Page 87 - Textos de Matemática Vol. 45

Chapter 7
Special Schreier and special homogeneous surjections
Coming back to the category Mon of monoids, in this chapter we will study a particular class of extensions with abelian kernel, obtaining results that are similar to some classical ones known for groups. For this purpose, we will make use of the categorical approach described in [14]. In particular, we shall show in this chapter that the above mentioned class of extensions with abelian kernel has a canonical abelian group structure, on the model of what happens in the category Gp of groups (see, for example, [24] for the case of groups). In order to do that, we shall use the general categorical process developed in [9, 11].
7.1 Special Schreier surjections in Mon
Let f : X Y be a surjective homomorphism in M on. It is now rather natural
to introduce the following:
Definition 7.1.1. The surjective homomorphism f is said special Schreier when its kernel equivalence relation R[f] is a Schreier one, and special homogeneous when the equivalence relation R[f] is homogeneous.
In this case the kernel K[f] of f, being isomorphic to the kernel of the projec- tion p0 : R[f] → X, is necessarily a group (because R[f] is a Schreier internal groupoid, see Proposition 3.3.2). This rather awkward terminology is compelled by the following observations:
1) a Schreier split epimorphism is not necessarily a special Schreier surjection; 2) by Proposition 3.1.12, a special Schreier surjection which is split is a Schreier split epimorphism.
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