Isomorphism Theorems for Groupoids and Some Applications

Jesús Ávila,Héctor Pinedo,Víctor Marín

doi:10.1155/2020/3967368

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.

Highlights

We present the isomorphism theorems for groupoids and their applications and obtain the corresponding version of the Zassenhaus Lemma and the Jordan-Holder theorem for groupoids
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. e Cayley theorem for groupoids is presented in eorem 3.1 of [5]
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 GBR associated with an ordered groupoid G and shows that it classifies partial actions of G 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]

Summary

Introduction

Let X be a nonempty set and X2 X × X. en X2 is a groupoid, where the product is given by: (y, z)(x, y) (x, z), for x, y, z ∈ X. en, the map f: G ∋ g ⟼ (d(g), r(g)) ∈ G20 is a strong groupoid homomorphism with kernel Iso(G). Proof (i) Consider the groupoid epimorphism φ: G ∋ g ⟼ gK ∈ G/K. en, by the definition of G/K the map φ is strong and Ker(φ) K. us, by (i) of Proposition 10 we get that KH φ− 1φ(H) is a subgroupoid of G.

Results

Conclusion