Star-regularity and regular completions

In this paper we establish a new characterisation of star-regular categories, using a property of internal reflexive graphs, which is suggested by a recent result due to O. Ngaha Ngaha and the first author. We show that this property is, in a suitable sense, invariant under regular completion of a category in the sense of A. Carboni and E. M. Vitale. Restricting to pointed categories, where star-regularity becomes normality in the sense of the second author, this reveals an unusual behaviour of the exactness property of normality (i.e. the property that regular epimorphisms are normal epimorphisms) compared to other closely related exactness properties studied in categorical algebra.Comment: 13 page

Epireflective subcategories and formal closure operators

On a category $\mathscr{C}$ with a designated (well-behaved) class $\mathcal{M}$ of monomorphisms, a closure operator in the sense of D. Dikranjan and E. Giuli is a pointed endofunctor of $\mathcal{M}$, seen as a full subcategory of the arrow-category $\mathscr{C}^\mathbf{2}$ whose objects are morphisms from the class $\mathcal{M}$, which "commutes" with the codomain functor $\mathsf{cod}\colon \mathcal{M}\to \mathscr{C}$. In other words, a closure operator consists of a functor $C\colon \mathcal{M}\to\mathcal{M}$ and a natural transformation $c\colon 1_\mathcal{M}\to C$ such that $\mathsf{cod} \cdot C=C$ and $\mathsf{cod}\cdot c=1_\mathsf{cod}$. In this paper we adapt this notion to the domain functor $\mathsf{dom}\colon \mathcal{E}\to\mathscr{C}$, where $\mathcal{E}$ is a class of epimorphisms in $\mathscr{C}$, and show that such closure operators can be used to classify $\mathcal{E}$-epireflective subcategories of $\mathscr{C}$, provided $\mathcal{E}$ is closed under composition and contains isomorphisms. Specializing to the case when $\mathcal{E}$ is the class of regular epimorphisms in a regular category, we obtain known characterizations of regular-epireflective subcategories of general and various special types of regular categories, appearing in the works of the second author and his coauthors. These results show the interest in investigating further the notion of a closure operator relative to a general functor. They also point out new links between epireflective subcategories arising in algebra, the theory of fibrations, and the theory of categorical closure operators.Comment: 18 pages. Updated version with many improvement

Every topos has an optimal noetherian form

The search, of almost a century long, for a unified axiomatic framework for establishing homomorphism theorems of classical algebra (such as Noether isomorphism theorems and homological diagram lemmas) has led to the notion of a `noetherian form', which is a generalization of an abelian category suitable to encompass categories of non-abelian algebraic structures (such as non-abelian groups, or rings with identity, or cocommutative Hopf algebras over any field, and many others). In this paper, we show that, surprisingly, even the category of sets, and more generally, any topos, fits under the framework of a noetherian form. Moreover, we give an intrinsic characterization of such noetherian form and show that it is very closely related to the known noetherian form of a semi-abelian category. In fact, we show that for a pointed category having finite products and sums, the existence of the type of noetherian form that any topos possesses is equivalent to the category being semi-abelian (this result is unexpected since only trivial toposes can be semi-abelian). We also show that these noetherian forms are optimal, in a suitable sense.Comment: 66 pages, submitted for publicatio

On difunctionality of class relations

For a given variety V of algebras, we define a class relation to be a binary relation R subset of S(2)which is of the form R = S-2 boolean AND K for some congruence class K on A(2), where A is an algebra in V such that S subset of A. In this paper we study the following property of V : every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal'tsev condition on the variety and in a suitable sense, it is a join of Chajda's egg-box property as well as Duda's direct decomposability of congruence classes.South African National Research FoundationNational Research Foundation - South AfricaCentre for Mathematics of the University of Coimbra - Portuguese Government through FCT/MEC [UID/MAT/00324/2019]European Regional Development Fund through the Partnership Agreement PT2020info:eu-repo/semantics/publishedVersio