Abstract
In this paper we introduce some algebraic properties of subgroupoids and normal subgroupoids. we define other things, we define the normalizer of a wide subgroupoid H of a groupoid G and show that, as in the case of groups, this normalizer is the greatest wide subgroupoid of G in which H is normal. Furthermore, we provide definitions of the center Z(G) and the commutator G' of the groupoid G and prove that both of them are normal subgroupoids. We give the notions of inner and partial isomorphism of G and show that the groupoid I(G) given by the set of all the inner isomorphisms of G is a normal subgroupoid of A(G), the set of all the partial isomorphisms of G. Moreover, we prove that I(G) is isomorphic to the quotient groupoid G/Z(G), which extends to groupoids the corresponding well-known result for groups.
Highlights
The notion of a (Brandt) groupoid was first introduced in [1] from an algebraic point of view
We show that the normalizer is the greatest wide subgroupoid of the groupoid G in which H is normal (Proposition 3.3)
We prove that I(G) is a normal subgroupoid of A(G) and that it is isomorphic to the quotient groupoid G/Z(G) (Proposition 5.2)
Summary
The notion of a (Brandt) groupoid was first introduced in [1] from an algebraic point of view. In [5, Definition 1.1], Ivan follows Ehresmann’s paper [2] and presents the notion of groupoid as a particular case of a universal algebra. He defines the notion of strong homomorphism for groupoids and proves the correspondence theorem (or the fourth isomorphism theorem), in this context. The purpose of the present paper is to introduce several concepts into the theory of groupoids which are analogous to those for groups, such as, the center, normalizer, commutator, and inner isomorphism. Ing.cienc., vol 16, no. 31, pp. 7–26, enero-junio. 2020
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