Maria Manuel Clementino’s research while affiliated with MIT Portugal and other places

The aim of this work is to further develop the calculus of (internal) relations for a regular Ord-category ℂ. To capture the enriched features of a regular Ord-category and obtain a good calculus, the relations we work with are precisely the ideals in ℂ. We then focus on an enriched version of the 1-dimensional algebraic 2-permutable (also called Mal’tsev) property and its well-known equivalent characterizations expressed through properties on ordinary relations. We introduce the notion of Ord-Mal’tsev category and show that these may be characterized through enriched versions of the above-mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional Mal’tsev category is necessarily an Ord-Mal’tsev category. We also give some examples of categories which are not Mal’tsev categories, but are Ord-Mal’tsev categories.

This paper investigates the interplay between properties of a topological space X, in particular of its natural order, and properties of the lax comma category $\mathsf{Top} \Downarrow X$, where $\mathsf{Top}$ denotes the category of topologicalspaces and continuous maps. Namely, it is shown that, whenever X is a topological $\bigwedge$-semilattice, the canonical forgetful functor $\mathsf{Top} \Downarrow X \to \mathsf{Top}$ is topological, preserves and reflects exponentials, and preserves effective descent morphisms. Moreover, under additional conditions on X, a characterisation of effective descent morphisms is obtained.

We study categorical properties of right-preordered groups, giving an explicit description of limits and colimits in this category, studying some exactness properties, and showing that it is a quasivariety. We show that, from an algebraic point of view, the category of right-preordered groups shares several properties with the one of monoids. Moreover, we describe split extensions of right-preordered groups, showing in particular that semidirect products of ordered groups always have a natural right-preorder.

We characterize effective descent morphisms of what we call filtered preorders, and apply these results to slightly improve a known result, due to the first author and F. Lucatelli Nunes, on the effective descent morphisms in lax comma categories of preorders. A filtered preorder, over a fixed preorder X, is defined as a preorder A equipped with a profunctor X→A$X\rightarrow A$ and, equivalently, as a set A equipped with a family (Ax)x∈X$(A_x)_{x\in X}$ of upclosed subsets of A with x′⩽x⇒Ax⊆Ax′$x'\leqslant x\Rightarrow A_x\subseteq A_{x'}$.

We present a characterization of effective descent morphisms in the lax comma category $\mathsf{Ord}//X$ when X is a locally complete ordered set with a bottom element.

The aim of this work is to further develop the calculus of (internal) relations for a regular Ord-category C. To capture the enriched features of a regular Ord-category and obtain a good calculus, the relations we work with are precisely the ideals in C. We then focus on an enriched version of the 1-dimensional algebraic 2-permutable (also called Mal'tsev) property and its well-known equivalent characterisations expressed through properties on ordinary relations. We introduce the notion of Ord-Mal'tsev category and show that these may be characterised through enriched versions of the above mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional Mal'tsev category is necessarily an Ord-Mal'tsev category. We also give some examples of categories which are not Mal'tsev categories, but are Ord-Mal'tsev categories.

We study the categorical properties of right-preordered groups, giving an explicit description of limits and colimits in this category, and studying some exactness properties. We show that, from an algebraic point of view, the category of right-preordered groups shares several properties with the one of monoids. Moreover, we describe split extensions of right-preordered groups, showing in particular that semidirect products of ordered groups have always a natural right-preorder.

The aim of this work is to compare the distinct notions of Mal’tsev object in the sense of Weighill and in the sense of Montoli–Rodelo–Van der Linden.