may be obtained by the restriction of the action of
$\eta$ to $X$? The main goal of this work is to give an affirmative answer for
this question.

Before we proceed, let us introduce the appropriate morphisms to form the
category of partial actions.

######  Definition 3.1.

Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and let
$\theta^{X}=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}^{X}\\}_{s\in\operatorname{\mathcal{S}}})$
and
$\theta^{Y}=(\\{Y_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}^{Y}\\}_{s\in\operatorname{\mathcal{S}}})$
be partial actions of $\operatorname{\mathcal{S}}$ on sets $X$ and $Y$,
respectively. A function $\varphi\colon X\rightarrow Y$ is said to be an
$\operatorname{\mathcal{S}}$_-function_ if $\varphi(X_{s})\subseteq Y_{s}$ for
every $s\in\operatorname{\mathcal{S}}$ and, moreover,
$\varphi\big{(}\theta_{s}^{X}(x)\big{)}=\theta_{s}^{Y}\big{(}\varphi(x)\big{)}$
for every $x\in X_{s^{*}}$.

Given an inverse semigroupoid $\operatorname{\mathcal{S}}$, the class of
$\operatorname{\mathcal{S}}$-sets together with the class of
$\operatorname{\mathcal{S}}$-functions form a category, which we denote by
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$. Likewise, the class of global
$\operatorname{\mathcal{S}}$-sets together with the class of
$\operatorname{\mathcal{S}}$-functions between then form a full subcategory of
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$, which we denote by
$\mathscr{A}(\operatorname{\mathcal{S}})$.

For example, if $(Y,\theta)$ is an $\operatorname{\mathcal{S}}$-set and
$X\subseteq Y$, it is easy to verify that the inclusion map $i\colon X\to Y$
is an $\operatorname{\mathcal{S}}$-function between the
$\operatorname{\mathcal{S}}$-sets $(X,\left.\theta\right|_{X})$ and
$(Y,\theta)$. Actually, $i$ is even more than an injective
$\operatorname{\mathcal{S}}$-function, it is an embedding in
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$ as follows.

######  Definition 3.2.

Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and let
$\theta^{X}=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}^{X}\\}_{s\in\operatorname{\mathcal{S}}})$
and
$\theta^{Y}=(\\{Y_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}^{Y}\\}_{s\in\operatorname{\mathcal{S}}})$
be partial actions of $\operatorname{\mathcal{S}}$ on sets $X$ and $Y$,
respectively. An injective $\operatorname{\mathcal{S}}$-function
$\varphi\colon X\rightarrow Y$ is said to be an _embedding_ if

$X_{s}=\varphi^{-1}\big{(}\theta_{s}^{Y}(\varphi(X)\cap Y_{s^{*}})\big{)}$

for every $s\in\operatorname{\mathcal{S}}$.

Notice that, if $\varphi\colon X\rightarrow Y$ is an embedding as in the
Definition above, then the action $\theta^{Y}$ of $\operatorname{\mathcal{S}}$
on $Y$ induces the action $\theta^{X}$ of $\operatorname{\mathcal{S}}$ on $X$
in the sense that, for given $x_{1},x_{2}\in X$ and
$s\in\operatorname{\mathcal{S}}$ such that $\varphi(x_{1})\in Y_{s^{*}}$ and
$\theta_{s}(\varphi(x_{1}))=\varphi(x_{2})$, then $x_{1}\in X_{s^{*}}$ and
$\theta_{s}(x_{1})=x_{2}$.

Actually, in order to an injective function $\varphi$ as in Definition 3.2 be
an embedding is necessary, and sufficient that, for given $x\in X$ and
$s\in\operatorname{\mathcal{S}}$, we have that $x\in X_{s^{*}}$ if, and only
if, $\varphi(x)\in Y_{s^{*}}$ and $\theta^{Y}_{s}(\varphi(x))\in\varphi(X)$
and, in the affirmative case,
$\theta^{Y}_{s}(\varphi(x))=\varphi(\theta_{s}^{X}(x))$. This amounts to say
that $\varphi$ co-restricts to an isomorphism $\varphi\colon X\to\varphi(X)$
between $(X,\theta^{X})$ and
$(\varphi(X),\left.\theta^{Y}\right|_{\varphi(X)})$ in
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$.

This discussion gives the appropriate definition for globalizations of partial
actions.

######  Definition 3.3.

Let $(X,\theta^{X})$ and $(Y,\theta^{Y})$ be $\operatorname{\mathcal{S}}$-sets
and $\varphi\colon X\to Y$ a $\operatorname{\mathcal{S}}$-function. The triple
$(\varphi,Y,\theta^{Y})$ is said to be

  1. (i)

a _globalization_ of $(X,\theta^{X})$ if $\varphi$ is an embedding and
$(Y,\theta^{Y})$ is a global $\operatorname{\mathcal{S}}$-set.

  2. (ii)

an _universal globalization_ of $(X,\theta^{X})$ if it is a globalization of
$(X,\theta^{X})$ and, for any globalization $(\psi,Z,\theta^{Z})$ of
$(X,\theta^{X})$, there exists a unique _mediating_
$\operatorname{\mathcal{S}}$-function $\sigma\colon Y\to Z$ such that the
diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{Z}$

commutes.

In the context of Definition 3.3 above, if the
$\operatorname{\mathcal{S}}$-function $\varphi$ is implicit in the context, we
may say simply that $(Y,\theta^{Y})$ is a globalization of $(X,\theta^{X})$.

We shall prove that every partial action admits a universal globalization.
Moreover, we shall prove that our construction gives a reflector from
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$ on
$\mathscr{A}(\operatorname{\mathcal{S}})$ (Theorem 3.13), whose definition we
recall below.

######  Definition 3.4.

A full subcategory $\mathcal{D}$ of a category $\mathcal{C}$ is said to be
_reflective_ if, for any $\mathcal{C}$-object $C$, there exists a
$\mathcal{D}$-object $R(C)$ and a $\mathcal{C}$-morphism $\varphi_{C}\colon
C\to R(C)$ such that, for every $\mathcal{D}$-object $D$ and
$\mathcal{C}$-morphism $\psi\colon C\to D$, there exists a unique mediating
$\mathcal{D}$-morphism $\sigma\colon R(C)\to D$ such that the diagram

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{C}}$$\scriptstyle{\psi}$$\textstyle{R(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{D}$

commutes. Moreover, in this case, the pair $(\varphi_{C},R(C))$ is said to be
a $\mathcal{D}$-_reflector_ of $C$.

In the context of Definition 3.4 above, if the morphism $\varphi_{C}$ is
implicit in the context, we may say simply that $R(C)$ is a
$\mathcal{D}$-reflector of $C$.

If $\mathcal{D}$ is a reflective subcategory of a category $\mathcal{C}$, then
the map $C\mapsto R(C)$ induces a _reflector_ functor
$R\colon\mathcal{C}\to\mathcal{D}$ which is right adjoint to the inclusion
functor $\mathcal{D}\to\mathcal{C}$.

From now on we fix an inverse semigroupoid $\operatorname{\mathcal{S}}$, a set
$X$ and a partial action

$\theta=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}}).$

Our goal now is to construct a global $\operatorname{\mathcal{S}}$-set
$(E,\eta)$ and an $\operatorname{\mathcal{S}}$-function $i\colon X\to E$ such
that $(i,E,\eta)$ is a globalization of $(X,\theta)$ which is a
$\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$. Notice
that, a globalization of $(X,\theta)$ which is also a
$\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$ is,
automatically, a universal globalization of $(X,\theta)$.

For reference, our construction is mainly inspired in [23]. We begin defining
a relation on a subset of $\operatorname{\mathcal{S}}\times X$ as follows.

######  Definition 3.5.

Let $D:=\\{(s,x)\in\operatorname{\mathcal{S}}\times X:x\in X_{s^{*}s}\\}$.
Define a relation $\sim$ on $D$ such that, for all $(s,x),(t,y)\in D$,
$(s,x)\sim(t,y)$ if either

  1. (R1)

$(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$ and
$\theta_{t^{*}s}(x)=y$;

  2. (R2)

or $s,t\in E(\operatorname{\mathcal{S}})$ and $x=y$.

Now, we point some observations about the relation $\sim$ defined above.

######  Remark 3.6.

First, in the case in which item (R2) is checked, we have $x\in X_{s}$ and
$y\in X_{t}$ and $\theta_{s}(x)=x=y=\theta_{t}(y)$, by (P1).

Still in the case in which item (R2) is checked, if $s$ and $t$ are
composable, then by Proposition 2.9, $X_{st}=X_{s}\cap X_{t}$ and, hence,
$(t,s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{st}$ and
$\theta_{st}(x)=x$ by (P1). This means that $(s,x)\sim(t,y)$ is also checked
by item (R1). In other words, item (R2) in Definition 3.5 plays a relevant
role only in the case $s$ and $t$ are idempotent elements over different
objects.

The set $E$ that we are looking for is going to be the quotient of $D$ by an
equivalence relation which identifies pairs satisfying (R1) or (R2).
Unfortunately, the relation $\sim$ in Definition 3.5 may fail to be an
equivalence relation. Indeed, in Example 2.11, we may notice that
$(a,1)\sim(aa^{*},4)$, $(aa^{*},4)\sim(bb^{*},4)$ but $(a,1)\nsim(bb^{*},4)$.
At least, $\sim$ is reflexive and symmetric.

######  Proposition 3.7.

The relation $\sim$ defined in 3.5 is reflexive and symmetric.

###### Proof.

Let $(s,x)\in D$. Since $x\in X_{s^{*}s}$ and $s^{*}s\in
E(\operatorname{\mathcal{S}})$, we have $\theta_{s^{*}s}(x)=x$ by (P1). This
means that $(s,x)\sim(s,x)$ by item (R1) of Definition 3.5. Hence, $\sim$ is
indeed reflexive.

It remains to show that $\sim$ is symmetric. Let $(s,x),(t,y)\in D$ and
suppose that $(s,x)\sim(t,y)$. If $x=y$ and $s,t\in
E(\operatorname{\mathcal{S}})$ it is clear that $(t,y)\sim(s,x)$. Otherwise,
assume that $(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$
and $\theta_{t^{*}s}(x)=y$. Then, we must have
$(s^{*},t)\in\operatorname{\mathcal{S}}^{(2)}$, $y\in X_{t^{*}s}$ and
$\theta_{s^{*}t}(y)=x$, which means that $(t,y)\sim(s,x)$ by (R2). ∎

Since $\sim$ may not be an equivalence relation, We turn around this problem
considering the equivalence relation generated by $\sim$, as follows.

######  Definition 3.8.

Let $\approx$ be the smallest equivalence relation on $D$ containing the
relation $\sim$ defined in 3.5. Explicitly, if $(s,x),(t,y)\in D$, then
$(s,x)\approx(t,y)$ if, and only if, there exist $n\in\mathbb{N}$ and
$(r_{1},z_{1}),\ldots,(r_{n},z_{n})\in D$ such that $(r_{1},z_{1})=(s,x)$,
$(r_{n},z_{n})=(t,y)$ and $(r_{i},z_{i})\sim(r_{i+1},z_{i+1})$ for all
$i\in\\{1,\ldots,n-1\\}$. We denote by $E$ the quotient of $D$ by the
equivalence relation $\approx$. The equivalence class of an element $(s,x)\in
D$ is denoted by $[s,x]$.

The next step is to define an action of $\operatorname{\mathcal{S}}$ on $E$.
The idea here is that, if an element $p\in\operatorname{\mathcal{S}}$ acts in
the class of an element $(s,x)\in D$ such that
$(p,s)\in\operatorname{\mathcal{S}}^{(2)}$, we should obtain the class of the
element $(ps,x)$. However, in order to $(ps,x)$ be an element of $D$ we must
have $x\in X_{(ps)^{*}(ps)}$. So, we now prove some results with the aiming to
obtain a well defined action of $\operatorname{\mathcal{S}}$ on $E$ in this
fashion.

######  Lemma 3.9.

Let $(s,x)$, $(t,y)\in D$ be such that $(s,x)\sim(t,y)$ and let
$p\in\operatorname{\mathcal{S}}$ be such that
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$. Then $x\in X_{s^{*}p^{*}ps}$
if, and only if, $y\in X_{t^{*}p^{*}pt}$. Moreover, when the memberships are
verified, we have $(ps,x)\sim(pt,y)$.

###### Proof.

Notice, at first, that the membership $x\in X_{(ps)^{*}(ps)}=X_{s^{*}p^{*}ps}$
is necessary (and sufficient) to guarantee that $(ps,x)$ lies in $D$.

Let $(s,x)$, $(t,y)\in D$ be such that $(s,x)\sim(t,y)$ and let
$p\in\operatorname{\mathcal{S}}$ be such that
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$. Notice that, if $s,t\in
E(\operatorname{\mathcal{S}})$, since
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$, we must have
$(s,t),(t,s)\in\operatorname{\mathcal{S}}^{(2)}$. Hence, by the second
paragraph of Remark 2.5, we may assume that $(s,x)\sim(t,y)$ is checked by
item (R1) of Definition 3.5. Then, assume that
$(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$ and
$\theta_{t^{*}s}(x)=y$. By symmetry of $\sim$, we have
$(s^{*},t)\in\operatorname{\mathcal{S}}^{(2)}$, $y\in X_{t^{*}s}$ and
$\theta_{s^{*}t}(y)=x$.

Assume now that $x\in X_{s^{*}p^{*}ps}$ and notice that $y$ must lie in the
domain

$\theta_{s^{*}t}^{-1}\big{(}X_{s^{*}t}\cap\theta_{s^{*}p^{*}ps}^{-1}(X_{s^{*}p^{*}ps}\cap
X_{s^{*}t})\big{)}=\theta_{s^{*}t}^{-1}(X_{s^{*}t}\cap X_{s^{*}p^{*}ps})$

of $\theta_{t^{*}s}\circ\theta_{s^{*}p^{*}ps}\circ\theta_{s^{*}t}$ since $x$
lies in both $X_{s^{*}t}$ and $X_{s^{*}p^{*}ps}$ and $\theta_{s^{*}t}(y)=x$.
Now, observing that

$(t^{*}s)(s^{*}p^{*}ps)(s^{*}t)=t^{*}ss^{*}p^{*}pss^{*}t=t^{*}ss^{*}p^{*}pt\leq
t^{*}p^{*}pt$

and using (E2) and (E3), we have that $y\in X_{t^{*}ss^{*}p^{*}pt}\subseteq
X_{t^{*}p^{*}pt}$ as desired.

Conversely, if $y\in X_{t^{*}p^{*}pt}$, just notice that

$(s^{*}t)(t^{*}p^{*}pt)(t^{*}s)=s^{*}tt^{*}p^{*}ptt^{*}s=s^{*}tt^{*}p^{*}ps\leq
s^{*}p^{*}ps$

and use a similar argument to deduce that $x\in X_{s^{*}p^{*}ps}$.

Finally, we assume that $x\in X_{s^{*}p^{*}ps}$ and $y\in X_{t^{*}p^{*}pt}$
and prove that $(ps,x)\sim(pt,y)$. For that purpose, notice that
$(t^{*}p^{*},ps)\in\operatorname{\mathcal{S}}^{(2)}$ and that

$(pt)^{*}(ps)=t^{*}p^{*}ps=t^{*}p^{*}pss^{*}s=t^{*}ss^{*}p^{*}ps=(t^{*}s)(s^{*}p^{*}ps),$

from which we can deduce by (P3) that

$x\in X_{s^{*}t}\cap
X_{s^{*}p^{*}ps}=\theta_{s^{*}p^{*}ps}^{-1}(X_{s^{*}t}\cap
X_{s^{*}p*ps})\overset{\ref{p3}}{=}X_{(t^{*}p^{*}ps)^{*}}\cap
X_{s^{*}p^{*}ps}\subseteq X_{(ps)^{*}(pt)},$

and, moreover, that

$\theta_{(pt)^{*}(ps)}(x)=\theta_{t^{*}s}\big{(}\theta_{s^{*}p^{*}ps}(x)\big{)}=\theta_{t^{*}s}(x)=y.$

Therefore, by item (R1) of Definition 3.5, $(ps,x)\sim(pt,y)$ as desired. ∎

We may also slightly modify Lemma 3.9 by changing the relation $\sim$ by the
equivalence relation $\approx$ generated by $\sim$. But first, we will need an
auxiliary result.

######  Lemma 3.10.

Let $(s,x)$, $(t,y)\in D$ such that $(s,x)\approx(t,y)$. Then $x\in X_{s^{*}}$
if, and only if, $y\in X_{t^{*}}$. Moreover, when the memberships are
verified, we have $\theta_{s}(x)=\theta_{t}(y)$.

###### Proof.

Let $(s,x)$, $(t,y)\in D$ and assume, initially, that $(s,x)\sim(t,y)$. If
$(s,x)\sim(t,y)$ by (R2) of Definition 3.5, then the result follows direct
from the comment in the first paragraph after Definition 3.5. Otherwise,
$(s,x)\sim(t,y)$ by (R1) of Definition 3.5 and, then, we must have
$(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$ and
$\theta_{t^{*}s}(x)=y$.

If $x\in X_{s^{*}}$, then $x\in X_{s^{*}}\cap X_{s^{*}t}$ which is the domain
of $\theta_{t^{*}}\circ\theta_{s}$, by (P3). Hence, again by (P3) we have that

$y=\theta_{t^{*}s}(x)=\theta_{t^{*}}\big{(}\theta_{s}(x)\big{)}$

which lies in $X_{t^{*}}$. Moreover, applying $\theta_{t}$ in the above
equality, we obtain that $\theta_{t}(y)=\theta_{s}(x)$.

If $y\in X_{t^{*}}$ we may use the symmetry of $\sim$ and a similar argument
to argue that $x\in X_{s^{*}}$ as well.

For the general case, there exist $n\in\mathbb{N}$ and
$(r_{1},z_{1}),\ldots,(r_{n},z_{n})\in D$ such that $(r_{1},z_{1})=(s,x)$,
$(r_{n},z_{n})=(t,y)$ and $(r_{i},z_{i})\sim(r_{i+1},z_{i+1})$ for all
$i\in\\{1,\ldots,n-1\\}$. Notice that, by the previous argument, $z_{i}\in
X_{r_{i}^{*}}$ if, and only if, $z_{i+1}\in X_{r_{i+1}^{*}}$ for all
$i\in\\{1\ldots,n-1\\}$ and, hence, $x\in X_{s^{*}}$ if, and only if, $y\in
X_{t^{*}}$. Moreover, assuming that $x\in X_{s^{*}}$ and $y\in X_{t^{*}}$,
then $z_{i}\in X_{r_{i}^{*}}$ for all $i\in\\{1\ldots,n\\}$ and, again by the
previous argument, $\theta_{r_{i}}(z_{i})=\theta_{r_{i+1}}(z_{i+1})$ for all
$i\in\\{1\ldots,n-1\\}$, which allow us to deduce that
$\theta_{s}(x)=\theta_{t}(y)$ as stated. ∎

######  Lemma 3.11.

Let $(s,x)$, $(t,y)\in D$ be such that $(s,x)\approx(t,y)$ and let
$p\in\operatorname{\mathcal{S}}$ be such that
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$. Then $x\in X_{s^{*}p^{*}ps}$
if, and only if, $y\in X_{t^{*}p^{*}pt}$. Moreover, when the memberships are
verified, we have $(ps,x)\approx(pt,y)$.

###### Proof.

Let $(s,x)$, $(t,y)\in D$ be such that $(s,x)\approx(t,y)$ and let
$p\in\operatorname{\mathcal{S}}$ be such that
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$.

Assume initially that $s\in E(\operatorname{\mathcal{S}})$. Then, since
$(s,x)\in D$, we have $x\in X_{s^{*}}$. By Lemma 3.10 and (P1), we have that
$y\in X_{t^{*}}$ and $\theta_{t}(y)=\theta_{s}(x)=x$. Since
$(tt^{*},t)\in\operatorname{\mathcal{S}}^{(2)}$, we deduce from the previous
sentence that

$(t,y)\sim(tt^{*},x)$ (4)

by item (R1) of Definition 3.5. Notice that, indeed, $(tt^{*},x)\in D$, since
$x=\theta_{t}(y)\in X_{t}$ and $X_{t}\subseteq X_{tt^{*}}$ by (P2).

Since $tt^{*}$ and $s$ are idempotent elements, we have that

$(tt^{*},x)\sim(s,x)$ (5)

by item (R2) of Definition 3.5.

Combining (4) and (5) with Lemma 3.9, we deduce that $x\in X_{s^{*}p^{*}ps}$
if, and only if, $y\in X_{t^{*}p^{*}pt}$ and, in the affirmative case, that
$(ps,x)\sim(ptt^{*},x)\sim(pt,y)$.

We can use a similar argument if $t\in E(\operatorname{\mathcal{S}})$. Hence,
we now assume that neither $s\in E(\operatorname{\mathcal{S}})$, nor $t\in
E(\operatorname{\mathcal{S}})$. Then, there exist $n\in\mathbb{N}$ and
$(r_{1},z_{1}),\ldots,(r_{n},z_{n})\in D$ such that $(r_{1},z_{1})=(s,x)$,
$(r_{n},z_{n})=(t,y)$ and $(r_{i},z_{i})\sim(r_{i+1},z_{i+1})$ for all
$i\in\\{1,\ldots,n-1\\}$.

Assume $x\in X_{s^{*}p^{*}ps}$. We proceed by induction over $n$. If $n=2$,
Lemma 3.9 applies directly. If $n>2$, assume that our claim holds for $m<n$.
Since $s\notin E(\operatorname{\mathcal{S}})$, we must have
$(s,x)\sim(r_{2},z_{2})$ by item (R1) of Definition 3.5 and, hence,
$(r_{2}^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$. This implies that
$(p,r_{2})\in\operatorname{\mathcal{S}}^{(2)}$ and, then, Lemma 3.9 applies.
Hence $z_{2}\in X_{r_{2}^{*}p^{*}pr_{2}}$ and $(ps,x)\sim(pr_{2},z_{2})$. If
$r_{2}\in E(\operatorname{\mathcal{S}})$, then $(r_{2},z_{2})\approx(t,y)$ and
we can use the initial argument to deduce that $y\in X_{t^{*}p^{*}pt}$ and
$(ps,x)\sim(pr_{2},z_{2})\approx(pt,y)$. If $r_{2}\notin
E(\operatorname{\mathcal{S}})$, then we can use the induction hypothesis to
infer that $y\in X_{t^{*}p^{*}pt}$ and
$(ps,x)\sim(pr_{2},z_{2})\approx(pt,y)$. In any such case, we obtain the
desired result.

Assuming that $y\in X_{t^{*}p^{*}pt}$, we can use a similar argument (this
time from right to left) to establish that $x\in X_{s^{*}p^{*}ps}$ and
$(ps,x)\approx(pt,y)$ as desired. ∎

We now begin to construct an action of $\operatorname{\mathcal{S}}$ on $E$ by
first defining a collection of functions between subsets of $D$. For this,
given $s\in\operatorname{\mathcal{S}}$, we set

$\displaystyle\zeta_{s}\colon\quad D_{s^{*}}$ $\displaystyle\longrightarrow
D_{s}$ (6) $\displaystyle(p,x)$ $\displaystyle\longmapsto(sp,x),$

in which

$\displaystyle D_{s^{*}}=\\{(p,x)\in
D:(s,p)\in\operatorname{\mathcal{S}}^{(2)}\text{ and }x\in
X_{p^{*}s^{*}sp}\\}.$ (7)

Notice that, $\zeta_{s}$ is well defined since
$(s^{*},sp)\in\operatorname{\mathcal{S}}^{(2)}$ and
$X_{(s^{*}sp)^{*}(s^{*}sp)}=X_{p^{*}s^{*}sp}$, which implies that $(sp,x)$
indeed lies in $D_{s}$.

Moreover, if $(p,x),(q,y)\in D_{s^{*}}$ are such that $(p,x)\approx(q,y)$,
then, by Lemma 3.11, $\zeta_{s}(p,x)=(sp,x)\approx(sq,y)=\zeta_{s}(q,y)$.
Thus, setting $E_{s}$ as the image of $D_{s}$ by the quotient map of $D$ by
the equivalence relation $\approx$, we obtain a well defined function
$\eta_{s}\colon E_{s^{*}}\to E_{s}$ such that

$\eta_{s}\big{(}[p,x])=[sp,x]$

for any $(p,x)\in D_{s^{*}}$. We, thus, obtain a pair

$\eta=(\\{E_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\eta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
(8)

which we now prove that is a global action of $\operatorname{\mathcal{S}}$ on
$E$. Before we proceed, just to emphasize, a point in $E$ lies in $E_{s}$ if,
and only if, it admits a representative in $D_{s}$.

######  Theorem 3.12.

The pair
$\eta=(\\{E_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\eta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$,
as in (8), is a global action of $\operatorname{\mathcal{S}}$ on $E$.

###### Proof.

We shall argue (E1), (E2), (E3) and (E4) to use Proposition 2.10.

Let $s\in\operatorname{\mathcal{S}}$ and $[p,x]\in E_{s^{*}}$. Since $[p,x]\in
E_{s^{*}}$, there exist $(q,y)\in D_{s^{*}}$ such that $(p,x)\approx(q,y)$. We
have already argued that $\zeta_{s}(q,y)=(sq,y)\in D_{s}$. Moreover, since
$(q^{*},s^{*}sq)\in\operatorname{\mathcal{S}}^{(2)}$, $y\in X_{q^{*}s^{*}sq}$
and $\theta_{q^{*}s^{*}sq}(y)=y$, we have that $(s^{*}sq,y)\sim(q,y)$ by item
(R1) of Definition 3.5. Therefore,

$\eta_{s^{*}}\big{(}\eta_{s}([p,x])\big{)}=\eta_{s^{*}}\big{(}\eta_{s}([q,y])\big{)}=\eta_{s^{*}}([sq,y])=[s^{*}sq,y]=[q,y]=[p,x],$

from where we infer that
$\eta_{s^{*}}\circ\eta_{s}=\operatorname{id}_{E_{s^{*}}}$. Similarly, we
obtain that $\eta_{s}\circ\eta_{s^{*}}=\operatorname{id}_{E_{s}}$ and, thus,
$\eta_{s}^{-1}=\eta_{s^{*}}$.

Moreover, let $[p,x]\in E$. This means that $x\in X_{p^{*}p}$ and, since,
$(p^{*},p)\in\operatorname{\mathcal{S}}^{(2)}$, we have that $[p,x]\in E_{p}$.
This completes the proof that $E=\cup_{s\in\operatorname{\mathcal{S}}}E_{s}$
and item (E1) is verified.

Next, we prove that $\eta_{s}\circ\eta_{t}=\eta_{st}$ to verify (E2) and (E4)
simultaneously. Let $(s,t)\in\operatorname{\mathcal{S}}^{(2)}$. In one hand,
let $[p,x]\in E_{(st)^{*}}$. Hence, there exists $(q,y)\in D_{(st)^{*}}$ such
that $(p,x)\approx(q,y)$. Since $(st,q)\in\operatorname{\mathcal{S}}^{(2)}$
and $y\in X_{q^{*}t^{*}s^{*}stq}$, we have that
$(s,tq)\in\operatorname{\mathcal{S}}^{(2)}$ and, thus, $(tq,y)\in D_{s^{*}}$.
Moreover, $(t,q)\in\operatorname{\mathcal{S}}^{(2)}$ and $y\in
X_{q^{*}t^{*}tq}$, since $X_{q^{*}t^{*}s^{*}stq}\subseteq X_{q^{*}t^{*}tq}$ by
Proposition 2.8, from where we deduce that $(q,y)\in D_{t^{*}}$. Therefore,
$[q,y]\in E_{t^{*}}$ and $\eta_{t}([q,y])=[tq,y]\in E_{s^{*}}$ which allow us
to infer that

$[p,x]=[q,y]\in\eta_{t}^{-1}(E_{t}\cap E_{s^{*}})$

and

$\eta_{st}([p,x])=\eta_{st}([q,y])=[stq,y]=\eta_{s}([tq,y])=\eta_{s}\big{(}\eta_{t}([q,y])\big{)}=\eta_{s}\big{(}\eta_{t}([p,x])\big{)}.$

On the other hand, let $[p,x]\in\eta_{t}^{-1}(E_{t}\cap E_{s^{*}})$. Thus,
$[p,x]\in E_{t^{*}}$ and $\eta_{t}([p,x])\in E_{s^{*}}$, from where we obtain
$(q,y)\in D_{t^{*}}$ and $(r,z)\in D_{s^{*}}$ such that $(p,x)\approx(q,y)$
and $\zeta_{t}(q,y)\approx(r,z)$. Since $(r,z)\in D_{s^{*}}$ and
$(s,tq)\in\operatorname{\mathcal{S}}^{(2)}$, by Lemma 3.11, we have
$(tq,y)=\zeta_{t}(q,y)\in D_{s^{*}}$. This means that $y\in
X_{q^{*}t^{*}s^{*}stq}$ and, thus, $(q,y)\in D_{(st)^{*}}$. Hence,
$[p,x]=[q,y]\in E_{(st)^{*}}$ and

$\eta_{s}\big{(}\eta_{t}([p,x])\big{)}=\eta_{s}\big{(}\eta_{t}([q,y])\big{)}=\eta_{s}([tq,y])=[stq,y]=\eta_{st}([q,y])=\eta_{st}([p,x]).$

This completes the proof that $\eta_{s}\circ\eta_{t}=\eta_{st}$ and, hence,
(E2) and (E4) are verified.

It remains to verify (E3). For that purpose, let
$s,t\in\operatorname{\mathcal{S}}$ such that $s\leq t$ and let $[p,x]\in
E_{s}$. Thus, there exist $(q,y)\in D_{s}$ such that $(p,x)\approx(q,y)$.
Since, $(q,y)\in D_{s}$, we have that
$(s^{*},q)\in\operatorname{\mathcal{S}}^{(2)}$ and $y\in X_{q^{*}ss^{*}q}$.
Since $s\leq t$, we have that $s^{*}\leq t^{*}$ and, hence,
$(t^{*},q)\in\operatorname{\mathcal{S}}^{(2)}$ and $y\in
X_{q^{*}ss^{*}q}\subseteq X_{q^{*}tt^{*}q}$, by Proposition 2.8. This means
that $(q,y)\in D_{t}$ and, hence, $[p,x]=[q,y]\in E_{t}$. This verifies (E3)
and closes this proof. ∎

We now prove that $(E,\eta)$ is a
$\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$.

######  Theorem 3.13.

Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid. The subcategory
$\mathscr{A}(\operatorname{\mathcal{S}})$ is reflective in
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$. More precisely, for any
$\operatorname{\mathcal{S}}$-set $(X,\theta)$, there exist a global
$\operatorname{\mathcal{S}}$-set $(E,\eta)$ and an
$\operatorname{\mathcal{S}}$-function $i\colon X\to E$ such that $(i,E,\eta)$
is a $\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$.

###### Proof.

By (P1), for all $x\in X$, there exists $e\in E(\operatorname{\mathcal{S}})$
such that $x\in X_{e}$. So, define $i\colon X\to E$ by $x\mapsto[e,x]$. Notice
that $i$ is well defined. In fact, if $e,f\in E(\operatorname{\mathcal{S}})$
are such that $x\in X_{e}$ and $x\in X_{f}$, by item (R2) of Definition 3.5 we
have $(e,x)\sim(f,x)$ and so $[e,x]=[f,x]$.

Moreover, $i$ is $\operatorname{\mathcal{S}}$-function. In fact, suppose that
$x\in X_{s^{*}}$. Then, by (P2), $x\in X_{s^{*}s}$ and, thus,
$i(x)=[s^{*}s,x]$. Since $(s,s^{*}s)\in\operatorname{\mathcal{S}}^{(2)}$ and
$x\in X_{s^{*}s}$, we have $(s^{*}s,x)\in D_{s^{*}}$ and
$\eta_{s}([s^{*}s,x])=[s,x]$. On the other hand, since $\theta_{s}(x)\in
X_{s}\subseteq X_{ss^{*}}$, then $(ss^{*},\theta_{s}(x))\in D$. That said,
notice that $(s,x)\sim(ss^{*},\theta_{s}(x))$ and so
$[s,x]=[ss^{*},\theta_{s}(x)]$. Finally,

$\eta_{s}\big{(}i(x)\big{)}=\eta_{s}([s^{*}s,x])=[s,x]=[ss^{*},\theta_{s}(x)]=i(\theta_{s}(x)).$

Next we prove that $(i,E,\eta)$ is a
$\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$. Indeed,
we already know by Theorem 3.12 that $(E,\eta)$ is a global
$\operatorname{\mathcal{S}}$-set. So, let $(Z,\omega)$ be a global
$\operatorname{\mathcal{S}}$-set and $j\colon X\to Z$ be an
$\operatorname{\mathcal{S}}$-function and set

$\displaystyle\sigma_{D}\colon\qquad D$ $\displaystyle\longrightarrow Z$
$\displaystyle(s,x)$ $\displaystyle\longmapsto\omega_{s}(j(x)).$

Notice that $\sigma_{D}$ is a well defined function. Indeed, given $(s,x)\in
D$, we have $x\in X_{s^{*}s}$. Since $j$ is an
$\operatorname{\mathcal{S}}$-function, we have $j(x)\in j(X_{s^{*}s})\subseteq
Z_{s^{*}s}$. Since $\omega$ is a global action of $\operatorname{\mathcal{S}}$
on $Z$, we have that $j(x)\in Z_{s^{*}}$ by (P4).

Furthermore, $\sigma_{D}$ factors trough $E$. As a matter of fact, it is
enough to prove that $\sigma_{D}(s,x)=\sigma_{D}(t,y)$ for $(s,x),(t,y)\in D$
such that $(s,x)\sim(t,y)$. If $(s,x)\sim(t,y)$ by (R1) of Definition 3.5,
then $(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$ and
$\theta_{t^{*}s}(x)=y$. Since $x\in X_{s^{*}t}$ and $j$ is an
$\operatorname{\mathcal{S}}$-function, we have that $j(x)\in
j(X_{s^{*}t})\subseteq Z_{s^{*}t}$ and
$\omega_{t^{*}s}(j(x))=j(\theta_{t^{*}s}(x))=j(y)$. Since $\omega$ is a global
action of $\operatorname{\mathcal{S}}$ on $Z$, by (E4), we have
$\omega_{t^{*}s}=\omega_{t^{*}}\circ\omega_{s}$ and, hence,
$\omega_{s}(j(x))=\omega_{t}(j(y))$. We, thus, have

$\sigma_{D}(s,x)=\omega_{s}(j(x))=\omega_{t}(j(y))=\sigma_{D}(t,y).$

If $(s,x)\sim(t,y)$ by (R2) of Definition 3.5, then $s,t\in
E(\operatorname{\mathcal{S}})$ and $x=y$. By (P1), we have

$\sigma_{D}(s,x)=\omega_{s}(j(x))=j(x)=j(y)=\omega_{t}(j(y))=\sigma_{D}(t,y).$

Therefore, we obtain a well-defined function

$\displaystyle\sigma\colon\qquad E$ $\displaystyle\longrightarrow Z$
$\displaystyle[s,x]$ $\displaystyle\longmapsto\omega_{s}(j(x)).$

We claim that $\sigma$ is an $\operatorname{\mathcal{S}}$-function. In fact,
let $s\in\operatorname{\mathcal{S}}$ and let $[p,x]\in E_{s^{*}}$. Thus, there
exists $(q,y)\in D_{s^{*}}$ such that $(p,x)\approx(q,y)$. Since $(q,y)\in
D_{s^{*}}$, we have that $(s,q)\in\operatorname{\mathcal{S}}^{(2)}$ and $y\in
X_{q^{*}s^{*}sq}$ and, moreover, $\eta_{s}([p,x])=\eta_{s}([q,y])=[sq,y]$.
Since $j$ is an $\operatorname{\mathcal{S}}$-function, we have that $j(y)\in
Z_{q^{*}s^{*}sq}$. By (P4), $Z_{q^{*}s^{*}sq}=Z_{q^{*}s^{*}}$ which is the
domain of $\omega_{sq}=\omega_{s}\circ\omega_{q}$ by (E4). Hence,

$\sigma([p,x])=\sigma([q,y])=\omega_{q}(j(y))\in Z_{s^{*}}.$

Moreover,

$\omega_{s}\big{(}\sigma([p,x])\big{)}=\omega_{s}\big{(}\omega_{q}(j(y))\big{)}=\omega_{sq}(j(y))=\sigma([sq,y])=\sigma\big{(}\eta_{s}([p,x])\big{)},$

and $\sigma$ is indeed an $\operatorname{\mathcal{S}}$-function.

Now, given $x\in X$, by (P1), choose $e\in E(\operatorname{\mathcal{S}})$ such
that $x\in X_{e}$. Notice that

$\big{(}\sigma\circ i\big{)}(x)=\sigma([e,x])=\omega_{e}(j(x))=j(x),$

which proves that the diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{j}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{Z}$

commutes.

It remains to prove the uniqueness part. For that, assume that
$\sigma^{\prime}:E\to Z$ is an $\operatorname{\mathcal{S}}$-function such that
$\sigma^{\prime}\circ i=j$. Let $[s,x]\in E$ and notice that $(s^{*}s,x)\in
D_{s^{*}}$, since $(s,s^{*}s)\in\operatorname{\mathcal{S}}^{(2)}$ and $x\in
X_{s^{*}s}$. Thus, $[s^{*}s,x]\in E_{s^{*}}$ and, being $\sigma^{\prime}$ an
$\operatorname{\mathcal{S}}$-function, we have that
$\sigma^{\prime}([s^{*}s,x])\in Z_{s^{*}}$ and moreover

$\sigma([s,x])=\omega_{s}(j(x))=\omega_{s}\big{(}(\sigma^{\prime}\circ
i)(x)\big{)}=\omega_{s}\big{(}\sigma^{\prime}([s^{*}s,x])\big{)}=\sigma^{\prime}\big{(}\eta_{s}([s^{*}s,x])\big{)}=\sigma^{\prime}([s,x]),$

from where we conclude that $\sigma^{\prime}=\sigma$, as desired. ∎

We finally prove that $(E,\eta)$ is a universal globalization of $(X,\theta)$.

######  Theorem 3.14.

Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid. For every
$\operatorname{\mathcal{S}}$-set $(X,\theta)$, there exist a global
$\operatorname{\mathcal{S}}$-set $(E,\eta)$ and an
$\operatorname{\mathcal{S}}$-function $i\colon X\to E$ such that $(i,E,\eta)$
is a universal globalization of $(X,\theta)$.

###### Proof.

Let $(i,E,\eta)$ be as in the proof of Theorem 3.13. So, it only remains to
prove that $i\colon X\to E$ is an embedding.

We first prove that $i$ is injective. Indeed, let $x,y\in X$ be such that
$i(x)=i(y)$. Let $e,f\in E(S)$ such that $x\in X_{e}$ and $y\in X_{f}$ and,
thus, we have $[e,x]=[f,y]$. Using Lemma 3.10, we infer that
$x=\theta_{e}(x)=\theta_{f}(y)=y$.

Finally, we prove that the action $\eta$ of $\operatorname{\mathcal{S}}$ on
$E$ induces the action $\theta$ of $\operatorname{\mathcal{S}}$ on $X$. Let
$x,y\in X$ and $s\in\operatorname{\mathcal{S}}$ such that $i(x)\in E_{s^{*}}$
and $\eta_{s}(i(x))=i(y)$. We need to show that $x\in X_{s^{*}}$ and
$\theta_{s}(x)=y$. Using (P1), choose $e,f\in E(\operatorname{\mathcal{S}})$
such that $x\in X_{e}$ and $y\in X_{f}$. Since $[e,x]=i(x)\in E_{s^{*}}$,
there exists $(r,z)\in D_{s^{*}}$ such that $(e,x)\approx(r,z)$. Hence,

$[f,y]=i(y)=\eta_{s}(i(x))=\eta_{s}([e,x])=\eta_{s}([r,z])=[sr,z].$

By Lemma 3.10, since $x\in X_{e}$, $y\in X_{f}$, $(e,x)\approx(r,z)$ and
$(f,y)\approx(sr,z)$, we have that $z\in X_{(sr)^{*}}\cap X_{r^{*}}$,
$\theta_{r}(z)=\theta_{e}(x)=x$ and $\theta_{sr}(z)=\theta_{f}(y)=y$. By (P3),
$x=\theta_{r}(z)\in X_{s^{*}}$ and

$\theta_{s}(x)=\theta_{s}(\theta_{r}(z))=\theta_{sr}(z)=y,$

which finishes the proof. ∎

We apply now our construction in the two examples in the end of section 2.

######  Example 3.15.

Let $\operatorname{\mathcal{S}}$ and $(X,\theta)$ be as in Example 2.11. In
this case, we have

$\displaystyle
D=\big{\\{}(b,1),(b,2),(b^{*}b,1),(b^{*}b,2),(b^{*},1),(b^{*},4),(bb^{*},1),(bb^{*},4),$
$\displaystyle(a,1),(a^{*}a,1),(a^{*},3),(a^{*},4),(aa^{*},3),(aa^{*},4)\big{\\}}.$

and five equivalence classes with respect to relation $\approx$:

$\displaystyle\overline{1}$
$\displaystyle=\\{(b,1),(a^{*},4),(b^{*},1),(a^{*}a,1),(b^{*}b,1),(bb^{*},1)\\},$
$\displaystyle\overline{2}$ $\displaystyle=\\{(b^{*}b,2),(b^{*},4)\\},$
$\displaystyle\overline{3}$ $\displaystyle=\\{(a^{*}a,3)\\},$
$\displaystyle\overline{4}$
$\displaystyle=\\{(b,2),(bb^{*},4),(a,1),(aa^{*},4)\\},$
$\displaystyle\overline{5}$ $\displaystyle=\\{(a^{*},3)\\}.$

The family $\\{D_{s}\\}_{s\in\operatorname{\mathcal{S}}}$, as in Equation (7)
is given by

$\displaystyle D_{a^{*}}$
$\displaystyle=D_{a^{*}a}=\\{(a^{*},3),(a^{*},4),(b,1),(b^{*},1),(a^{*}a,1),(bb^{*},1),(b^{*}b,1)\\},$
$\displaystyle D_{a}$
$\displaystyle=D_{aa^{*}}=\\{(a,1),(aa^{*},3),(aa^{*},4)\\},$ $\displaystyle
D_{b^{*}}$
$\displaystyle=D_{b^{*}b}=\\{(a^{*},3),(a^{*},4),(a^{*}a,1),(b,1),(b^{*}b,1),(b^{*}b,2),(b^{*},1),(b^{*},4),(bb^{*},1)\\},$
$\displaystyle D_{b}$
$\displaystyle=D_{bb^{*}}=\\{(a^{*},3),(a^{*},4),(b,1),(b,2),(b^{*},1),(a^{*}a,1),(bb^{*},1),(bb^{*},4),(b^{*}b,1)\\}.$

Thus, the globalization $(E,\eta)$ of $(X,\theta)$ is given by
$\eta=(\\{E_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\eta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
in which

$\displaystyle E_{a^{*}}=E_{a^{*}a}=\\{\overline{1},\overline{5}\\},$
$\displaystyle E_{a}=E_{aa^{*}}=\\{\overline{3},\overline{4}\\},$
$\displaystyle
E_{b^{*}}=E_{b^{*}b}=\\{\overline{1},\overline{2},\overline{5}\\},$
$\displaystyle E_{b}=E_{bb^{*}}=\\{\overline{1},\overline{4},\overline{5}\\},$

$\eta_{b}\colon E_{b^{*}}\to E_{b}$ and $\eta_{a}\colon E_{a^{*}}\to E_{a}$
are given by

$\begin{array}[]{rcl}\eta_{b}(\overline{1})&=&\eta_{b}([b^{*}b,1])=[b,1]=\overline{1}\\\
\eta_{b}(\overline{2})&=&\eta_{b}([b^{*}b,2])=[b,2]=\overline{4}\\\
\eta_{b}(\overline{5})&=&\eta_{b}([a^{*},3])=[a^{*},3]=\overline{5}\end{array},\qquad\begin{array}[]{rcl}\eta_{a}(\overline{1})&=&\eta_{a}([a^{*}a,1])=[a,1]=\overline{4}\\\
\eta_{a}(\overline{5})&=&\eta_{a}([a^{*},3])=[aa^{*},3]=\overline{3}\\\
&&\end{array},$

and $\eta_{a^{*}}=\eta_{a}^{-1}$, $\eta_{b^{*}}=\eta_{b}^{-1}$ and
$\eta_{e}=\operatorname{id}_{E_{e}}$ for $e\in
E(\operatorname{\mathcal{S}})=\\{b^{*}b,bb^{*},a^{*}a,aa^{*}\\}$. The
$\operatorname{\mathcal{S}}$-function $i\colon X\to E$ is given by
$i(1)=\overline{1}$, $i(2)=\overline{2}$, $i(3)=\overline{3}$ e
$i(4)=\overline{4}$.

In the next example, we first restrict the global action of Example 2.12 and
then compare the globalization of the restricted action with the original
partial action.

######  Example 3.16.

Let $\operatorname{\mathcal{S}}$ and $(Y,\theta^{Y})$ be as in Example 2.12.
If we restrict the action $\theta^{Y}$ to $X=\\{1,2\\}$, we obtain a partial
action
$\theta=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
of $\operatorname{\mathcal{S}}$ on $X$ such that $X_{a^{*}}=\\{1\\}$,
$X_{a}=\\{2\\}$, $\theta_{a}\colon X_{a^{*}}\to X_{a}$ and
$\theta_{a^{*}}\colon X_{a}\to X_{a^{*}}$ are evident, $X_{s}=X$ and
$\theta_{s}=\operatorname{id}_{X}$ if $s\neq a,a^{*}$.

It is clear that $(j,Y,\theta^{Y})$, in which $j\colon X\to Y$ is the
inclusion map, is a globalization of $(X,\theta)$. However, this is not a
universal globalization of $(X,\theta)$.

Indeed, applying our construction in this case, we obtain
$D=\operatorname{\mathcal{S}}\times X$ and we have four equivalence classes
with respect to relation $\approx$:

$\displaystyle\overline{1}$
$\displaystyle=\\{(b^{*}b,1),(bb^{*},1),(a^{*}a,1),(aa^{*},1),(b,1),(b^{*},1),(a^{*},2)\\},$
$\displaystyle\overline{2}$
$\displaystyle=\\{(b^{*}b,2),(bb^{*},2),(a^{*}a,2),(aa^{*},2),(b,2),(b^{*},2),(a,1)\\},$
$\displaystyle\overline{3}$ $\displaystyle=\\{(a,2)\\},$
$\displaystyle\overline{4}$ $\displaystyle=\\{(a^{*},1)\\}.$

The family $\\{D_{s}\\}_{s\in\operatorname{\mathcal{S}}}$, as in Equation (7)
is simply given by

$D_{s^{*}}=\\{(p,x)\in D:(s,p)\in\operatorname{\mathcal{S}}^{(2)}\\}$

and, thus, the globalization $(E,\eta)$ of $(X,\theta)$ is given by
$\eta=(\\{E_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\eta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
in which

$\displaystyle
E_{a^{*}}=E_{a^{*}a}=\\{\overline{1},\overline{2},\overline{4}\\},$
$\displaystyle E_{a}=E_{aa^{*}}=\\{\overline{1},\overline{2},\overline{3}\\},$
$\displaystyle
E_{b^{*}}=E_{b^{*}b}=\\{\overline{1},\overline{2},\overline{4}\\},$
$\displaystyle E_{b}=E_{bb^{*}}=\\{\overline{1},\overline{2},\overline{4}\\},$

$\eta_{b}\colon E_{b^{*}}\to E_{b}$ and $\eta_{a}\colon E_{a^{*}}\to E_{a}$
are given by

$\begin{array}[]{rcl}\eta_{b}(\overline{1})&=&\eta_{b}([b^{*}b,1])=[b,1]=\overline{1}\\\
\eta_{b}(\overline{2})&=&\eta_{b}([b^{*}b,2])=[b,2]=\overline{2}\\\
\eta_{b}(\overline{4})&=&\eta_{b}([a^{*},1])=[a^{*},1]=\overline{4}\end{array},\qquad\begin{array}[]{rcl}\eta_{a}(\overline{1})&=&\eta_{a}([a^{*}a,1])=[a,1]=\overline{2}\\\
\eta_{a}(\overline{2})&=&\eta_{a}([a^{*}a,2])=[a,2]=\overline{3}\\\
\eta_{a}(\overline{4})&=&\eta_{a}([a^{*},1])=[aa^{*},1]=\overline{1}\end{array},$

and $\eta_{a^{*}}=\eta_{a}^{-1}$, $\eta_{b^{*}}=\eta_{b}^{-1}$ and
$\eta_{e}=\operatorname{id}_{E_{e}}$ for $e\in
E(\operatorname{\mathcal{S}})=\\{b^{*}b,bb^{*},a^{*}a,aa^{*}\\}$. Moreover,
the $\operatorname{\mathcal{S}}$-function $i\colon X\to E$ is given by
$i(1)=\overline{1}$ and $i(2)=\overline{2}$.

Notice that, considering the globalization $(j,Y,\theta^{Y})$, the mediating
function $\sigma\colon E\to Y$ given by the universal property of $(i,E,\eta)$
is given by

$\begin{array}[]{rcl}\sigma(\overline{1})&=&\sigma([b^{*}b,1])=\theta_{b^{*}b}^{Y}(j(1))=\theta_{b^{*}b}^{Y}(1)=1\\\
\sigma(\overline{2})&=&\sigma([b^{*}b,2])=\theta_{b^{*}b}^{Y}(j(2))=\theta_{b^{*}b}^{Y}(2)=2\\\
\sigma(\overline{3})&=&\sigma([a,2])=\theta_{a}^{Y}(j(2))=\theta_{a}^{Y}(2)=3\\\
\sigma(\overline{4})&=&\sigma([a^{*},1])=\theta_{a^{*}}^{Y}(j(1))=\theta_{a^{*}}^{Y}(1)=3\end{array}$

and hence, $\sigma$ is not injective. Moreover, the reader can check that any
function $\sigma^{\prime}\colon Y\to E$ such that $\sigma^{\prime}\circ j=i$
is not an $\operatorname{\mathcal{S}}$-function.

The last example shows that the mediating function given by the universal
property of the globalization $(E,\eta)$ may not be injective (and, hence, may
not be an embedding). However, we can guarantee its injectivity in some
especial subsets of $E$.

Indeed, for each $u\in\operatorname{\mathcal{S}}^{(0)}$, set

$\displaystyle D_{u}=\\{(s,x)\in D:c(s)=u\\}.$ (9)

and let $E_{u}$ be the image of $D_{u}$ by the quotient map of $D$ by the
equivalence relation $\approx$.

We end this paper by showing that the mediating function given by the
universal property of $(i,E,\eta)$ is injective in each $E_{u}$.

######  Proposition 3.17.

Let $(X,\theta)$ be an $\operatorname{\mathcal{S}}$-set and $(j,Z,\omega)$ a
globalization of $(X,\theta)$. Given any
$u\in\operatorname{\mathcal{S}}^{(0)}$, the mediating function $\sigma\colon
E\to Z$ given by the universal property of the globalization $(i,E,\eta)$ is
injective in $E_{u}$.

###### Proof.

Let $[s,x],[t,y]\in E_{u}$ such that $\sigma([s,x])=\sigma([t,y])$. With no
loss in generality, we may assume $(s,x),(t,y)\in D_{u}$. This means that

$\omega_{s}(j(x))=\sigma([s,x])=\sigma([t,y])=\omega_{t}(j(y))$

and, hence, $\omega_{s}(j(x))\in Z_{t}\cap Z_{s}$. Since $c(s)=c(t)=u$, we
have $(t^{*},s)\in S^{(2)}$ and, since $\omega$ is a global action, we have
$\omega_{t^{*}s}=\omega_{t^{*}}\circ\omega_{s}$ by (E4). Thus,
$j(x)\in\omega_{s}^{-1}(Z_{t}\cap Z_{s})=Z_{s^{*}t}$ and

$\omega_{t^{*}s}(j(x))=\omega_{t^{*}}\big{(}\omega_{s}(j(x))\big{)}=\omega_{t^{*}}\big{(}\omega_{t}(j(y))\big{)}=j(y).$

Since $j$ is an embedding, $x\in X_{s^{*}t}$ and $\theta_{t^{*}s}(x)=y$, which
implies that $(s,x)\sim(t,y)$ by (R2). ∎

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# Discriminant Analysis in Contrasting Dimensions for Polycystic Ovary
Syndrome Prognostication

Abhishek M. Gupta
Bachelor of Engineering - EXTC
University of Mumbai
Mumbai, MH, IN
<EMAIL_ADDRESS>
Himanshu H. Soni
Bachelor of Engineering - EXTC
University of Mumbai
Mumbai, MH, IN
<EMAIL_ADDRESS>
Raunak M. Joshi
Mentor
Master of Engineering
Mumbai, MH, IN
<EMAIL_ADDRESS>
Ronald Melwin Laban
Assistant Professor - Dept of EXTC
St. John College of Engineering and Management
Palghar, MH, IN
<EMAIL_ADDRESS>

###### Abstract

A lot of prognostication methodologies have been formulated for early
detection of Polycystic Ovary Syndrome also known as PCOS using Machine
Learning. PCOS is a binary classification problem. Dimensionality Reduction
methods impact the performance of Machine Learning to a greater extent and
using a Supervised Dimensionality Reduction method can give us a new edge to
tackle this problem. In this paper we present Discriminant Analysis in
different dimensions with Linear and Quadratic form for binary classification
along with metrics. We were able to achieve good accuracy and less variation
with Discriminant Analysis as compared to many commonly used classification
algorithms with training accuracy reaching 97.37% and testing accuracy of
95.92% using Quadratic Discriminant Analysis. Paper also gives the analysis of
data with visualizations for deeper understanding of problem.

_K_ eywords Dimensionality Reduction $\cdot$ Linear Discriminant Analysis
$\cdot$ Quadratic Discriminant Analysis

##  1 Introduction

The Polycystic Ovary Syndrome also known as PCOS [1] is a symptom which causes
disorder in women ranging from reproductive age to their later stages in life.
The commonly associated disorder with PCOS is irregularity in menstrual cycles
and skin diseases. The high blood pressure and high cholesterol abide to the
PCOS symptoms making things worse. High blood sugar is also observed in the
later stages of the symptom. The early detection of this major disorder is a
necessary issue. Machine Learning can be leveraged to the advantage of early
detection. The medical expenses cannot be borne by all the women. Some also
feel insecurity for disclosing such a matter even to doctors. A functional
prognostication system can be very helpful in such a situation. Many people
have proved to be successful in using Machine Learning with the right data for
this problem. The PCOS is a binary classification problem. It clearly focuses
on the result of whether the woman is suffering from PCOS or not. There is no
distinction in the choices. Many machine learning algorithms can be used for
such a problem. The first algorithm taken into consideration is Logistic
Regression [2] being the binary classification problem. A lot of work has been
done on Logistic Regression so far now. Very brief implementation between
Logistic Regression and Bayesian Classifier has been performed [3]. The
results obtained were having a Bayesian Classifier which was able to beat the
performance of Logistic Regression with almost 2% improvement. Trying to give
more in depth implementation with machine learning inclination, various
algorithms like KNN, CART and SVM were used [4]. A detailed result section
gives the metrics for the best algorithm. The improvements simultaneously were
even extended with the boosting algorithms [5]. Many different boosting
algorithms were used in an implementation for PCOS where the CatBoost was able
to outrun all the boosting algorithms [6]. This was a compendious comparison
of all the advanced boosting algorithms like AdaBoost, LGBM, XGBoost and
CatBoost.

All methods that have been implemented by now are purely prognostication
oriented with machine learning. What we present in this paper is to use
Discriminant Analysis [7] for getting a deeper understanding of the problem.
Discriminant Analysis is an extension of dimensionality reduction with
classification. Dimensionality reduction has supervised as well as
unsupervised learning approaches [8]. The most commonly used dimensionality
reduction technique is Principal Component Analysis [9]. It is an unsupervised
learning algorithm. The most commonly used supervised dimensionality reduction
algorithm on the other hand is Linear Discriminant Analysis [10]. It maximizes
the separability between the classes. An increase in dimensions gives us the
Quadratic Discriminant Analysis [11]. It is a type of generative model.

There are some differences between Linear Discriminant Analysis and Quadratic
Discriminant Analysis. The observable difference is that Linear Discriminant
Analysis does not have class-specific covariance matrices, but one shared
covariance matrix among the classes. The shared covariance matrix is basically
the covariance of all the inputs given. The benefit of Discriminant Analysis
is that for the purpose of classification it uses the canonical variables.

##  2 Implementation

###  2.1 Linear Discriminant Analysis

The classification aspect of the Linear Discriminant Analysis works on maximum
separability. Working with higher dimensions is necessary to generate a better
understanding of the data for the final outcome. The higher dimensions not
necessarily can give better understanding of the data points. So in order to
maintain the feature dimensions and not increase them more than required,
Linear Discriminant Analysis minimizes the dimensions. It transforms the
features to one axis which is called Linear Discriminant. The common working
of the algorithm considers the maximum separability by maximizing the
difference between the means and minimizing the variance. This minimizing
variation is called scatter in LDA The covariance matrix is the underlying
source of the LDA. It approaches the problem by using conditional probability
density functions which fall under normal distribution mean parameters and
covariance parameters. For this the Bayesian Optimization [12] is used which
gives the log of likelihood for the ratios with threshold to classify the
right class.

###  2.2 Quadratic Discriminant Analysis

The QDA [11] is a variant of LDA [10] which considers an covariance matrix
that is individual. It is estimated for all the classes of observations. If
the individual classes orchestrates distinct covariances, in such a scenario
QDA can prove to be better as there is an involvement of the prior knowledge.
The effective parameters increase for QDA as compared to LDA. The most common
consideration for the binary classification based Quadratic Discriminant
Analysis or Linear Discriminant Analysis is given by a multivariate Gaussian
Distribution [13]. The formula is given as follows

$P(x|y=k)=\frac{1}{(2\pi)^{d/2}|\Sigma_{k}|^{1/2}}\exp\left(-\frac{1}{2}(x-\mu_{k})^{t}\Sigma_{k}^{-1}(x-\mu_{k})\right)$
(1)

The discriminant function for the LDA in Quadratic can be represented as
follows

$\delta_{k}(x)=-\frac{1}{2}\log|\Sigma_{k}|-\frac{1}{2}(x-\mu_{k})^{T}\Sigma_{k}^{-1}(x-\mu_{k})+\log\pi_{k}$
(2)

We try to estimate $\Sigma_{k}$ in 2 for each and every class instead of
assuming like we do in Linear Discriminant Analysis.

##  3 Results

This section constitutes our work with data. The implementation phase has many
facets of results. We have presented numerous results below.

###  3.1 Analysis

This section of the paper clearly focuses on the analysis of the entire data.
Many graphical representations given below provide an idea of different
features in the data. The analysis section was taken into consideration to see
the interaction of the different features with respect to age. The age is the
parameter that influences the entire other parameters. The entire data can be
checked for correlation using heat map. It gives a distinct distribution of
the dataset features with valuation.

Figure 1: Heatmap for all the features

The Figure 1 is a very larger overview of all the features. It gives you an
idea of how all the features are correlated from each other. More improvements
can be bought into the analysis section. We can perform analysis for intricate
observations. Boxplot [14] is known to be one of the most common form of
visual representation to give a better edge in analysis of data.

Figure 2: Categorical Plot for Exercise Level with respect to Age for Labels

The Figure 2 gives a very good analysis of the effect of exercise levels with
respect to age. The labels are taken into consideration. The visual
representation of the features gives inference that exercise is an influencing
factor taken into consideration for the prevention of the symptom. Another
very important consideration is the irregularities in the menstrual cycles. It
gives a very deep understanding about the problem. The Figure 3 gives a very
detailed analysis about the distribution of the occurrence and effect of
irregularities in menstrual cycles.

Figure 3: Categorical Plot for Periods with respect to Age for Labels

###  3.2 Train and Test Accuracy

The accuracies give the basic analogy of any algorithm. It the most primary
metric taken into consideration. It gives a baseline effect of how well the
model performs with the data.

Table 1: Comparison of LDA and QDA Training & Testing Accuracy Algorithm | Training Accuracy | Testing Accuracy
---|---|---
Linear Discriminant Analysis | 86.84 % | 81.63 %
Quadratic Discriminant Analysis | 97.37 % | 95.92 %

In Table 1 we can clearly see that the variation in the accuracy of Quadratic
Discriminant Analysis is comparatively less than Linear Discriminant Analysis.

###  3.3 Confusion Matrix

The Type-I and Type-II Error are the errors which give clear distinction of
interacting variables with the test data. The basic aspect of Confusion Matrix
[15] is observation of predicted class with target class. It gives the errors
which are targeted for specific problems. The Type-II Error works with False
Negatives which is a type of error we want to target. The importance of this
error is that if the person does have symptoms of PCOS, the system should
detect the symptom. The table 2 below gives a comparison of False Positives
and False Negatives between LDA and QDA.

Table 2: Confusion Matrix for LDA & QDA Algorithm | False Negatives | False Positives
---|---|---
Linear Discriminant Analysis | 6 | 3
Quadratic Discriminant Analysis | 0 | 0

###  3.4 Metric Scores

The two major metrics we are focusing in this paper are the Average Precision
Score [16] and Precision Recall Curve [17]. The Precision and Recall are
individually calculated. Precision is a measure of quality of predictions. The
Precision is given by formula

$Precision={\frac{TruePositives}{TruePositives+FalsePositives}}$ (3)

and Recall is given by formula

$Recall={\frac{TruePositives}{TruePositives+FalseNegatives}}$ (4)

Recall is a measure of the quantity of correctly classified predictions. Since
our problem works with binary classification problem [18], the values are
distinguished Yes/No and do not have any threshold values.

Average Precision considers a Precision-Recall curve as the weighted mean of
all the Precision achieved at each threshold, with the increase in recall from
the previous threshold used as the weight. The formula given below gives a
precise representation for the binary classification problem. If the data is
available with thresholds the graph considers an integral spread over the
points. The discrete data on the other hand gives the summation spread over
the points. The formula is given as

$AveragePrecision=\sum^{n}_{k=1}(Recall_{k}-Recall_{k-1}).Precision_{k}$ (5)

The below are given graphs that give clear distinction of above specified
metrics in the form of visualization. These graphs plot the precision and
recall score with respect to average precision. In Figure 4 we can see the
interaction of the Precision and Recall Curve. The sudden drop can be seen
after some points. Whereas in Figure 5 we can a sharp drop which goes beyond
the performance of the Linear Discriminant Analysis.

Figure 4: Precision Recall Curve with Average Precision for LDA Figure 5:
Precision Recall Curve with Average Precision for QDA

###  3.5 Receiver Operator Characteristic and Area Under Curve

Receiver Operator Characteristic Curve [19] also known as RoC Curve is used
for performance measurement of classification models. It outputs comprehensive
probability fluctuations throughout the predictions. This is computed on the
basis of two functions known as True Positive Rate (TPR) and False Positive
Rate (FPR). True Positive Rate is also known as Sensitivity and works much
similar to Recall. It is denoted by formula as

$TruePositiveRate={\frac{TruePositives}{TruePositives+FalseNegatives}}$ (6)

After True Positive Rate we have to work our way towards False Positive Rate,
but the issue is that the calculation cannot be done directly. One needs to
first calculate Specificity. It is a measure for finding negative classes
without anomalies. This is represented by formula as

$Specificity={\frac{TrueNegatives}{TrueNegatives+FalsePositives}}$ (7)

After this specificity can help compute the False Positive Rate. It can be
also denoted with a formula

$FalsePositiveRate={\frac{FalsePositives}{FalsePositives+TrueNegatives}}$ (8)

Apart from RoC Curve we require Area Under Curve which is also known as AUC
[20]. This AUC supports specifying threshold settings. It gives degree or
measure of separability. Higher the value of AUC, better is the prediction.
The higher the value on the scale, the more better algorithmic performance.

Figure 6: RoC Curve with AUC for LDA and QDA

In Figure 6 the Red Line indicates the boundary of threshold for the Area
Under Curve. The metrics for the models need to be above the Red Line. In our
paper, the Quadratic Discriminant Analysis performs better than Linear
Discriminant Analysis. The QDA covers much more area than LDA and the
inference can be derived that more area for classification is considered.

###  3.6 Separability

Considering Discriminant Analysis the representation of the points is given
over plots with dimensions.

Figure 7: The Separability of the Data with respect to Labels

In Figure 7 we can see the separation in both the labels of their respective
distributions. The blue ellipse represents the Yes labels and red ellipse
represents the No labels. The Linear Discriminant Analysis has ellipses that
are separated in Linear dimensional form while Quadratic Discriminant Analysis
has ellipses in quadratic form, with much more increased dimensions.

##  4 Conclusion

This paper presents a very intricate area of Machine Learning domain which is
not much worked with. Our aim in this paper was highlighting the point that
classification with dimensionality reduction is as important as other
classification parametric methods. We used supervised dimensionality reduction
algorithm Linear Discriminant Analysis. We worked on increasing the dimension
of the same algorithm which resulted in Quadratic Discriminant Analysis.
Definitely this paper promotes more ideas and new improvements can be done by
researchers in the future. With our best belief and knowledge we conclude this
paper.

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