Isomorphism Theorems for Groupoids and Some Applications

Abstract

Using an algebraic point of view we present an introduction to the groupoid theory; that is, we give fundamental properties of groupoids as uniqueness of inverses and properties of the identities and study subgroupoids, wide subgroupoids, and normal subgroupoids. We also present the isomorphism theorems for groupoids and their applications and obtain the corresponding version of the Zassenhaus Lemma and the Jordan-Hölder theorem for groupoids. Finally, inspired by the Ehresmann-Schein-Nambooripad theorem we improve a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups.

1. Introduction

The concept of groupoid from an algebraic point of view appeared for the first time in [1]. From this setting, a (Brandt) groupoid can be seen as a generalization of a group, that is, a set with partial multiplication on it that could contain many identities.

Brandt groupoids were generalized by Ehresmann in [2], where the author added further structures such as topological and differentiable structures. Other equivalent definitions of groupoids and their properties are given in [3], where a groupoid is defined as a small category where each morphism is invertible.

In Definition 1.1 of [4], the author follows the definition given by Ehresmann and presents the notion of groupoid as a particular case of universal algebra, and he defines strong homomorphism for groupoids and proves the correspondence theorem in this context. The Cayley theorem for groupoids is also presented in Theorem 3.1 of [5].

Recently, some applications of groupoids to the study of partial actions are presented in different branches, for instance, in [6] the author constructs a Birget-Rhodes expansion associated with an ordered groupoid and shows that it classifies partial actions of on sets, in the topological context in [7] is treated the globalization problem, connections between partial actions of groups and groupoids are given in [8, 9]. Also, ring theoretic and cohomological results of global and partial actions of groupoids on algebras are obtained in [10–16]. Galois theoretic results for groupoid actions are obtained in [12, 17–19]. Finally, the globalization problem for partial groupoid actions has been considered in [7, 20, 21].

In [19], Paques and Tamusiunas give some structural definitions in the context of groupoid such as abelian groupoid, subgroupoid, and normal subgroupoid and showed necessary and sufficient conditions for a subgroupoid to be normal. Furthermore, they built quotient groupoids.

Due to the applications of the groupoids to partial actions and their usefulness, we will give an elementary introduction to the theory of groupoids from an axiomatic definition following Lawson [22].

Our principal goal in this work is to continue the algebraic development of a groupoid theory. The paper is organized as follows. After the introduction, in Section 2, we present groupoids from an axiomatic point and show some properties of them. In Section 3 we recall the notions of some substructures of groupoids, such as subgroupoid, wide subgroupoid, and normal subgroupoid. In Section 4, we prove the correspondence and isomorphism theorems for groupoids. In the final section we show an application of section four, we prove the Zassenhaus Lemma and Hölder theorem for groupoids, and we improve Theorem 4.2 of [23] using the Ehresmann-Schein-Nambooripad theorem.

It is important to note that the notion of the groupoid can be presented from categories, algebraic structures, and universal algebra. In the last setting, the isomorphism theorems are valid, but the idea is to do an algebraic presentation and verify which assumptions are necessary. So it is possible to reach a wider audience.

2. Groupoids

Now, we give two definitions of groupoids from an algebraic point of view.

Definition 1. (see [22], p. 78). Let be a set equipped with a partial binary operation on which is denoted by concatenation. If and the product is defined, we write . An element is called an identity ifThe set of identities of is denoted by . Then is said to be a groupoid if the following axioms hold:(i), if and only if and (ii), if and only if and (iii)For each , there are unique identities and such that and (iv)For each , there is an element such that , , , and The following definition of groupoid is presented in Definition 1.1 of [24].

Definition 2. A groupoid is a set endowed with a product mapwhere the set is called the set of composible pairs and an inverse map such that for all the following relations are satisfied. (G1)  (G2) If then and  (G3) and if then  (G4) and if then We shall check that Definitions 1 and 2 are equivalent. First, we need a couple of lemmas.

Lemma 1 (see [25], Lemma 1.1.4). Suppose that is a groupoid in the sense of Definition 1. Let Then , if and only if .

Proof. Let such that . By (iv) of Definition 1, we have that , , and . Since , then . That is, . Now, since and are identities, then . Conversely, if , then , and since we have . Whence by (ii) of Definition 1, we have that .

Lemma 2. Suppose that is a groupoid in the sense of Definition 1. Then, the element in (iv) is unique and .

Proof. For each , assume that there exists such that , , , , , and . Notice that which implies that , which is defined by (ii) of Definition 1, and then by associativity, . Thus, , so . It is analogous for . In particular, the inverse is unique.
Finally, the equality follows from the uniqueness of the inverse of
We give the following.

Proposition 1. Let be a set. Then, it is a groupoid in the sense of Definition 1, if and only if it is a groupoid in the sense of Definition 2.

Proof. Let . By using (iv) of Definition 1, we define . Then, by Lemma 2 this map is well defined. We shall check – of Definition 2: (G1) It is the second assumption in Lemma 2. (G2) If , then and . By (i) and (ii), and that means and  (G3) By item (iv), we get that . Let with . By Lemma 1, we get that , and by using (iii), we obtain  (G4) This is proved analogously to the previous item.Conversely, suppose that is a set. We define a partial binary operation on by if and only if and We shall check that properties (i)–(iv) in Definition 1 hold:(i)Let such that . Then and by , and Thus, and by , and . In particular, We conclude that and by using , we get that Conversely, suppose that . Then, and by , we have that and . Thus, and by , . Finally, since we obtain, again by , that . Hence, (ii)This is shown analogously to the previous items.(iii)(iv) If , then . Thus, we set and . Hence, by , , and , , and the equalities hold.

Remark 1. The interested reader can find another two equivalent definitions of groupoids in [26, 27].
From now on in this work denotes a groupoid.
For the sake of completeness, we give the proof of some known consequences of Definition 1.

Proposition 2 (see [25], Lemma 1.1.4). For each we have:(i)If then and .(ii), if and only if , and in this case .

Proof.  (i)For the first equality, we prove that satisfies the axiom (iii) from Definition 1. Indeed, assume that . Then, , , and In a similar way, it is possible to show that .(ii)We have that , if and only if . Notice that for any we have thatThen, . That is, . Furthermore,Therefore, by the uniqueness of the inverse element, we get that .
The following statements also follow from the definition of groupoid.

Proposition 3. Let . Then, the following statements hold:

Proof.  (i)This is (4)(ii), where the last equality follows from (i) of Proposition 2(iii), where the last equality also follows from (i) of Proposition 2Items (iv) and (v) are proved analogously.

Remark 2. Let be a groupoid. In ([12], p. 3660), Bagio and Paques called an element an identity if , for some .

Proposition 4. Let be a groupoid. An element e of is an identity in the sense of Bagio and Paques, if and only if it satisfies (1).

Proof. Suppose that is an identity in the sense of Bagio and Paques, for some . By (i) of Proposition 3, Now, let such that and By Lemma 1 and (ii)–(v) of Proposition 3, we have that , then and . Therefore, e satisfies (1).
Conversely, suppose that satisfies (1). By (iii) of Definition 1, we get and Thus , and it follows that e is an identity in the sense of Bagio and Paques.

Remark 3. It follows from the proof of Proposition 4 that , , and for any . Moreover, note that the elements of are the unique idempotents of . In fact, if and , then and so . Since , it follows that .

Proposition 5. Let . Then, the set is a group.

Proof. By Remark 3, we have that Thus If , then , and so thanks to Lemma 1. Now, (i) of Proposition 2 implies that and . Hence, . If , then by Lemma 1, and and we have that and . Therefore, e is the identity element of . Finally, let . By Proposition 3, and . Hence, , and we conclude that is a group.

Definition 3. The group is called the isotropy group associated with The isotropy subgroupoid (see Definition 5) or the group bundle associated to is defined by the disjoint union

Remark 4. A concept of abelian groupoid was presented in ([19], p. 111) as follows: a groupoid is abelian if for each ; and , for all with .
We have the following.

Proposition 6. A groupoid is abelian in the sense of Paques and Tamusiunas, if and only if and is abelian for all
In the light of Proposition 6, we prefer to use the following definition of the abelian groupoid.

Definition 4. (see [28], Definition 1.1). A groupoid is called abelian if all its isotropy groups are abelian.
Note that if is abelian in the sense of Paques and Tamusiunas, then it is abelian in the sense of Definition 4. Now, consider the groupoid with . Then, we have that and . That is, is an abelian groupoid in the sense of Definition 4, but it is not a union of abelian groups.

3. Normal Subgroupoids, the Quotient Groupoid and Homomorphisms

In this section, we present a theory of substructures in a groupoid. We follow the definition of subgroupoid given in [19].

Definition 5. Let be a groupoid and a nonempty subset of . is said to be a subgroupoid of if it satisfies: for all ,(i)(ii)If , then If is a subgroupoid of then it is called wide if .

Remark 5. It is clear that if is a subgroupoid of , then it is a groupoid with the product (2), restricted to

Example 1. Let be a groupoid.(1)Take such that . The set is a subgroupoid of . Indeed, first of all note that by assumption . If , then , , and . Since is a group, then and That is, . If , then . Hence, we have that and since Observe that this example generalizes the concept of centralizer in groups.(2)Suppose that is abelian and . Then the set is a subgroupoid of . If , then for some . If then , and this implies that and thus for some . Then, , and so . Now, if , then for some . Thus, . Finally, note that for , . Hence, , and we conclude that is wide.(3)Suppose that is abelian. Then, the set is a wide subgroupoid of . First, it is clear that . If , then for some and some . Thus, we obtain that and . If , then and thus and since is an abelian groupoid. Then , that is, . Now, since we have , and hence . We conclude that is a wide subgroupoid of . Note that if we take a fixed and define the set , then is a wide subgroupoid of and . That is, is a subgroupoid of . Observe that this example generalizes the concept of torsion subgroup in abelian groups.

Proposition 7. Let be a groupoid and , subgroupoids of . Then,(i)If is non-empty, then is a subgroupoid of , if and only if (ii)If and are wide and is a subgroupoid, then is wide

Proof. The proof of (i) is similar to the group case. To prove (ii), it is enough to observe that and if , then
Now, we present the notion of a normal subgroupoid and prove several properties of them, which generalize well-known results in group theory. We follow the definition given in [19].

Definition 6. Let be a groupoid. The subgroupoid of is said to be normal, denoted by , if and for all , where

Remark 6. By the proof of Lemma 3.1 of [19], one has that if and only if is wide. Also the assertion is equivalent to for all
Several examples of normal groupoids are presented in [19], p. 110-111.
Given a wide subgroupoid of , in [19], Paques and Tamusiunas define a relation on as follows: for every Furthermore, they prove that this relation is a congruence, which is an equivalence relation that is compatible with products. The equivalence class of containing is the set . This set is called the left coset of in containing . Then, we have the following.

Proposition 8. (see [19], Lemma 3.12). Let be a normal subgroupoid of and let be the set of all left cosets of in . Then, is a groupoid such that , if and only if and the partial binary operation is given by .

The groupoid in Proposition 8 is called the quotient groupoid of by .

Now, we present the notion of groupoid homomorphism and prove several properties of them, which generalize well-known results in homomorphisms of groups.

Definition 7. Let and be groupoids. A map is called groupoid homomorphism if for all , implies that , and in this case .
Notice that defined by for all , is a surjective groupoid homomorphism.

Definition 8. Let be a homomorphism of groupoids. We define the following sets:(i)For , write , the direct image of In particular, the set is called the image of ϕ.(ii), the kernel of ϕ.(iii)Let , , the inverse image of by ϕ.(iv)ϕ is called a monomorphism if it is injective, an epimorphism if it is surjective, and an isomorphism if it is bijective.

Remark 7. If is abelian and is a subgroupoid of then it is not difficult to show that is abelian. Moreover, if is another groupoid, such that there is a groupoid epimorphism then is also abelian.

Proposition 9. Let be a groupoid homomorphism. Then,(i)For each , , and .(ii)If is a subgroupoid of , then is a subgroupoid of Moreover, if is wide then is wide, and it contains .(iii)If , then and . In particular, .

Proof. (i)Let . Since then and . Thus, by the uniqueness of the identities . Analogously, . Finally, since and , then and . Moreover, Which implies that .(ii)It is not difficult to show that is a subgroupoid of Now suppose that is wide. By item (i), we know that , that is, . Finally, if , then and hence as desired.(iii)By item (ii), it is enough to see that for all . Indeed, let with and . Then, and thus . We haveThen, , and since and we obtain thatFinally, to show that , it is enough to observe that and is normal in .

3.1. Strong Isomorphism Theorems for Groupoids

In this section, we present a special type of groupoid homomorphism, called the strong groupoid homomorphism. Using these homomorphisms, we show the correspondence theorem and the isomorphism theorems for groupoids. This notion of strong groupoid homomorphism has been considered before by several authors (see [4], Remark 2.2).

Definition 9. Let be a groupoid homomorphism. ϕ is called strong if for all , implies that .

Example 2. Let X be a nonempty set and . Then is a groupoid, where the product is given by: for Then, the map is a strong groupoid homomorphism with kernel

Proposition 10. Let be a strong groupoid homomorphism. Then,(i)If , then and . In particular, and are subgroupoids of and respectively.(ii)If , then .(iii)([29], Proposition 9) ϕ is an injective homomorphism, if and only if .(iv)(The Correspondence Theorem for Groupoids) There exists a one-to-one correspondence between the sets and . Moreover, this correspondence preserves normal subgroupoids.

Proof. (i)It is clear that . Let and suppose that . Then, for some . Since ϕ is strong, we have that . Thus, . Now, if , then for some , and we have . Now, we check the equality If , then there exists with . Since ϕ is strong, we get that and . Hence, . The other inclusion is clear.(ii)-(iii)These are similar to the group case.(iv)First, define the functions by for each , and by for each . By (i) of Proposition 9 and (ii) of Proposition 10, it has that and . That is, α is a bijective function. The remaining proof follows from item of Proposition 9 and (ii) above.Now, we use strong homomorphisms to extend to the groupoid context, a well-known result concerning the product of groups.

Proposition 11. Let and be subgroupoids of . If is normal then(i) is a subgroupoid of (ii)If is normal, then is a normal subgroupoid of (iii)If is wide, then is a normal subgroupoid of

Proof. (i)Consider the groupoid epimorphism . Then, by the definition of the map ϕ is strong and . Thus, by (i) of Proposition 10 we get that is a subgroupoid of . Hence, the result follows from Proposition 7.(ii)By the previous item, is a subgroupoid of . Moreover, it is clear that is wide. Let and . Then, , with , , and . Thus, and we have that that is, is a normal subgroupoid of .(iii)It is clear that is a wide subgroupoid of . Let and with . Then and by assumptions it follows that .Next we present the isomorphism theorems for groupoids.

Theorem 1. (the first isomorphism theorem). Let be a surjective strong groupoid homomorphism. Then there exists a strong isomorphism such that , where j is the canonical homomorphism of onto .

Proof. Let . We define as , for each . First of all, we show that is a well defined function. Indeed, assume that . Then and . That is for some , and then . Since ϕ is surjective, then , for some . Multiplying the above equation by , we have thatThen, . So , whence . Hence, is well defined.
Now, note that is a surjective strong homomorphism. Finally, we prove that is injective. Indeed, assume that , that is, . Then, as ϕ is strong we have that . Thus, and we have that .

Example 3. (1)Consider the identity function of the groupoid . Then, it is clear that is a surjective strong homomorphism and . Thus, by the first isomorphism theorem, we obtain that .(2)Consider the function , defined by for all . For suppose that . Then, , , and Now, let such that . Then, which implies that and since we obtain . In conclusion, θ is a strong homomorphism, with and . Whence, by the first isomorphism theorem, we obtain .(3)Let and be a groupoids. The set is a groupoid with the product defined by iff , and in this case . Moreover, note that . If and , then and . Indeed, it is clear that For the second affirmation, define by , and note that ψ is a strong homomorphism. Moreover,Thus, by the first isomorphism theorem the result follows.

Theorem 2. (the second isomorphism theorem). Let be a groupoid, a wide subgroupoid of , and a normal subgroupoid of . Then, and

Proof. First, note that by (i) of Proposition 11, is a subgroupoid of . Moreover, since we have . Also, it is clear that .
We consider given by for all . Then, it is clear that ψ is a strong homomorphism. Furthermore, if , then . Thus, ψ is surjective. Now,On the other hand, . Indeed, if , then and thus . For the other inclusion, if , then and for some . Thus, and . That is, , and we have . Finally, by Theorem 1, we conclude that as desired.

Remark 8. Given M and N as in Theorem 2, we saw in the proof of the same theorem that which implies that Indeed, let . By Proposition 4, there is such that . Since with then Conversely, the condition clearly implies that From this, we conclude that for and subgroupoids of we have that if and only if

Theorem 3. (the third isomorphism theorem). Let be a groupoid, and with . Then, and

Proof. Define by . First of all, we show that φ is a well-defined function. Indeed, if , then and . Since , we have , and hence . Now,Thus, and Theorem 1 implies that

4. Normal and Subnormal Series for Groupoids

In this section, we present some applications of the isomorphism theorems of groupoids to normal and subnormal series. In particular, we show that the Jordan-Hölder Theorem is also fulfilled in the context of groupoids. First, we introduce the following natural definitions.

Definition 10. Let be a groupoid. Then,(i)A subnormal series of a groupoid , is a chain of subgroupoids such that is normal in for . The factors of the series are the quotient groupoids . The length of the series is the number of strict inclusions. A subnormal series such that is normal in for all i, is called normal.(ii)Let be a subnormal series. A one-step refinement of this series is any series of the form or , where N is a normal subrgoupoid of and is normal in N (if ). A refinement of a subnormal series S is any subnormal series obtained from S by a finite sequence of one-step refinements. A refinement of S is called to be proper if it is larger than the length of S.(iii)A subnormal series is a composition series if each factor is simple; that is, its only normal subgroupoids are and and it is solvable if each factor is abelian.

Remark 9. It follows from (iv) of Proposition 10 that if is a normal subgroupoid of a groupoid , every normal subgroupoid of is of the form where is a normal subroupoid of , which contains . Thus, if , then is simple, if and only if is a maximal element in the set of all the normal subgroupoids of , such that .

Proposition 12. Let be a groupoid. Then,(i)If is finite, then it has a composition series(ii)Every refinement of a solvable series is a solvable series(iii)A subnormal series is a composition series, if and only if it has no proper refinements

Proof. (i)Let be a maximal normal subgroupoid of . Then, is simple by (iv) of Proposition 10. Let be a maximal normal subgroupoid of , and so on. Now, since is finite, this process must end with . Thus, is a composition series.(ii)Here we use Remark 7 to observe that if is abelian and , then is abelian since it is a subgroupoid of . Moreover, is abelian since it is isomorphic to by Theorem 3.(iii)It follows from (iv) of Proposition 10 and that a subnormal series has a proper refinement, if and only if there is a subgroupoid such that for some with proper in and proper in .

Definition 11. Two subnormal series and of a groupoid are equivalent, if there is a one-to-one correspondence between the nontrivial factors of and the nontrivial factors of , such that the corresponding factors are isomorphic groupoids.

Lemma 3. If is a composition series of a groupoid , then any refinement of is equivalent to .

Proof. Let . By Proposition 12 (iii), has no proper refinement. Thus, the only possible refinements of are obtained by inserting additional copies of each . Whence, any refinement of has exactly the same nontrivial factors as . Therefore, it is equivalent to .

Lemma 4. (Zassenhaus theorem for groupoids). Let be wide subgroupoids of a groupoid such that(i) is normal in (ii) is normal in Then, and are subgroupoids of such that(i) is a normal subgroupoid of (ii) is a normal subgroupoid of (iii)

Proof. (i) Since is normal in , is a normal subgroupoid of thanks to (iii) of Proposition 11; similarly, is normal in . Then, is a normal subgroupoid of by (ii) of Proposition 11. Also, by this same proposition, we have that and are subgroupoids of and respectively. Now, we definefor all . The map τ is well defined since with , implies thatwhence . The map τ is clearly a strong epimorphism, and the equality is shown in an analogous way to the group case.
Thus, Proposition 9 (iv) implies that is normal in , and by the first isomorphism theorem, we get .
A symmetric argument shows that is normal in and . Whence (iii) follows.

Proposition 13. (Schreier theorem for groupoids). Any two subnormal (resp. normal) series of a groupoid have subnormal (resp. normal) refinement, which are equivalent.

Proof. It follows from Lemma 4, (ii) of Proposition 11, and Proposition 7 (1).

Proposition 14. (Jordan-Hölder theorem for groupoids). Any two composition series of a groupoid are equivalent.

Proof. It follows from Proposition 13 and Lemma 3.

4.1. Some Remarks on the Equivalence between Inductive Groupoids and Inverse Semigroups

Recall that an inverse semigroup, is a semigroup S such that for any there is a unique such that and Now, let X be a set and consider the inverse semigroup

We recall the following.

Definition 12. Let S be a semigroup. An action of S on X is a semigroup homomorphism
It follows from Theorem 4.2 of [23]) that partial actions of a group on X are in one-to-one correspondence with actions of on where is the semigroup generated by the symbols under the following relations: for ,The semigroup was introduced in [23], and it is called the Exel semigroup of.

Remark 10. Now, we present some facts about (1)The semigroup is a monoid with (2)([23], Proposition 2.5). For each let . Then, is an idempotent of , each element may be uniquely written (up to the order of the ’s) as for some with and for From equation (25), it follows that any idempotent in has the form for some (uniquely) (3)([23], Theorem 3.4) The set is an inverse semigroup. In particular, the idempotents of commute (see [22], Theorem 3)).Given an inverse semigroup S and , one defines the restricted productIt follows from Proposition 3.1.4 and Proposition 4.1.1 that is an inductive groupoid (see [22], p. 108), where is the natural partial order defined on Then, by using the restricted product in , we have thatand is a groupoid with the product given by composition of maps restricted to Moreover,
With respect to the semigroup , we have the following result.

Proposition 15. Let and where are as in (25) of Remark 10. Then if and only if

Proof. We have that Then,where the last equivalence follows from (2) of Remark 10.
Using the restricted product to provide with a groupoid structure we get by Proposition 15 that,and
From [7], it follows that a global action β of a groupoid on X is a family of bijections such that(i)(ii) for all (iii) for all Then according to ([7], Proposition 10), global actions of on X correspond to groupoid homomorphism On the other hand, in the case when is a group we obtain the definition of a partial group action on a set (see [2], Definition 1.2)
If is an inductive groupoid, then Proposition 4.1.7 of [22]) implies that is an inverse semigroup, where denotes the pseudo product defined on (see [22], p. 112).
Then, we have the following.

Proposition 16. For every group G and any set X, there is a one-to-one correspondence between(1)Partial actions of on X(2)Unital semigroup actions of on (3)Groupoid homomorphisms (4)Groupoid actions of on

Proof. We have already observed that there is a one-to-one correspondence between partial actions of on X and semigroup actions of on and between groupoid homomorphisms and global actions of on Moreover, given a semigroup action then let and Since , one has that the family is a global action of on Conversely, given a global action of on let . Then, φ is an action of on Indeed, if , then by Proposition 4.1.7, we have that and as desired.

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Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.