Luis Merino

LOCALIZATION IN COALGEBRAS. STABLE LOCALIZATIONS AND PATH COALGEBRAS

We study localizing and colocalizing subcategories of a comodule category of a coalgebra C over a field, using the correspondence between localizing subcategories and equivalence classes of idempotent elements in the dual algebra C*. In this framework, we give a useful description of the localization functor by means of the Morita–Takeuchi context defined by the quasi-finite injective cogenerator of the localizing subcategory. Applying this description; first we characterize that a localizing subcategory 𝒯, with associated idempotent element e ∈ C*, is colocalizing if and only if eC is a quasi-finite eCe-comodule and, in addition, 𝒯 is perfect whenever eC is injective. And second, we prove that a localizing subcategory 𝒯 is stable if and only if e is a semicentral idempotent element of C*. We apply the theory to path coalgebras and obtain, in particular, that the “localized” coalgebra of a path coalgebra is again a path coalgebra.

Strongly normalizing extensions

Abstract In this paper, we introduce strongly normalizing extensions as a natural generalization of centralizing extensions between rings. We show that these extensions behave in a much nicer way than normalizing extensions, both from the geometric and localization theoretic point of view.

Rough ideals under relations associated to fuzzy ideals

In this paper we deal with the theory of rough ideals started in (Davvaz, 2004). We show that the approximation spaces built from an equivalence relation compatible with the ring structure, i.e. associated to a two-sided ideal, are too naive in order to develop practical applications. We propose the use of certain crisp equivalence relations obtained from fuzzy ideals. These relations make available more flexible approximation spaces since they are enriched with a wider class of rough ideals. Furthermore, these are fully compatible with the notion of primeness (semiprimeness). The theory is illustrated by several examples of interest in Engineering and Mathematics.

Decomposition of comodules

In this note we develop a decomposition theory of comodules over a general coalgebra. We relate it with other known decomposition theories and apply it to the decomposition of categories.

Full and folded forms: a compact review of the formulation of tensegrity structures:

This piece of work presents a simple and compact overview of the design problem of tensegrity structures in two and three dimensions. The main aim of this study is to present the design and calculation of tensegrity structures in their simplest form, avoiding unnecessary simplifications that can rule out solutions, as has happened up until now. As a result of the simplicity of the procedure, two types of tensegrity structures are obtained for the same initial topology: full and folded forms. Several examples are shown.

Embeddings between lattices of fuzzy sets: An application of closed-valued fuzzy sets

Abstract In this paper we deal with the problem of extending Zadehs operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to SVFSs. Alternatively, if the former singleton viewpoint is required, we reformulate the intersection on hesitant fuzzy sets and introduce what we have called closed-valued fuzzy sets. This new type of fuzzy sets extends standard union and intersection on FSs. In addition, it allows handling together membership degrees of different nature as, for instance, closed intervals and finite sets. Finally, all these constructions are viewed as T2FSs forming a chain of lattices.

Hilbert's Theorem 90 for Hopf Galois Extensions

For a faithfully flat extension A/B and a right A-module M, we give a new characterization of the set of descent data on M. Assuming that B is a simple Artinian ring and A/B is H-Galois, for a certain finite dimensional Hopf algebra H, we prove that Sweedlers noncommutative cohomology H 1(H*, A) is trivial as a pointed set.

A structure sheaf on noetherian rings

Several different constructions of a structure sheaf over noetherian noncommutative rings have been made in the recent past; however the problem of functoriality remains open in the general case. Here we present a new approach to the problem. Since the two-sided structure of the ring is fundamental in finding a solution of the above problem it seems natural to evaluate the sheaf functors on (normalizing) bimod-ules. In this context, biradicals arise as a useful tool as well as classical localization at prime ideals and cliques. The main results are valid for noetherian rings satisfying the strong second layer condition in which every clique is classically localizable. This paper can be considered an extension of [5], and develops its results outside of FBN rings.