Percorrer por autor "Sobral, Manuela"
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- Baer sums of special Schreier extensions of monoidsPublication . Martins-Ferreira, Nelson; Montoli, Andrea; Sobral, ManuelaWe show that the special Schreier extensions of monoids, with abelian kernel, admit a Baer sum construction, which generalizes the classical one for group extensions with abelian kernel. In order to do that, we characterize the special Schreier extensions by means of factor sets.
- Monoids and pointed S-protomodular categoriesPublication . Bourn, Dominique; Martins-Ferreira, Nelson; Montoli, Andrea; Sobral, ManuelaWe investigate the notion of pointed S-protomodular category, with respect to a suitable class S of points, and we prove that these categories satisfy, relatively to the class S, many partial aspects of the properties of Mal’tsev and protomodular categories, like the split short five lemma for S-split exact sequences, or the fact that a reflexive S-relation is transitive. The main examples of S-protomodular categories are the category of monoids and, more generally, any category of monoids with operations, where the class S is the class of Schreier points.
- On categories with semidirect productsPublication . Martins-Ferreira, Nelson; Sobral, ManuelaNecessary and sufficient conditions for a pointed category to admit semidirect products, in the sense of Bourn and Janelidze (1998) [3], are provided and interpreted in terms of protomodularity and exactness of appropriate split chains.
- On some categorical-algebraic conditions in S-protomodular categoriesPublication . Martins-Ferreira, Nelson; Montoli, Andrea; Sobral, ManuelaIn the context of protomodular categories, several additional conditions have been considered in order to obtain a closer group-like behavior. Among them are locally algebraic cartesian closedness and algebraic coherence. The recent notion of S-protomodular category, whose main examples are the category of monoids and, more generally, categories of monoids with operations and Jo\'{o}nsson-Tarski varieties, raises a similar question: how to get a description of S-protomodular categories with a strong monoid-like behavior. In this paper we consider relative versions of the conditions mentioned above, in order to exhibit the parallelism with the "absolute" protomodular context and to obtain a hierarchy among S-protomodular categories.
- On the classification of Schreier extensions of monoids with non-abelian kernelPublication . Martins-Ferreira, Nelson; Montoli, Andrea; Patchkoria, Alex; Sobral, ManuelaWe show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel Φ: M → End(A)/Inn(A) . If an abstract kernel factors through SEnd(A)/Inn(A) , where SEnd(A) is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group U(Z(A)) of invertible elements of the center Z(A) of A, on which M acts via Φ. An abstract kernel Φ: M → SEnd(A)/Inn(A) (resp. Φ: M → Aut(A)/Inn(A) ) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero.We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel Φ: M → SEnd(A)/Inn(A) (resp. Φ: M → Aut(A)/Inn(A) ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U(Z(A)).
- Semidirect products and crossed modules in monoids with operationsPublication . Martins-Ferreira, Nelson; Montoli, Andrea; Sobral, ManuelaWe describe actions, semidirect products and crossed modules in categories of monoids with operations. Moreover we characterize, in this context, the internal categories corresponding to crossed modules. Concrete examples in the cases of monoids, semirings and distributive lattices are given.
- The Nine Lemma and the push forward construction for special Schreier extensions of monoids with operationsPublication . Martins-Ferreira, Nelson; Montoli, Andrea; Sobral, ManuelaWe show that the Nine Lemma holds for special Schreier extensions of monoids with operations. This fact is used to obtain a push forward construction for special Schreier extensions with abelian kernel. This construction permits to give a functorial description of the Baer sum of such extensions.