arXiv subjects

N. Martins-Ferreira

Publications and source records attributed to N. Martins-Ferreira.

12 recordsLinked to original sources

arXiv2026

On the zero-classes of monoid semi-congruences

This paper studies the zero-classes of monoid semi-congruences, understood as internal reflexive relations on a monoid. Classical examples include normal submonoids, which arise as zero-classes of congruences, and positive cones, which are the zero-classes of preorders; both admit well-known syntactic characterizations via the Eilenberg syntactic equivalence and the syntactic preorder introduced by Pin, respectively. Beyond these cases, however, no general notion of a syntactic object characterizing zero-classes of semi-congruences, also called clots, has been established. We address this gap by introducing a syntactic relation that is reflexive and characterizes clots whenever it is compatible with the monoid operation, a property that is not automatic in contrast to the congruence and preorder settings. We further develop a hierarchy of conditions on submonoids of a given monoid that induce syntactic relations with progressively stronger properties: first ensuring compatibility with the monoid operation, and subsequently enforcing additional relational properties such as transitivity and symmetry. Several illustrative situations are discussed, including the cases of positive cones and normal submonoids.

math.CT↗

arXiv2022

A refinement of ternary Boolean algebras

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 to distinct subvarieties. Rings and near-rings of characteristic 2 are used to illustrate the procedure.

math.RA↗

arXiv2021

A new look at Ternary Boolean Algebras

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.

math.LO↗

arXiv2021

Affine mobi spaces

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 pointed affine mobi spaces over a mobi algebra R as soon as R is a unitary ring in which 2 is an invertible element.

math.RA↗

arXiv2021

Mobility spaces and geodesics for the n-sphere

math.GT↗

arXiv2020

Mobility spaces and their geodesic paths

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 constants and a ternary operation. In the case of the closed unit interval $A=[0,1]$, the three constants are 0, 1 and 1/2 while the ternary operation is $p(x,y,z)=x-yx+yz$. A mobility space is a set $X$ together with a map $q\colon{X\times A\times X\to X}$ with the meaning that $q(x,t,y)$ indicates the position of a particle moving from point $x$ to point $y$ at the instant $t\in A$, along a geodesic path within the space $X$. A mobility space is thus defined with respect to a mobility algebra, in the same way as a module is defined over a ring. We introduce the axioms for mobility spaces, investigate the main properties and give examples. We also establish the connection between the algebraic context and the one of spaces with geodesic paths. The connection with affine spaces is briefly mentioned.

math.GM↗

arXiv2019

Cancellative conjugation semigroups and monoids

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 from X to B which induce a reflexive graph, an internal category or an internal groupoid. We describe Schreier split epimorphisms in terms of external actions and consider the notions of precrossed semimodule, crossed semimodule and crossed module in the context of cancellative conjugation monoids. In this category we prove that a relative version of the so-called "Smith is Huq" condition for Schreier split epimorphisms holds as well as other relative conditions.

math.CT↗

arXiv2018

Mobi algebra as an abstraction to the unit interval and its comparison to rings

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 from a mobi algebra. We prove several results on the interplay between mobi algebras, IMM algebras and unitary rings. It turns out that every unitary ring with one half uniquely determines and is uniquely determined by a mobi algebra with one double. This paper is the second of a planned series of papers dedicated to the study of geodesic paths from an algebraic point of view.

math.RA↗

arXiv2016

What is an ideal a zero-class of?

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.

math.CT↗

arXiv2014

On the notion of pseudocategory internal to a category with a 2-cell structure

The notion of pseudocategory, as considered in [11], 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 form internal transformations in internal categories, conjugations in groups, derivations in crossed-modules or homotopies in abelian chain complexes are studied in this context, namely their behaviour as abstract 2-cells in a 2-cell structure. Issues such as naturality of a 2-cell structure are investigated. This article is intended as a preliminary starting work towards the study of the geometrical aspects of the 2-cell structures from an algebraic point of view.

math.CT↗

arXiv2013

From A-spaces to arbitrary spaces via spatial fibrous preorders

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.

math.CT↗

arXiv2009

Internal precategories relative to split epimorphisms

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. We study the additive case, where the split epimorphisms are "coproduct projections", and the semi-additive case where split epimorphisms are "semi-direct product projections". The result is a generalization of the well known equivalence between precategories and 2-chain complexes. We also consider an abstract setting, containing, for example, strongly unital categories.

math.CT↗