Browsing by Author "Montoli, Andrea"
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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.
- 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 categorical behaviour of preordered groupsPublication . Clementino, Maria Manuel; Martins-Ferreira, Nelson; Montoli, AndreaWe study the categorical properties of preordered groups. We first give a description of limits and colimits in this category, and study some classical exactness properties. Then we point out a strong analogy between the algebraic behaviour of preordered groups and monoids, and we apply two different recent approaches to relative categorical algebra to obtain some homological properties of preordered groups.
- 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)).
- 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.