Center for Mathematics, University of Coimbra

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The Nine Lemma and the push forward construction for special Schreier extensions of monoids with operations

Publication . Martins-Ferreira, Nelson; Montoli, Andrea; Sobral, Manuela

We 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.

2018-08-10Journal article

Open access

On the categorical behaviour of preordered groups

Publication . Clementino, Maria Manuel; Martins-Ferreira, Nelson; Montoli, Andrea

We 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.

2019Journal article

Open access

On some categorical-algebraic conditions in S-protomodular categories

Publication . Martins-Ferreira, Nelson; Montoli, Andrea; Sobral, Manuela

In 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.

2017-09-01Journal article

Open access

On the “Smith is Huq” Condition in S-Protomodular Categories

Publication . Martins-Ferreira, Nelson; Montoli, Andrea

We study the so-called “Smith is Huq” condition in the context of S-protomodular categories: two S-equivalence relations centralise each other if and only if their associated normal subobjects commute. We prove that this condition is satisfied by every category of monoids with operations equipped with the class S of Schreier split epimorphisms. Some consequences in terms of characterisation of internal structures are explored.

2017-02Journal article

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Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

Publication . Branquinho, A.; Moreno, A. Foulquié; Mendes, A.

In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a com plex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, general ized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattice.

2016-11-11Journal article

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