Related papers: A note on the atomicity of arithmeticity
Matrix conditions extend linear Mal'tsev conditions from Universal Algebra to exactness properties in Category Theory. Some can be stated in the finitely complete context while, in general, they can only be stated for regular categories. We…
We introduce the notion of a majority category --- the categorical counterpart of varieties of universal algebras admitting a majority term. This notion can be thought to capture properties of the category of lattices, in a way that…
Matrix properties are a type of property of categories which includes the ones of being Mal'tsev, arithmetical, majority, unital, strongly unital and subtractive. Recently, an algorithm has been developed to determine implications…
This paper is concerned with the taxonomy of finitely complete categories, based on 'matrix properties' - these are a particular type of exactness properties that can be represented by integer matrices. In particular, the main result of the…
In universal algebra, it is well known that varieties admitting a majority term admit several Mal'tsev-type characterizations. The main aim of this paper is to establish categorical counterparts of some of these characterizations for…
This paper provides a short introduction to the notion of regular category and its use in categorical algebra. We first prove some of its basic properties, and consider some fundamental algebraic examples. We then analyse the algebraic…
In a recent paper, an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices -- the so called `matrix properties'. In this paper we extend this algorithm to…
The goal of this note is to present Kaplansky's proof of the Regular Element Property and to explain how this argument can be adapted to the case of a coherent, strongly discrete and Noetherian (with an inductive definition of Noetherian)…
In this note we give an elementary demonstration of the fact that AB=I implies BA=I for square matrices A,B with coefficients in a field K. By elementary we mean that our proof follows from the very definitions of matrix and product of a…
In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set X which takes values in a suitable non-Archimedean field, and satisfies the same formal properties of finite…
This note presents several conditions to characterize real matrix similarity between a Hurwitz matrix (and then more generally, a real square matrix) and a diagonal dominant matrix.
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…
We are interested in investigating some definitions and assumptions stated in [4], in particular the notions of measurability and atomicity that the two authors used in order to give a representation for multiplicative linear functionals…
We introduce and explore the Uniform Izumi-Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if $R$ is a normal domain…
In these notes, we present a general result concerning the Lipschitz regularity of a certain type of set-valued maps often found in constrained optimization and control problems. The class of multifunctions examined in this paper is…
We first establish several general properties of modality of algebraic group actions. In particular, we introduce the notion of a modality-regular action and prove that every visible action is modality-regular. Then, using these results, we…
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…
We study atom canonicity for several varieties of cylindric like algebras that contain properly the variety of representable algebras. The algebras in such varieties have relativized representations, and we thereby obtain many omitting…
Following and generalizing a construction by Kontsevich, we associate a zeta function to any matrix with entries in a ring of noncommutative Laurent polynomials with integer coefficients. We show that such a zeta function is an algebraic…
The goal of this note is to provide a geometric setting in which generalized arithmetic means are best predictors in an appropriate metric. This characterization provides a geometric interpretation to the concept of certainty equivalent.…