In the fuzzy theory of sets and groups, the use of α-levels is a standard to translate problems f... more In the fuzzy theory of sets and groups, the use of α-levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong α-levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. The main result of this paper is to identify the class of fuzzy sets, respectively, fuzzy groups, with subcategories of the functorial categories Set (0, 1], resp., Gr (0, 1]. In this line, the algebraic potential of this theory will be reached, in forthcoming papers.
The categorical treatment of fuzzy modules presents some problems, due to the well known fact tha... more The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod−P, which is a Grothendieck category. To do that, first we consider the preadditive category P, defined by the interval P=(0,1], to build a torsionfree class J in Mod−P, and a hereditary torsion theory in Mod−P, to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G,F), of decreasing gradual submodules of M, where G belongs to J, satisfying G=Fd, and ∪αF(α) is a disjoint union of F(1) and F(α)\G(α), where α is running in (0,1].
This piece of work presents a simple and compact overview of the design problem of tensegrity str... more 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.
In fuzzy theory of sets and groups, the use of $\alpha$--levels is a standard to translate proble... more In fuzzy theory of sets and groups, the use of $\alpha$--levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong $\alpha$--levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. The main result of this paper is to identify the class of fuzzy sets, respectively fuzzy groups, with subcategories of the functorial categories $\mathcal{S}\textit{et}^{(0,1]}$, resp. $\mathcal{G}\textit{r}^{(0,1]}$.
Ь б г и з лгж з иг з гл гл знб га гбдйи и гв в йз иг д ж гжб бйаи к ж и Ф ж в И Р жб и в ж г ви ж... more Ь б г и з лгж з иг з гл гл знб га гбдйи и гв в йз иг д ж гжб бйаи к ж и Ф ж в И Р жб и в ж г ви ждга и гв в ад йз иг й а бгж ж а зи ви ждга и в йв и гвзК и ж и гж и а вижг й и гв в л л в ано и гбда м ин г и б и г л з аа г йз гйж ии ви гв гв дда и гвзК ... Хйаи к ж и ви ждга и гв гвз зиз в к в ...
Abstract. A study of finiteness in a kind of finitely presented quotient algebras is displayed in... more Abstract. A study of finiteness in a kind of finitely presented quotient algebras is displayed in this paper. The relations generating the ideals are given by monomials or binomials of same length, in order to obtain homogeneous computations. These ideals are parametrized by 3-tuples (a, b, c), being a the number of variables, b the length of monomials and c the number of relations conforming the ideal. We focus on the analysis of the (2, 3, 4)-family, constituted by 58905 elements, compute all the possible ideals, obtain the corresponding Gröbner- ...
Proceedings of the Edinburgh Mathematical Society, 1997
In this note we propose an effective method based on the computation of a Gröbner basis of a left... more In this note we propose an effective method based on the computation of a Gröbner basis of a left ideal to calculate the Gelfand-Kirillov dimension of modules.
We develop the notion of primeness in coalgebras (over a commutative field). In particular, in th... more We develop the notion of primeness in coalgebras (over a commutative field). In particular, in this work we focus our attention on the study and characterization of prime subcoalgebras of path coalgebras of quivers and, by extension, of prime pointed coalgebras.
We establish the so-called Invariance under Twisting Theorems, stating that twisted tensor produc... more We establish the so-called Invariance under Twisting Theorems, stating that twisted tensor products built up from different algebras that relate in certain ways are canonically isomorphic. These results generalize (and provide categorical versions of) several well-known results in Hopf algebra theory, such as the invariance of the smash product under Drinfeld twisting, or the isomorphism between the Drinfeld double and a certain smash product for quasitriangular Hopf algebras.
Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H ... more Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z(H,M). The cyclic homology of Z(H,M) extends the usual cyclic homology of the algebra structure of H, and the relative cyclic homology of an H-Galois extension. For a Hopf subalgebra K we compute, under some assumptions, the cyclic homology of an induced modular crossed module. As a direct application of this computation, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the (usual) cyclic homology of group algebras and quantum tori. Finally, when H is the enveloping algebra of a Lie algebra, we construct a spectral sequence that converges to the cyclic homology of H with coefficients in an arbitrary modular crossed module. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of H with coe...
For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a heredi... more For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a hereditary torsion theory $\sigma$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $\sigma$-noetherian ring, then $\sigma$ is of finite type, and that totally $\sigma$-noetherian is a local property.
In the fuzzy theory of sets and groups, the use of α-levels is a standard to translate problems f... more In the fuzzy theory of sets and groups, the use of α-levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong α-levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. The main result of this paper is to identify the class of fuzzy sets, respectively, fuzzy groups, with subcategories of the functorial categories Set (0, 1], resp., Gr (0, 1]. In this line, the algebraic potential of this theory will be reached, in forthcoming papers.
The categorical treatment of fuzzy modules presents some problems, due to the well known fact tha... more The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod−P, which is a Grothendieck category. To do that, first we consider the preadditive category P, defined by the interval P=(0,1], to build a torsionfree class J in Mod−P, and a hereditary torsion theory in Mod−P, to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G,F), of decreasing gradual submodules of M, where G belongs to J, satisfying G=Fd, and ∪αF(α) is a disjoint union of F(1) and F(α)\G(α), where α is running in (0,1].
This piece of work presents a simple and compact overview of the design problem of tensegrity str... more 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.
In fuzzy theory of sets and groups, the use of $\alpha$--levels is a standard to translate proble... more In fuzzy theory of sets and groups, the use of $\alpha$--levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong $\alpha$--levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. The main result of this paper is to identify the class of fuzzy sets, respectively fuzzy groups, with subcategories of the functorial categories $\mathcal{S}\textit{et}^{(0,1]}$, resp. $\mathcal{G}\textit{r}^{(0,1]}$.
Ь б г и з лгж з иг з гл гл знб га гбдйи и гв в йз иг д ж гжб бйаи к ж и Ф ж в И Р жб и в ж г ви ж... more Ь б г и з лгж з иг з гл гл знб га гбдйи и гв в йз иг д ж гжб бйаи к ж и Ф ж в И Р жб и в ж г ви ждга и гв в ад йз иг й а бгж ж а зи ви ждга и в йв и гвзК и ж и гж и а вижг й и гв в л л в ано и гбда м ин г и б и г л з аа г йз гйж ии ви гв гв дда и гвзК ... Хйаи к ж и ви ждга и гв гвз зиз в к в ...
Abstract. A study of finiteness in a kind of finitely presented quotient algebras is displayed in... more Abstract. A study of finiteness in a kind of finitely presented quotient algebras is displayed in this paper. The relations generating the ideals are given by monomials or binomials of same length, in order to obtain homogeneous computations. These ideals are parametrized by 3-tuples (a, b, c), being a the number of variables, b the length of monomials and c the number of relations conforming the ideal. We focus on the analysis of the (2, 3, 4)-family, constituted by 58905 elements, compute all the possible ideals, obtain the corresponding Gröbner- ...
Proceedings of the Edinburgh Mathematical Society, 1997
In this note we propose an effective method based on the computation of a Gröbner basis of a left... more In this note we propose an effective method based on the computation of a Gröbner basis of a left ideal to calculate the Gelfand-Kirillov dimension of modules.
We develop the notion of primeness in coalgebras (over a commutative field). In particular, in th... more We develop the notion of primeness in coalgebras (over a commutative field). In particular, in this work we focus our attention on the study and characterization of prime subcoalgebras of path coalgebras of quivers and, by extension, of prime pointed coalgebras.
We establish the so-called Invariance under Twisting Theorems, stating that twisted tensor produc... more We establish the so-called Invariance under Twisting Theorems, stating that twisted tensor products built up from different algebras that relate in certain ways are canonically isomorphic. These results generalize (and provide categorical versions of) several well-known results in Hopf algebra theory, such as the invariance of the smash product under Drinfeld twisting, or the isomorphism between the Drinfeld double and a certain smash product for quasitriangular Hopf algebras.
Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H ... more Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z(H,M). The cyclic homology of Z(H,M) extends the usual cyclic homology of the algebra structure of H, and the relative cyclic homology of an H-Galois extension. For a Hopf subalgebra K we compute, under some assumptions, the cyclic homology of an induced modular crossed module. As a direct application of this computation, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the (usual) cyclic homology of group algebras and quantum tori. Finally, when H is the enveloping algebra of a Lie algebra, we construct a spectral sequence that converges to the cyclic homology of H with coefficients in an arbitrary modular crossed module. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of H with coe...
For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a heredi... more For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a hereditary torsion theory $\sigma$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $\sigma$-noetherian ring, then $\sigma$ is of finite type, and that totally $\sigma$-noetherian is a local property.
Uploads
Papers