Page 10 - Textos de Matemática Vol. 45

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However the flexibility in question will be showed to maintain some im- portant aspects of the Mal’tsev structural organization. For instance we shall show that in Mon there are pairs of equivalence relations which centralize each other, and that a certain class of equivalence relations admits centralizers. Ex- actly in the same way of what happens in Gp we shall show, as well, that certains classes of exact sequences with abelian kernel in Mon are endowed with a natural structure of abelian group.
Beyond this, recall that the category Rng of non-commutative rings also satisfies a property dealing with classification of split exact sequences, namely action accessibility [15], of which action representability appeared to be a par- ticular case. A semiring is only a commutative monoid with an associative and distibutive multiplication. We shall show, in the same way, that the category SRng of semirings inherits a notion of Schreier split epimorphism which cap- tures, as well, the tracks of the property of action accessibility of the category Rng. Again, from this, we shall get results about centrality and centralizers of equivalence relations and again an abelian group structure on certain classes of extensions with trivial kernel. So that our approach gives a strong structural meaning to the intuitive proportion:
Mon = SRng Gp Rng
Up to now the structural attempt to characterize a group (resp. a ring), as an object in Mon (resp. in SRng), by an intrincsic property inside the category Mon (resp. in the category SRng) has failed. The notion of Schreier split epimorphism allows us to characterize groups (resp. rings) inside Mon (resp. SRng). Unfortunately the notion of Schreier split epimorphism itself is not yet showed to be quite intrinsic to Mon (resp. to SRng), it seems to need the use of the forgetful functor Mon → Set (resp. SRng → Set); among other things, we do hope that this work will constitute a step towards this expected intrinsic characterization.
This work is organized along the following lines: in Chapter 1 we briefly re- call the notion of unital category and its main properties related to the commu- tation of subobjects. In Chapter 2 we introduce Schreier (resp. homogeneous) split epimorphisms, and in Chapter 3 we introduce Schreier (resp. homoge- neous) equivalence relations, internal categories and groupoids, and in both chapters we study the first stability properties of these new tools. In Chapter 4 we investigate their remaining associated Mal’tsev aspects in terms of cen- tralization of equivalence relations. Chapter 5 is devoted to the Schreier and homogeneous representabilities and their consequences dealing with the exis- tence of centralizers. Chapter 6 develops the same kind of structural analysis for the semirings and investigates Schreier accessibility. Chapter 7 is devoted to