Showing papers in "Quaestiones Mathematicae in 1997"
TL;DR: This paper sets forth in detail point-set lattice-theoretic or poslat foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and pre-image of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure and probability theory, and topology.
Abstract: This paper sets forth in detail point-set lattice-theoretic or poslat foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and pre-image of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure and probability theory, and topology. In particular, those aspects of fuzzy sets, hinging around (crisp) powersets of fuzzy subsets and around powerset operators between such powersets lifted from ordinary functions between the underlying base sets, are examined and characterized using point-set and lattice-theoretic methods. The basic goal is to uniquely derive the powerset operators and not simply stipulate them, and in doing this we explicitly distinguish between the “fixed-basis” case (where the underlying lattice of membership values is fixed for the sets in question) and the “variable-basis” case (where the underlying lattice of membership values is allowed to change). Applications to fuzzy sets/logic include: development a...
98 citations
TL;DR: A construction by the second author of generating fuzzy topologies from a decreasing chain of fuzzyTopologies is generalized by considering the lattice of all lower sets of a completely distributive lattice as a range space of a fuzzy topology.
Abstract: A construction by the second author of generating fuzzy topologies from a decreasing chain of fuzzy topologies is generalized by considering the lattice of all lower sets of a completely distributive lattice as a range space of a fuzzy topology.
64 citations
TL;DR: In Zermelo-Fraenkel set theory without the Axiom of Choice AC, results such as the following are established: The Boolean Prime Ideal Theorem BPI is equivalent to the statement: ZFE Every zero-filter is contained in a maximal one.
Abstract: In Zermelo-Fraenkel set theory ZF without the Axiom of Choice AC, results such as the following are established: The Boolean Prime Ideal Theorem BPI is equivalent to the statement: ZFE Every zero-filter is contained in a maximal one. The Boolean Prime Ideal Theorem is properly weaker than the statement: CFE Every closed filter is contained in a maximal one. The Axiom of Choice is equivalent to the conjunction of CFE and the Axiom of Countable Choice CC. The Axiom of Countable Choice is equivalent to the statement: C=SC Functions between metric spaces are continuous iff they are seqentially continuous.
14 citations
TL;DR: This work unify notation and language and compare the different frameworks in the traditional point-set context and uses only the real unit interval as a lattice in order to define fuzzy sets.
Abstract: We consider old and new categories of fuzzy topological spaces where objects associated to arbitrary fuzzy sets are considered and fuzzy topological subspaces are characterized as particular subobjects. Also fuzzy homeomorphisms are characterized in the considered categories. We unify notation and language and compare the different frameworks in the traditional point-set context. We use only the real unit interval as a lattice in order to define fuzzy sets.
13 citations
TL;DR: Sobriety in the setting of fuzzy topological spaces and its relation to the fuzzy Hausdorff concept(s) is discussed.
Abstract: Sobriety in the setting of fuzzy topological spaces and its relation to the fuzzy Hausdorff concept(s) is discussed
12 citations
TL;DR: In this article, Fuzzy SYNTOPOGENEOUS STRUCTURES are used to describe the structure of a polygonal lattice, and the lattice is shown to be polytope.
Abstract: (1997). FUZZY SYNTOPOGENEOUS STRUCTURES. Quaestiones Mathematicae: Vol. 20, No. 3, pp. 431-461.
12 citations
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TL;DR: A criterion for a given family of prefilters on a set X to generate a fuzzy filter 𝔉 on X with {F α)αe I0 as its family of α-level prefilter, that is 𝓂α = F α is found and extended to super uniformities.
Abstract: A new, generalized form, of uniformity, the so called super uniformity is defined and studied. It is based on the concept of fuzzy filter, as introduced by Eklund and Gaaler [EG]. To each super uniformity, a fuzzy α-uniformities system can be associated. They will be called α-levels. These α-levels are fuzzy uniformities in the sense of Lowen, for α=1, and α-modifications with pleasant properties, for α≠1. The *-version of super uniformities is related, at level 1, with T-uniformities, as defined by Hohle [Ho]. A criterion for a given family of prefilters {F α}αeI0 on a set X to generate a fuzzy filter 𝔉 on X with {F α)αe I0 as its family of α-level prefilters, that is 𝔉α = F α is found, and extended to super uniformities. Finally, super uniformities are related with fuzzy topologies in the sense of Sostak.
12 citations
TL;DR: In this paper, it was shown that if R is a *-prime ring which is not a prime ring, then R is essentially a direct product of two prime rings, and if P is a prime ideal of R, then X is minimal among prime ideals of R containing P, P = X ∩ X* and either: (1) P is essential in X and X is essential on R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R.
Abstract: Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.
11 citations
TL;DR: In this paper, the authors consider the problem of finding a 4-dimensional smooth manifold such that the anticanonical divisor is ample and generates Div(V) for which values of (K V )4 and h 0(V, O V (k V )) do such a manifold exist?
Abstract: Let V be a Fano 4-fold of index one and Picardnumber one, i.e. a 4-dimensional smooth variety such that the anticanonical divisor—K V is ample and generates Div(V). We consider problems concerning the “geography” of these manifolds: for which values of (—K V )4 and h 0(V, O V (—K V )) do such V exist? We describe various classes of examples, discuss some of their properties and state several open problems.
10 citations
TL;DR: In this article, the notion of computational Cauchy net and an appropriate notion of strong convergence were proposed to obtain Smyth completeness for non-symmetric spaces with non-uniform topologies.
Abstract: Smyth completeness is the appropriate notion of completeness for quasi-uniform spaces carrying an additional topology to serve as domains of computation [2, 3]. The goal of this paper is to provide a better understanding of Smyth completeness by giving a characterization in terms of nets. We develop the notion of computational Cauchy net and an appropriate notion of strong convergence to get the result that a space is Smyth complete if and only if every computational Cauchy net strongly converges. As we are dealing with typically non-symmetric spaces, this is not an instance of the classical net-filter translation in general topology.
9 citations
TL;DR: In this paper, a universal class U of hemirings is studied and necessary and sufficient conditions for equality are given for the equality of two radicals ϱ∞(S) and ϱe(S).
Abstract: In this article we continue investigations on a Kurosh-Amitsur radical theory for a universal class U of hemirings as introduced by O.M. Olson et al. We give some necessary and sufficient conditions that such a universal class U consists of all hemirings. Further we consider special and weakly special subclasses M of U which yield hereditary radical classes P = um of U. In this context we correct some statements in the papers of Olson et al. Moreover, a problem posed there concerning the equality of two radicals ϱ∞(S) and ϱe(S) and two similar ideals β∞ (S) and βe(S) is widely solved. We prove ϱ∞(S) ⊇ ϱe(S) = β∞(S) = βe(S) and give necessary and sufficient conditions for equality in the first inclusion. This yields in particular that the weakly special class Me(U) is always semisimple, a result which is not true for the special class M∞(U).
TL;DR: In this article, a generalization of the connectedness-disconnectedness Galois connection (also called torsion-torsion free in algebraic contexts) was shown to factor through the class of all closure operators on X with respect to M.
Abstract: Let X be an arbitrary category with an (E, M)-factorization structure for sinks. A notion of constant morphism that depends on a chosen class of monomorphisms was previously used to provide a generalization of the connectedness-disconnectedness Galois connection (also called torsion-torsion free in algebraic contexts). This Galois connection was shown to factor through the class of all closure operators on X with respect to M. Here, properties and implications of this factorization are investigated. In particular, it is shown that this factorization can be further factored. Examples are provided.
TL;DR: The basic theory of function spaces is shown to extend to the fuzzy setting and conditions for completeness and compactness of fuzzy subsets of Yx are established.
Abstract: It is shown that if (Y, D) is a fuzzy uniform space in the sense of Lowen and X is a set, then a fuzzy uniformity D π, of pointwise convergence and a fuzzy uniformity D u, of uniform convergence can be defined on Yx . These are natural generalisations of the uniformities of pointwise and uniform convergence. The basic theory of function spaces is shown to extend to the fuzzy setting. In particular, conditions for completeness and compactness of fuzzy subsets of Yx are established.
TL;DR: In this paper, it was shown that the category SUS, of super uniform spaces also enjoys this property and furthermore, the categories FUS and GUS can be embedded into SUS.
Abstract: The category US of uniform spaces has been generalised in various ways. The category FUS, of fuzzy uniform spaces and the category GUS, of generalised uniform spaces have both been shown to be good extensions in the sense that US can be embedded into them. We show here that the category SUS, of super uniform spaces also enjoys this property and furthermore, the categories FUS and GUS can be embedded into SUS.
TL;DR: Two special kinds of fuzzy topologies in the sense of the third author: the so called even and supereven fuzzyTopologies are introduced and their role in the (general) theory of fuzzytopologies is discussed.
Abstract: Two special kinds of fuzzy topologies in the sense of the third author: the so called even and supereven fuzzy topologies are introduced. Some properties of even fuzzy topologies are established; their role in the (general) theory of fuzzy topologies is discussed. Besides, proximal and uniform counterparts of (super) even fuzzy topologies are considered.
TL;DR: In this paper, the authors investigated the σ-injectivity of O. Goldman and other related types of relative injectivity when σ is an arbitrary torsion radical, and when the associated filter satisfies some standard finiteness conditions.
Abstract: We investigate the σ-injectivity of O. Goldman and other related types of relative injectivity when σ is an arbitrary torsion radical, and when the associated filter satisfies some standard finiteness conditions.
TL;DR: In this article, a categorical formulation of the neighborhood axioms of topological spaces including a characterization of the corresponding interior operators of interior operators is presented. And properties such as Hausdorff's separation axiom, compactness, etc.
Abstract: This paper presents a categorical formulation of the neighborhood axioms of topological spaces including a characterization by the corresponding axioms of interior operators. Properties as Hausdorff's separation axioms, compactness are discussed, and various links to internal topologies of topoi, fuzzy topologies, etc. are given.
TL;DR: In this article, it was shown that α-compactness of fuzzy topological spaces is the categorical compactness (in the sense of Herrlich, Salicrup and Strecker) which arises from a factorization structure on the category of fuzzy spaces and fuzzy continuous maps.
Abstract: It is shown that α-compactness (due to Gantner, Steinlage and Warren) of fuzzy topological spaces is the categorical compactness (in the sense of Herrlich, Salicrup and Strecker) which arises from a factorization structure on the category of fuzzy topological spaces and fuzzy continuous maps. Other characterizations of α-compactness are given, including characterizations in terms of fuzzy nets and fuzzy filters. The separatedness notion corresponding to the factorization structure is also identified.
TL;DR: In this paper, it was shown that the main problem left open in [2] can be solved using the Banach spaces Zα recently constructed by Kalton [1], which gives an example of a complex operator ideal that has no real analogue.
Abstract: We show that the main problem left open in [2] can be solved using the Banach spaces Zα recently constructed by Kalton [1]. This gives an example of a complex operator ideal that has no real analogue. It thus shows the richer structure of complex operator ideals compared with the real ones.
TL;DR: In this paper, the relationship between sober spaces and lower separation axioms is examined; every hereditarily sober space satisfies axiom T D (the derived set of every set is closed).
Abstract: A T 0 space is called sober provided the only irreducibly closed sets are the closures of singletons; a closed set is irreducibly closed if it cannot be written as a union of two of its proper closed subsets. The relationship between hereditarily sober spaces and the lower separation axioms is examined; e.g., every hereditarily sober space satisfies axiom T D (the derived set of every set is closed). For T 1 spaces, hereditary sobriety is much weaker than Hausdorff, however an hereditarily sober T 1 topology on a countably infinite set has cardinality of the continumn.
TL;DR: In this paper, the authors give new conditions on Banach spaces E and F which ensure that all polynomials P:E→F are completely continuous (i.e., send weakly converging sequences into norm-converging sequences).
Abstract: In this paper the authors give new conditions on Banach spaces E and F which ensure that all polynomials P:E→F are completely continuous (ie, send weakly converging sequences into norm-converging sequences) Among them are the following: (i) E has the Dunford-Pettis property and any Dunford-Pettis subset of F is relatively compact; (ii) all weakly null sequences of E are limited and F has the Gelʹfand-Phillips property These results complement similar ones by M Gonzalez and J M Gutierrez del Alamo [Arch Math (Basel) 63 (1994), no 2, 145–151;Glasgow Math J 37 (1995), no 2, 211–219;] It is also shown that complete continuity of all polynomials in the Taylor expansion at a of a holomorphic function f does not imply the complete continuity of f, but that such a condition is equivalent to the "local'' complete continuity of f at a: Whenever a sequence (xn) converges weakly to a and ∥xn−a∥≤c
TL;DR: In this paper, the authors examined the approximation of linear functionals of the form Ω = ⟨x, g⟩, where x is the solution of the linear operator equation Ax = f, with x,g and f elements of Hilbert Space H and A is an invertible operator on H.
Abstract: We examine the approximation of linear functionals of the form Ω = ⟨x, g⟩, where x is the solution of the linear operator equation Ax = f, with x,g and f elements of Hilbert Space H and A is an invertible operator on H. We show that the results from the literature all involve the zero order approximation to the inverse operator.
TL;DR: This paper showed that a strengthened form of a property of Roth-berger implies that a certain polarized partition relation holds for the ω-covers of a space and that this partition relation implies a strengthened version of Menger's property.
Abstract: We show that a strengthened form of a property of Roth-berger implies that a certain polarized partition relation holds for the ω-covers of a space and that this partition relation implies a strengthened form of a property of Menger.
TL;DR: In this article, the authors prove that if X and Y are Banach spaces such that X* has the weak Radon-Nikodym property (WRNP), Y has the RadonNikoda property (RNP) and Y is complemented in its bidual, then the space N(X,Y) of all nuclear operators from X to Y has a WRNP.
Abstract: We prove that if X and Y are Banach spaces such that X* has the weak Radon-Nikodym property (WRNP), Y has the Radon-Nikodym property (RNP) and Y is complemented in its bidual, then the space N(X,Y) of all nuclear operators from X to Y has the WRNP. If moreover X* has the RNP, then N(X,Y) has the RNP.
TL;DR: The monopole oscillator problem is a subset of the more general system with equation of motion [ruml] + λr -3L + (μr -4 + ω2)r = 0 which possesses sl(2, R) ⊕ so(3) symmetry.
Abstract: The monopole oscillator problem is a subset of the more general system with equation of motion [ruml] + λr -3L + (μr -4 + ω2)r = 0 which possesses sl(2, R) ⊕ so(3) symmetry. All of the first integrals are computed from the symmetry generators.
TL;DR: In this article, a closed differential ideal that is generated by 1-forms that are solutions of an exterior differential is constructed, which does not satisfy the Frobenius theorem because there are no solutions that satisfy the necessary independence condition.
Abstract: Explicit solutions of certain exterior differential systems of degrees one and two are constructed on a star-shaped region of a differentiable manifold. A closed differential ideal that is generated by 1-forms that are solutions of an exterior differential is constructed. This ideal does not satisfy the Frobenius theorem because there are no solutions that satisfy the necessary independence condition, although a modified version of the Frobenius theorem is exhibited in this case. Solutions of certain exterior differential systems whose associated curvature 2-forms have compact support in a star-shaped region are also constructed.
TL;DR: In this article, the authors introduced graded Artin algebras whose category of graded modules is locally of finite representation type, and studied the representation theory of such algesbras.
Abstract: Graded Artin algebras whose category of graded modules is locally of finite representation type are introduced. The representation theory of such algebras is studied. In the hereditary case and in the stably equivalent to hereditary case, such algebras are classified.
TL;DR: In this paper, the authors generalize a result of L. Weis and V. V. Shevchik about strict singularity of the Fourier transform to a large class of locally integral operators.
Abstract: In this note we generalize a result of L. Weis and V. V. Shevchik about strict singularity of the Fourier transform to a large class of locally integral operators on function spaces.
TL;DR: In this paper, the authors studied the Dieudonne-Kothe spaces of Lusin-measurable functions with values in a locally convex space and derived good permanence results for reflexivity when the density condition holds.
Abstract: We study Dieudonne-Kothe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.