Nelson Martins-Ferreira

• PhD in Mathematics
• Professor (Assistant) at Polytechnic Institute of Leiria

A generalization to the categorical notion of biproduct, called semibiproduct, which in the case of groups covers classical semidirect products, has recently been analysed in the category of monoids with surprising results in the classification of weakly Schreier extensions. The purpose of this paper is to extend the study of semibiproducts to the...

Although algebraic structures are frequently analyzed using unary and binary operations, they can also be effectively defined and unified through ternary operations. In this context, we introduce structures that contain two constants and a ternary operation. We demonstrate that these structures are isomorphic to various significant algebraic system...

We study fibrations arising from indexed categories of the following form: fix two categories $\mathcal{A},\mathcal{X}$ and a functor $F : \mathcal{A} \times \mathcal{X} \longrightarrow\mathcal{X} $, so that to each $F_A=F(A,-)$ one can associate a category of algebras $\mathbf{Alg}_\mathcal{X}(F_A)$ (or an Eilenberg-Moore, or a Kleisli category if...

We introduce a novel concept of action for unitary magmas, facilitating the classification of various split extensions within this algebraic structure. Our method expands upon the recent study of split extensions and semidirect products of unitary magmas conducted by Gran, Janelidze, and Sobral. Building on their research, we explore split extensio...

When a category is equipped with a 2-cell structure it becomes a sesquicategory but not necessarily a 2-category. It is widely accepted that the latter property is equivalent to the middle interchange law. However, little attention has been given to the study of the category of all 2-cell structures (seen as sesquicategories with a fixed underlying...

The category of internal groupoids (in an arbitrary category) is shown to be equivalent to the full subcategory of so called involutive-2-links that are unital and associative.

Additive manufacturing (AM), also known as three-dimensional (3D) printing, allows the fabrication of complex parts, which are impossible or very expensive to produce using traditional processes. That is the case for dinnerware and artworks (stoneware, porcelain and clay-based products). After the piece is formed, the greenware is fired at high tem...

Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to define and study similar classes of split extensions in general algebraic structures (parameterised by a term θ\documentclass[12pt]{minimal} \usepacka...

Regardless of its environment, the category of internal groupoids is shown to be equivalent to the full subcategory of involutive-2-links that are unital and associative. The new notion of involutive-2-link originates from the study of triangulated surfaces and their application in additive manufacturing and 3d-printing. Thus, this result establish...

While surveying some internal categorical structures and their applications, it is shown that triangulations and internal groupoids can be unified as two different instances of the same common structure, namely a multi-link. A brief survey includes the categories of directed graphs, reflexive graphs, links, multi-links, triangulations, trigraphs, m...

An answer to the question investigated in this paper brings a new characterization of internal groupoids such that: (a) it holds even when finite limits are not assumed to exist; (b) it is a full subcategory of the category of involutive-2-links, that is, a category whose objects are morphisms equipped with a pair of interlinked involutions. This r...

Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to define and study similar classes of split extensions in general algebraic structures (parameterised by a term $\theta$). These generalise weakly Schre...

In the category of monoids we characterize monomorphisms that are normal, in an appropriate sense, to internal reflexive relations, preorders or equivalence relations. The zero-classes of such internal relations are first described in terms of convenient syntactic relations associated to them and then through the adjunctions associated with the cor...

A generalization to the categorical notion of biproduct, called semibiproduct, which in the case of groups covers classical semidirect products, has recently been analysed in the category of monoids with surprising results in the classification of weakly Schreier extensions. The purpose of this paper is to extend the study of semibiproducts to the...

We define the product of admissible abstract kernels of the form [Formula: see text], where [Formula: see text] is a monoid, [Formula: see text] is a group and [Formula: see text] is a monoid homomorphism. Identifying [Formula: see text]-equivalent abstract kernels, where [Formula: see text] is the center of [Formula: see text], we obtain that the...

Semibiproducts of monoids are introduced here as a common generalization to biproducts (of abelian groups) and to semidirect products (of groups) for exploring a wide class of monoid extensions. More generally, abstract semibiproducts exist in any concrete category over sets in which map addition is meaningful thus reinterpreting Mac Lane's relativ...

The category of mobi algebras has been introduced as a model to the unit interval of real numbers. The notion of mobi space over a mobi algebra has been proposed as a model for spaces with geodesic paths. In this paper we analyse the particular case of affine mobi spaces and show that there is an isomorphism of categories between modules over unita...

An algebraic structure with two constants and one ternary operation, which is not completely commutative, is put forward to accommodate ternary Boolean algebras. When the ternary operation is interpreted as Church's conditioned disjunction, Boolean algebras are characterized as a subvariety. Different interpretations for the ternary operation lead...

The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spatial. A special class of spatial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary...

We present a new approach to ternary Boolean algebras in which negation is derived from the ternary operation. The key aspect is the replacement of complete commutativity by other axioms that do not require the ternary operation to be symmetric.

It is shown that the category of \emph{semi-biproducts} of monoids is equivalent to the category of \emph{pseudo-actions}. A semi-biproduct of monoids is a new notion, obtained through generalizing a biproduct of commutative monoids. By dropping commutativity and requiring some of the homomorphisms in the biproduct diagram to be merely identity-pre...

The category of mobi algebras has been introduced as a model to the unit interval of real numbers. The notion of mobi space over a mobi algebra has been proposed as a model for spaces with geodesic paths. In this paper we analyse the particular case of affine mobi spaces and show that there is an isomorphism of categories between R-modules and poin...

The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A special class of spacial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary...

We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. We show that there is a strong connection between modules over a ri...

Properties of preordered monoids are investigated and important subclasses of such structures are studied. The corresponding full subcategories are related between them by appropriate functors as well as with the categories of preordered sets and of monoids. Schreier split extensions are described in the full subcategory of preordered monoids whose...

We show that the category of cancellative conjugation semigroups is weakly Mal’tsev and give a characterization of all admissible diagrams there. In the category of cancellative conjugation monoids we describe, for Schreier split epimorphisms with codomain B and kernel X, all morphisms \(h:X\rightarrow B\) which induce a reflexive graph, an interna...

Properties of preordered monoids are investigated and important subclasses of such structures are studied. The corresponding full subcategories of the category of preordered monoids are functorially related between them as well as with the categories of preordered sets and monoids. Schreier split extensions are described in the full subcategory of...

In this paper we give unified characterizations of categories defined by variations of the Mal’tsev property.

It is shown that the category of semi-biproducts in monoids is equivalent to a category of pseudo-actions. A semi-biproduct in monoids is at the same time a generalization of a semi-direct product in groups and a biproduct in commutative monoids. Every Schreier extension of monoids can be seen as an instance of a semi-biproduct; namely a semi-bipro...

We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel {\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}} . If an abstract kernel factors through {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} , where {\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A...

We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. Mobility algebras consist on a set $A$ together with three constant...

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 algebr...

This work is concerned with 3D printing. It's main goal is to establish a conceptualsetting in which the theory of 3D printing can be developed. Following the analogy that aUniversal Turing Machine, in a veryprecise and specific way, computes everything that is computable, we propose to the development for the notion of a Universal PrintingMachine....

A classification theorem for three different sorts of Mal'tsev categories is proven. The theorem provides a classification for Mal'tsev category, naturally Malt'sev category, and weakly Mal'tsev category in terms of classifying classes of spans. The class of all spans characterizes naturally Mal'tsev categories. The class of relations (i.e. jointly...

We show that the category of conjugation semigroups with cancellation is weakly Mal'tsev and give a characterization of all admissible diagrams there. In the subcategory of conjugation monoids with cancellation we describe, for Schreier split epimorphisms with codomain B and kernel X, all morphisms h from X to B which induce a reflexive graph, an i...

We introduce and compare several new exactness conditions involving what we call split cubes. These conditions are motivated by their special cases, some which become familiar, in the pointed context, once we reformulate them with split cubes replaced with split extensions.

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.

This paper presents a computational method to extract optimum rectangular Regions of Interest (RoI) in images with an associated saliency map. Although saliency maps provide an individual relevance measure for each pixel, to find the sub-image (i.e., rectangular region) that contains the set of the most relevant pixels requires an optimisation proc...

In this work we study the notion of Whitehead sequence in the category of crossed modules and actions of crossed modules. As expected, Whitehead sequences in that context are the same as crossed squares. We investigate under which conditions a Whitehead sequence of crossed modules gives rise to an internal groupoid in the category of crossed module...

We begin by introducing an algebraic structure with three constants and one ternary operation to which we call mobi algebra. This structure has been designed to capture the most relevant properties of the unit interval that are needed in the study of geodesic paths. Another algebraic structure, called involutive medial monoid (IMM), can be derived...

We begin by introducing an algebraic structure with three constants and one ternary operation to which we call mobi algebra. This structure has been designed to capture the most relevant properties of the unit interval that are needed in the study of geodesic paths. Another algebraic structure, called involutive medial monoid (IMM), can be derived...

In this paper we propose a procedural approach to the problem of image segmentation. This procedural approach may be parameterized with optimized values for a specific type of data. It uses an algorithm which is inspired by the superposition principle of Quantum Mechanics, in the sense that each particular pixel has a certain probability of being o...

A detailed description of a normalized internal bicategory in the category of groups is derived from the general description of internal bicategories in weakly Mal’tsev categories endowed with a V-Mal’tsev operation in the sense of Pedicchio. The example of bicategory of paths in a topological abelian group is presented.

A mathematical model is being considered within the context of the movement of water molecules in space. It aims to identify the three different phases in terms of their position in space and it is related with the hydrogen bonding effect.

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...

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...

We 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.

Traditionally, additive manufacturing (AM) uses digital information from a computer-aided design (CAD) file that is later converted to a stereolithography (STL) file format in order to produce a physical part. The sur- faces of a 3D computer model are approximated by a set of triangular facets and then sliced to generate a series of layers for fabr...

In this paper, we introduce and study a new exactness property in the sense of categorical algebra, which can be seen as a natural strengthening of the well-known Mal’tsev property. A variety of universal algebras has this exactness property if and only if its algebraic theory contains binary terms a1,…,an and an (n + 1)-ary term d satisfying d(a1(...

We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the "Smith is Huq" condition: two surjective left split relations commute as soon as their zero-classes commute.

We 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...

Traditionally, direct digital manufacturing (DDM) uses digital information from a computer-aided design (CAD) file that is later converted to a stereo-lithography (STL) file format in order to produce a physical part. The surfaces of a 3D computer model are approximated by a set of triangular facets and then sliced to generate a series of layers fo...

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. Som...

A detailed description of internal bicategory in the category of groups is derived from the general description of internal bicategories in weakly Mal'tsev sesquicategories. The example of bicategory of paths in a topological abelian group is presented, as well as a description of internal bicategories in weakly Mal'tsev categories with a $V$-Mal't...

We introduce the notion of Whitehead sequence which is defined for a base category together with a system of abstract actions over it. In the classical case of groups and group actions the Whitehead sequences are precisely the crossed-modules of groups. For a general setting we give sufficient conditions for the existence of a categorical equivalen...

We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the Smith is Huq condition: two surjective left split relations commute as soon as their zero-classes commute.

The notion of pseudocategory is extended from the context of a 2-category to the more general one of a sesquicategory, which is considered as a category equipped with a 2-cell structure. Some particular examples of 2-cells arising from internal transformations in internal categories, con- jugations in groups, derivations in crossed-modules or homot...

In this work we present a vectorized Matlab algorithm for the decomposition of an endomap into its finite orbits.

Given an arbitrary region on the plane, modeled as a graph with a symmetry, we describe a procedure to find the best orientation for slicing, via a zigzag tool-path trajectory based curve, in order to minimize its discontinuities. We also develop some directions on how to generalize the procedure up to the level of optimizing tool-path trajectories...

ScienceDirect 2212-0173 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of CENTERIS 2014. Abstract We present a method for extracting a set of descriptors from snail images tha...

We decompose the weighted subobject commutator of M. Gran, G. Janelidze and A. Ursini as a join of a binary and a ternary commutator.

This article considers the category of commutative medial magmas with cancellation, a structure that generalizes midpoint algebras and commutative semigroups with cancellation. In this category each object admits at most one internal monoid structure for any given unit. Conditions for the existence of internal monoids and internal groups, as well a...

The following classes of categories are shown to be weakly Mal’tsev in the sense of the author: (i) a suitable class of algebras with cancellation; (ii) the dual of any quasi-adhesive category; (iii) the dual of any extensive category with pullback-stable epimorphisms; (iv) the dual of any solid quasi-topos. The examples in (i) include all the Mal’...

We study the difference between internal categories and internal groupoids in terms of generalised Mal'tsev properties---the weak Mal'tsev property on the one hand, and n-permutability on the other. In the first part of the article we give conditions on internal categorical structures which detect whether the surrounding category is naturally Mal't...

We explore some properties of Schreier split epimorphisms between monoids, which correspond to monoid actions. In particular, we prove that the split short five lemma holds for monoids, when it is restricted to Schreier split epimorphisms, and that any Schreier reflexive relation is transitive, partially recovering in monoids a classical property o...

We prove that in a regular category all reflexive and transitive relations are symmetric if and only if every internal category is an internal groupoid. In particular, these conditions hold when the category is n-permutable for some n.

We compare the 'Smith is Huq' condition (SH) with three commutator conditions in semi-abelian categories: first an apparently weaker condition which arose in joint work with Bourn and turns out to be equivalent with (SH), then an apparently equivalent condition which takes commutation of non-normal subobjects into account and turns out to be strong...

This is a progress report of a project called BioFab Toolbox, funded by the Portuguese National Science Foundation, consisting of the development of a collection of software tools to assist in the process of rapid prototyping and rapid manufacturing of new products. This project is especially designed to produce scaffolds, grafts, implants and othe...

Given any two regions A, B in the plane, defined by polygonal (simple, closed and oriented) curves, associated with their respective boundaries, we describe a procedure to compute the symmetric difference A ⊕ B. The output is also presented in the form of polygonal curves, where in particular the curves describing the union A∪B, the intersection A∩...

This is a report on the progress of the project BioFab Toolbox (PTDC/EME-CRO/120585/2010), funded by the Portuguese National Science Foundation (FCT). It consists in the development of a collection of software tools to assist the process of rapid prototyping and rapid manufacturing of new products. It is especially designed to produce scaffolds, gr...

Agile-CAD is a reverse engineering computer tool that allows the recovering of outer and inner 3D shapes of existing objects of revolution with constant thickness, using silhouettes computed from one or two digital views. In this context, some improved algorithms were developed in order to provide more accurate and robust tridimensional reconstruct...

In this paper we study a generalization of the notion of categorical semidirect product, as defined in [6], to a non-protomodular context of categories where internal actions are induced by points, like in any pointed variety. There we define semidirect products only for regular points, in the sense we explain below, provided the Split Short Five L...

The well known equivalence between preorders and Alexandrov spaces is extended to an equivalence between arbitrary topological spaces and spatial fibrous preorders, a new notion to be introduced.

We 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.

We give a characterisation of the ``Smith is Huq'' condition for a pointed Mal'tsev category $\mathbb C$ by means of a property of the fibration of points $\P_{\mathbb C}\colon{\Pt(\mathbb C)\rightarrow \mathbb C}$, namely: any change of base functor $h^*\colon {\Pt_Y(\mathbb C) \rightarrow \Pt_X(\mathbb C)}$ reflects commuting of normal subobjects...

Given multiple identical polyhedral objects and a parallelepiped container, how should one place the objects so that the largest number fits inside the container? This simple question is important in many applications, yet the answer is elusive. In fact, we know of no published solution for this very general formulation. Still, in many circumstance...

We characterize those varieties of universal algebras where every split epimorphism considered as a map of sets is a product projection. In addition we obtain new characterizations of protomodular, unital and subtractive varieties as well as varieties of right omega-loops and biternary systems.

We prove that a variety of lattices is weakly Mal’tsev if and only if it is a variety of distributive lattices.

Necessary 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.

We define a strong relation in a category C to be a span which is "orthog-onal" to the class of jointly epimorphic pairs of morphisms. Under the presence of finite limits, a strong relation is simply a strong monomorphism R → X × Y . We show that a category C with pullbacks and equalizers is a weakly Mal'tsev category if and only if every reflexive...

We show that two known conditions which arose naturally in commutator theory and in the theory of internal crossed modules coincide: every star-multiplicative graph is multiplicative if and only if every two effective equivalence relations commute as soon as their normalisations do. This answers a question asked by George Janelidze.

The slicing and tool-path generation are two fundamental processing steps in rapid prototyping/rapid manufacturing (RP/RM), as well as in CAM. Nevertheless, the specific requirements of the diverse RP/RM techniques are not always addressed by existing tools for CAM. We present a set of efficient and robust algorithms for processing STL files in RP/...

We describe the (tetra) category of pseudo-categories, pseudo-functors, natural transformations, pseudo-natural transformations, and modifications, as introduced in Martins-Ferreira (JHRS 1:47–78, 2006), internal to an additive 2-category with kernels, as formalized in Martins-Ferreira (Fields Inst Commun 43:387–410, 2004). In the context of a 2-Ab...

We show that two known conditions which naturally arose in commu- tator theory and in the theory of internal crossed modules coincide: every star- multiplicative graph is multiplicative if and only if every two eective equivalence

Resumo. Neste artigo debruçar-nos-emos sobre o problema do varrimento de uma região compacta por trajectórias de traço contínuo. Após apresen-tar uma denição rigorosa do problema concreto, identicamos uma estru-tura abstracta, derivada do problema geral, a qual permitirá obter, por um lado soluções simplicadas para o problema, e proporcionando por...

Protomodularity, in the pointed case, is equivalent to the Split Short Five Lemma. It is also well known that this condition implies that every internal category is in fact an internal groupoid. In this work, this is condition (II) and we introduce two other conditions denoted (I) and (III). Under condition (I), every multiplicative graph is an int...

For a given category B we are interested in studying internal categorical structures in B. This work is the starting point, where we consider reflexive graphs and precategories (i.e., for the purpose of this note, a simplicial object truncated at level 2). We introduce the notions of reflexive graph and precategory relative to split epimorphisms. W...

For a given (fixed) category, we consider the category of all 2-cell structures (over it) and study some naturality properties. A category with a 2-cell structure is a sesquicategory; we use additive notation for the vertical composition of 2-cells; instead of a law for horizontal composition we consider a relation saying which pairs of 2-cells can...

A PRESENTE INVENÇÃO RELATA UM NOVO PROCESSO E EQUIPAMENTO PARA FABRICO RÁPIDO POR BIOEXTRUSÃO, DESTINADO À PRODUÇÃO DE MATRIZES DE SUPORTE (SCAFFOLDS) MONO E MULTI-MATERIAL CONTENDO OU NÃO CÉLULAS E FACTORES DE CRESCIMENTO E DESTINADA A APLICAÇÕES MÉDICAS NO CAMPO DA ENGENHARIA DE TECIDOS. A OBTENÇÃO DAS MATRIZES DE SUPORTE É EFECTUADA ATRAVÉS DO P...

We introduce a notion of weakly Mal'cev category, and show that: (a) every internal reflexive graph in a weakly Mal'tsev category admits at most one multiplicative graph structure in the sense of [10] (see also [11]), and such a structure always makes it an internal category; (b) (unlike the special case of Mal'tsev categories) there are weakly Mal...