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Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $\mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category $\mathcal{E}$, and can be constructed as the heart $\mathcal{LH}(\mathcal{E})$ of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$ or as the localization of the category of monomorphisms in $\mathcal{E}.$ However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of $\operatorname{LB}$-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. These categories can be characterized as pre-torsionfree subcategories of abelian categories. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category $\mathcal{E}$ can be found as the heart of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$, or as the localization of the category of monomorphisms of $\mathcal{E}$. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category., 35 pages, comments welcome

La investigación tuvo como objetivo determinar la influencia de los textos lúdicos en la Producción de textos en los estudiantes del III Ciclo, Mención: Lengua y literatura de la Facultad de Educación de la UNSLG, realizamos el diagnóstico que permitió evidenciar las limitaciones para producir textos de los estudiantes y proponer el uso de los textos lúdicos de manera recreativa convencidos que el juego es un recurso de aprendizaje renovador y motivador, planteamos nuestro problema general y específicos. La investigación fue de tipo aplicada con diseño cuasi experimental, método experimental, la muestra: 55 estudiantes distribuidos en grupo experimental, 30 estudiantes y grupo de control, 25 estudiantes, luego de aplicar los instrumentos, los resultados obtenidos comprobaron la hipótesis: Los textos lúdicos influyen significativamente en la Producción de textos en los estudiantes del III Ciclo de la mención de Lengua y Literatura de la Facultad de Educación. En la evaluación de entrada con respecto a la producción de textos el grupo de control obtuvo 80% en la categoría deficiente, 15% en la categoría regular y 5% en la categoría muy deficiente; el grupo experimental en la categoría deficiente obtuvo 95% y 5% en la categoría muy deficiente. En la evaluación de salida el grupo de control obtuvo 70% en la categoría deficiente, 30% en la categoría regular y el grupo experimental obtuvo 75% en la categoría regular, 15% en la categoría bueno y 10% en la categoría deficiente.

Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free regular category FRg(T) on a set T. From this point of view, regular theories are certain monoidal 2-functors from a suitable 2-category of contexts -- the 2-category of relations in FRg(T) -- to that of posets. Such functors assign to each context the set of formulas in that context, ordered by entailment. We refer to such a 2-functor as a regular calculus because it naturally gives rise to a graphical string diagram calculus in the spirit of Joyal and Street. We shall show that every natural category has an associated regular calculus, and conversely from every regular calculus one can construct a regular category., Comment: In Proceedings ACT 2019, arXiv:2009.06334. Condensed version of arXiv:1812.05765

As a first objective, we characterise those essentially algebraic categories which satisfy properties like being unital, strongly unital, n-permutable, subtractive or protomodular. For each such property, we obtain a Mal'tsev condition as an equivalent condition. Using the language of Janelidze matrix conditions, we treat many of these properties together. As a second objective, using these characterisations, we prove some embedding theorems for those properties in a regular context in the same style as we did in the companion paper [23] . Concrete examples of how to use these embedding theorems are given. Finally, to extend those embedding theorems to the exact context, we show that these properties are stable under the exact completion of a regular category.

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 regular categories. We prove a categorical version of Bergman's Double-projection Theorem: a regular category is a majority category if and only if every subobject $S$ of a finite product $A_1 \times A_2 \times \cdots \times A_n$ is uniquely determined by its two-fold projections. We also establish a categorical counterpart of the Pairwise Chinese Remainder Theorem for algebras, and characterize regular majority categories by the classical congruence equation $\alpha \cap (\beta \circ \gamma) = (\alpha \cap \beta) \circ (\alpha \cap \gamma)$ due to A.F.~Pixley., Comment: 29 pages

Evidence from multiple category learning studies suggest that exceptions to a category rule are remembered better than the items that follow that rule (Davis, Love, & Preston, 2012; Palmeri & Nosofsky, 1995; Sakamoto & Love, 2004). Based on differences in recognition memory, it has been suggested that category exceptions may be represented separately from regular category members. Here, we present 4 experiments investigating representations of regular and exceptional category members as well as potential developmental changes in these representations. Although 4-year-olds and adults demonstrated different memory patterns, both age groups showed (a) higher memory sensitivity for regular members of the category and (b) isomorphic memory patterns for regular and exception items. Additionally, we report important developmental differences in generalization patterns. In children, features of regulars and of exceptions contributed to categorization of both regular and exception items. In contrast, an asymmetry was found in adults: features of regulars contributed to categorization of both regular and exception items, whereas features of exceptions contribution only to categorization of exceptions. These findings challenge the hypothesis that items that violate known knowledge structures have a special status in memory and suggest that exceptions can be represented jointly with regular items early in development and as a subset of regular items later in development. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

We construct an exact completion for regular categories enriched in the cartesian closed category $\mathsf{Pos}$ of partially ordered sets and monotone functions by employing a suitable calculus of relations. We then characterize the embedding of any regular category into its completion and use this to obtain examples of concrete categories which arise as such completions. In particular, we prove that the exact completion in this enriched sense of both the categories of Stone and Priestley spaces is the category of compact ordered spaces of L. Nachbin. Finally, we consider the relationship between the enriched exact completion and categories of internal posets in ordinary categories., Comment: 42 pages. To appear in Journal of Pure and Applied Algebra

Mal’tsev categories turned out to be a central concept in categorical algebra. On the one hand, the simplicity and the beauty of the notion is revealed by the wide variety of characterizations of a markedly different flavour. Depending on the context, one can define Mal’tsev categories as those for which ‘any reflexive relation is an equivalence’; ‘any relation is difunctional’; ‘the composition of equivalence relations on the same object is commutative’; ‘each fibre of the fibration of points is unital’ or ‘the forgetful functor from internal groupoids to reflexive graphs is saturated on subobjects’. For a variety of universal algebras, these are also equivalent to the existence in its algebraic theory of a Mal’tsev operation, i.e. a ternary operation p(x, y, z) satisfying the axioms p(x, x, y) = y and p(x, y, y) = x. On the other hand, Mal’tsev categories have been shown to be the right context in which to develop the theory of centrality of equivalence relations, Baer sums of extensions, and some homological lemmas such as the denormalized 3 × 3 Lemma, whose validity in a regular category is equivalent to the weaker ‘Goursat property’, which has also turned out to be of wide interest.

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 ana-lyse the algebraic properties of the categories satisfying the additional Mal'tsev axiom, and then the weaker Goursat axiom. These latter contexts can be seen as the categorical counterparts of the properties of 2-permutability and of 3-permutability of congruences in universal algebra. Mal'tsev and Goursat categories have been intensively studied in the last years: we present here some of their basic properties, which are useful to read more advanced texts in categorical algebra.

We study those exactness properties of a regular category C that can be expressed in the following form: for any diagram of a given ‘finite shape’ in C , a given canonical morphism between finite limits built from this diagram is a regular epimorphism. The main goal of the paper is to characterize essentially algebraic categories which satisfy this property via (essential versions of) linear Mal'tsev conditions, which are known to correspond to the so-called matrix properties. We then apply this characterization, along with our earlier work on preservation of exactness properties by pro-completions, to prove that these exactness properties can be reduced to matrix properties already in the general setting of regular categories.

We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids.

In this paper, we obtain a non-abelian analogue of Lubkin's embedding theorem for abelian categories. Our theorem faithfully embeds any small regular Mal'tsev category $\mathbb{C}$ in an $n$-th power of a particular locally finitely presentable regular Mal'tsev category. The embedding preserves and reflects finite limits, isomorphisms and regular epimorphisms, as in the case of Barr's embedding theorem for regular categories. Furthermore, we show that we can take $n$ to be the (cardinal) number of subobjects of the terminal object in $\mathbb{C}$.

In this paper we explore the snail lemma in a pointed regular category. In particular, we show that under the presence of cokernels of kernels, the validity of the snail lemma is equivalent to subtractivity of the category. As a corollary, this gives that in the more restrictive context of a normal category the validity of the snail lemma is equivalent to the validity of the snake lemma.

In the context of regular unital categories we introduce an intrinsic version of the notion of a Schreier split epimorphism, originally considered for monoids. We show that such split epimorphisms satisfy the same homological properties as Schreier split epimorphisms of monoids do. This gives rise to new examples of S-protomodular categories, and allows us to better understand the homological behaviour of monoids from a categorical perspective., 20 pages

Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosick\'y, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Considering a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosick\'y., Comment: Reviewed version: changed statement of Theorem 3.3; changed proof of Lemma 8.2; added minor comments. To appear in the Journal of Pure and Applied Algebra: https://doi.org/10.1016/j.jpaa.2019.106268

Background and Objectives: The aim of the present research is to study the questions used in the 2018 MIR exam (a test that allows access to specialized medical training in Spain), describe their psychometric properties, and evaluate their quality. Materials and Methods: This analysis is performed with the help of classical test theory (CTT) and item response theory (IRT). The answers given to the test questions by a total of 3868 physicians are analyzed. Results: According to CTT, the average difficulty index for all of the test questions was 0.629, which falls into the acceptable category. The average difficulty index with correction for random effects was 0.515, which corresponds to a value within the optimal range. The mean discrimination index was 0.277, which is in the good category, while the mean point biserial correlation coefficient, with a value of 0.275 fits in the regular category. The values of difficulty and discrimination calculated according to the model of two parameters of the IRT seem adequate with average values of &minus, 0.389 and 0.677. The Cronbach alpha score obtained for the overall test was 0.944. This value is considered as very good. Conclusions: A decrease was observed in the average values of discrimination in the last three calls, which may be related to the greater proportion of Spanish graduates that take the exam in the same year of finalization of their studies in Medicine.

The notion of 𝒟-mutation pairs of subcategories in an abelian category is defined in this article. When (𝒵,𝒵) is a 𝒟-mutation pair in an abelian category 𝒜, the quotient category 𝒵∕𝒟 carries natura...

It is well known that the abelianization of a group G can be computed as the cokernel of the diagonal morphism (1G, 1G): G → G × G in the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism.

The notion of an M -coextensive object is introduced in an arbitrary category C , where M is a distinguished class of morphisms from C . This notion allows for a categorical treatment of the strict refinement property in universal algebra, and highlights its connection with extensivity in the sense of Carboni, Lack and Walters. If M is the class of product projections in a category C with finite products, then M -coextensivity is closely related to the notion of a Boolean category in the sense of E. Manes: C is co-Boolean if and only if its product projections are pushout stable, and every object is M -coextensive. We show that if C is a variety of algebras then the M -coextensive objects are precisely those algebras which have the strict refinement property, when M is the class of product projections. If M is the class of surjective homomorphisms in the variety, then the M -coextensive objects are those algebras which have directly-decomposable (or factorable) congruences. Moreover, these results are proved for any object with global support in a regular category. We also show that in exact Mal'tsev categories, every centerless object with global support is projection-coextensive, i.e., has the strict refinement property. We will also show that in every exact majority category, every object with global support has the strict refinement property.

We prove that Mal'tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences $R$, $S$ and $T$, and characterises congruence modular varieties. We first show that a regular category $\mathcal C$ is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in $\mathcal C$. Moreover, we prove that a regular category $\mathcal C$ is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation $S$ and reflexive and positive relations $R$ and $T$ in $\mathcal C$. In particular this provides a new characterisation of $2$-permutable and $3$-permutable varieties and quasi-varieties of universal algebras., 12 pages, minor changes in this revised version

We characterize categories with weak finite limits whose regular completions give rise to Goursat categories.

We introduce and study relative Rickart objects and dual relative Rickart objects in abelian categories. We show how our theory may be employed in order to study relative regular objects and (dual) relative Baer objects in abelian categories. We also give applications to module and comodule categories.

Beligiannis and Marmaridis in 1994, constructed the one-sided triangulated structures on the stable categories of additive categories induced from some homologically finite subcategories. We extend their results to slightly more general settings. As an application of our results, we give some new examples of one-sided triangulated categories arising from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via those of stable categories.

We study morphisms in varieties of ordered universal algebras. We prove that (i) monomorphisms are precisely the injective homomorphisms and that (ii) every regular monomorphism is an order embedding, but the converse is not true in general. We also give a necessary and sufficient condition for a morphism to be a regular epimorphism. Finally, we discuss factorizations in such varieties.

Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. In this paper, we consider some category-theoretic properties of the category of all soft sets, with morphisms between them. We also characterize several kinds of epimorphisms and monomorphisms.

For a pomonoid S, let us denote Pos-S the category ofS-posets and S-poset maps. In this paper, we consider the slice cate-gory Pos-S/B for an S-poset B,and study some categorical ingredients.We first show that there is no non-trivial injective object in Pos-S/B.Then we investigate injective objects with respect to the class of regu-lar monomorphisms in this category and show that Pos-S/B has enoughregular injective objects. We also prove that regular injective objects areretracts of exponentiable objects in this category. One of the main aimsof the paper is to draw attention to characterizing injectivity in the cate-gory Pos-S/Bunder a particular case where Bhas trivial action. Amongother things, we also prove that the necessary condition for a map (an ob-ject) here to be regular injective is being convex and present an exampleto show that the converse is not true, in general. 1. Introduction and preliminariesA slice category is a construction in category theory which provides anotherway of looking at morphisms: instead of simply relating objects of a categoryto one another, morphisms become objects in their own right. This notion wasintroduced in 1963 by F. W. Lawvere, although the technique did not becomegenerally known until many years later.One of the very useful notions in many categories as well as in homologicalalgebra is the injectivity of objects with respect to a class M of morphisms. Forexample, in the category Pos of partially ordered sets and monotone mappings,injective objects, with respectto the classofall order-embeddings,coincidewiththe complete lattices (see [4]). Injective objects in slice categories have beeninvestigated in detail (see [1, 16]), especially in relation with weak factorizationsystems, a concept used in homotopy theory, in particular for model categories.