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.

The Notions of Center, Commutator and Inner Isomorphism for Groupoids

“…The other isomorphism theorems also remain valid in our context of groupoids. The proofs of these theorems and several examples of them can be found in [11]. Definition 3.4.…”

“…In this sense, Paques and Tamusiunas [10] give necessary and sufficient conditions for a subgroupoid to be a normal subgroupoid and they construct the quotient groupoid. In [11], the isomorphism theorems are proven and one application of them to the normal series is presented.…”

“…In the theory of groupoids it is important to characterize those cases for which the product of two elements exists. It can be proved that if x, y ∈ G, then ∃xy if, and only if, d(x) = r(y) [11,Lemma 2.3]. The following proposition states several important properties that are fulfilled by groupoids.…”

“…[11, Proposition 2.8]. If G is a groupoid and g ∈ G, thend(g) = r(g −1 ), d(d(g)) = d(g) = r(d(g)),and d(r(g)) = r(g) = r(r(g)).…”

On functor-quotients and their isomorphism theorems

“…Related research on this area includes studies on hyperrings [11,23], polygroups [12], hypermodules [24] and general hyperalgebras [13]. Discussion on isomorphism theorems in other structures can be found in [3,14,17]. Inspired by these advances, this paper aims to study these notions and obtain standard results from universal algebra [7,10] in the context of model theory and category theory.…”