Strong Cauchy completeness in uniform frames

We show that for uniform frames, with the underlying frame being Boolean, uniform paracompactness and strong Cauchy completeness are equivalent conditions. Certain aspects of uniform paracompactness are also considered. We then introduce the pointfree notion of preparacompactness and show that the completion of a preparacompact uniform frame is strongly Cauchy complete. We also formulate pointfree filter characterizations of Lindelöfness for regular frames in analogy to their classical topological counterparts.     

Cauchy completions of nearness frames

A nearness frame is Cauchy complete if every regular Cauchy filter on the nearness frame is convergent and we show that the categoryCCNFrm of Cauchy complete nearness frames is coreflective in the categoryNFrmC of nearness frames and Cauchy homomorphisms and that the coreflection of a nearness frame is given by the strict extension associated with regular Cauchy filters on the nearness frame. Using the same completion, we show that the categoryCCSNFrm of Cauchy complete strong nearness frames is coreflective in the categorySNFrm of strong nearness frames and uniform homomorphisms.     

CAUCHY POINTS OF UNIFORM AND NEARNESS FRAMES

Abstract

The notion of Cauchy point (= regular Cauchy filter) and the corresponding Cauchy spectrum, for a nearness frame (= uniform without the star-refinement condition) are investigated in various directions, including basic motivation, several functorial aspects, and the recognition of the Cauchy spectrum as the ordinary spectrum of the completion, after the unique existence of the latter is obtained as a central new result in this context.     

A note on precompact uniform frames

Precompactness or total boundedness for uniform frames is usually distinguished by a cover approach. In this note, we provide alternate characterizations of precompact uniform frames. In particular, we formulate pointfree filter analogues of various classical topological results on precompactness. We also revisit the notion of convergence and clustering of filters in a frame and introduce weakly Cauchy filters and strong Cauchy completeness in the setting of uniform frames.     

Cauchy Complete Nearness Spaces

A nearness space is Cauchy complete if every regular Cauchy filter on the space is convergent. We show that the category CCNear ₂ of Cauchy complete N ₂ spaces is reflective in the category Near ₂ C of N ₂-spaces and Cauchy maps and that the reflection of an N ₂-space is given by the strict extension associated with regular Cauchy filters on the space.     

Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in Frith and Schauerte (Quaest Math 33:507–530, 2010) a completion (called a quasi-completion) was constructed for such structures and in Frith and Schauerte (Quaest Math, 2012) completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this paper questions of functoriality for this quasi-completion are considered and one sees that having enough regular Cauchy bifilters plays an important rôle. The quasi-complete strong quasi-nearness biframes with enough regular Cauchy bifilters are seen to form a coreflective subcategory of the strong quasi-nearness biframes with enough regular Cauchy bifilters. Here a significant difference between the symmetric and asymmetric cases emerges: a strong (even quasi-uniform) quasi-nearness biframe need not have enough regular Cauchy bifilters. The Cauchy filter quotient leads to further characterizations of those quasi-nearness biframes having enough regular Cauchy bifilters. The fact that the Cauchy filter quotient of a totally bounded quasi-nearness biframe is compact shows that any totally bounded quasi-nearness biframe with enough regular Cauchy bifilters is in fact quasi-uniform. The paper concludes with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases.     

Generalized Cauchy Spaces

In this paper a unified theory of Cauchy spaces is presented including the classical cases of filter and sequence Cauchy spaces. To by-pass a lattice-theoretical barrier the notion of Urysohn modification of a functor is introduced. Employing this notion for many types of generalized Cauchy spaces a completion method is given.     

On preparacompactness of uniform frames

We present a pointfree characterization of paracompactness via strong Cauchy completeness. We also provide a filter characterization of separability in uniform frames and determine those uniform frames that have a Lindelöf and compact completion using the notion of preparacompactness. Further, as an application of preparacompactness, we provide filter conditions for the Lindelöfness of the Hewitt realcompactification υL of a completely regular frame L.     

S-fuzzy Version of Stone's Theorem for Distributive Lattices

In this paper,we initiate a study of S-fuzzy ideal(filter) of a lattice where S stands for a meet semilattice.A S-fuzzy prime ideal(filter) of a lattice is defined and it is proved that a S-fuzzy ideal(filter) of a lattice is S-fuzzy prime ideal(filter) if and only if any non-empty α-cut of it is a prime ideal(filter).Stone's theorem for a distributive lattice is extended by considering S-fuzzy ideals(filters).     

On fuzzy fantastic filters of lattice implication algebras

Fuzzification of a fantastic filter in a lattice implication algebra is considered. Relations among a fuzzy filter, a fuzzy fantastic filter, and fuzzy positive implicative filter are stated. Conditions for a fuzzy filter to be a fuzzy fantastic filter are given. Using the notion of level set, a characterization of a fuzzy fantastic filter is considered. Extension property for fuzzy fantastic filters is established. The notion of normal/maximal fuzzy fantastic filters and complete fuzzy fantastic filters is introduced, and some related properties are investigated.     

Dynamical Locality and Covariance: What Makes a Physical Theory the Same in all Spacetimes?

The question of what it means for a theory to describe the same physics on all spacetimes (SPASs) is discussed. As there may be many answers to this question, we isolate a necessary condition, the SPASs property, that should be satisfied by any reasonable notion of SPASs. This requires that if two theories conform to a common notion of SPASs, with one a subtheory of the other, and are isomorphic in some particular spacetime, then they should be isomorphic in all globally hyperbolic spacetimes (of given dimension). The SPASs property is formulated in a functorial setting broad enough to describe general physical theories describing processes in spacetime, subject to very minimal assumptions. By explicit constructions, the full class of locally covariant theories is shown not to satisfy the SPASs property, establishing that there is no notion of SPASs encompassing all such theories. It is also shown that all locally covariant theories obeying the time-slice property possess two local substructures, one kinematical (obtained directly from the functorial structure) and the other dynamical (obtained from a natural form of dynamics, termed relative Cauchy evolution). The covariance properties of relative Cauchy evolution and the kinematic and dynamical substructures are analyzed in detail. Calling local covariant theories dynamically local if their kinematical and dynamical local substructures coincide, it is shown that the class of dynamically local theories fulfills the SPASs property. As an application in quantum field theory, we give a model independent proof of the impossibility of making a covariant choice of preferred state in all spacetimes, for theories obeying dynamical locality together with typical assumptions.     

Strong regularity and generalized inverses in jordan systems

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a?J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a?P(a)²J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a?J with generalized inverse b the subtriple generated by {a, b} is strongly associative     

Sobriety in Terms of Nets

Sobriety is a subtle notion of completeness for topological spaces: A space is sober if it may be reconstructed from the lattice of its open subsets. The usual criterion to check sobriety involves either irreducible closed subsets or completely prime filters of open sets. This paper provides an alternative possibility, thus trying to make sobriety easier to understand. We define the notion of observative net, which, together with an appropriate convergence notion, characterizes sobriety. As the filter approach does not involve just usual (topological) convergence, this is not an instance of the classical net-filter translation in general topology.     

A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation

In this paper, we solve the Cauchy problem for an inhomogeneous Helmholtz-type equation with homogeneous Dirichlet and Neumann boundary condition. The proposed problem is ill-posed. Up to now, most investigations on this topic focus on very specific cases, and with Dirichlet boundary condition. Recently, we solve this problem in 2D for an inhomogeneous modified Helmholtz equation (2012). This work is a continuous expansion of our previous results. Herein we introduce a general filter regularization (GFR) method, and then from the GFR we deduce two concrete filters, which are a foundation to implement a numerical procedure. In addition, we develop a numerical model for solving this problem in three dimensional region. The proposed filter method has been verified by numerical experiments.     

Completion of topological loops

A Hausdorff topological group equipped with the right uniformity admits a group completion iff the inversion mapping preserves Cauchy filters, cf. [1], III. §3, No.5, Théorème 1. Up until today a general theorem on the completion of topological loops is not available, for partial results see [9], [10]. This is among others due to the fact that topological loops will not necessarily have a compatible right uniformity. The main results (6–8) of this paper are the following: All topological loops are locally uniform in the sense of [11], and, provided the notion of “Cauchy filter” is suitably chosen, they can be completed. An analogue of the completion theorem for groups cited above holds for topological loops. According to these aims the theory of completion of locally uniform spaces is developped in 1–5 of this paper.     

Weakly Cauchy Filters and Quasi-Uniform Completeness

It is well-known that the notion of a Smyth complete quasi-uniform space provides an appropriate notion of completeness to study many interesting quasi-metric spaces which appear in theoretical computer science. We observe that several of these spaces actually possess a stronger form of completeness based on the use of weakly Cauchy filters in the sense of H. H. Corson and we develop a theory of completion and completeness for this kind of filters. In parallel, we also study a more general notion of completeness based on the use of certain stable filters. Thus our results extend and generalize important theorems of Á. Császár, J. R. Isbell and N. R. Howes on uniform completeness.     

Radical of filters in BL‐algebras

In this paper, the notion of the radical of a filter in BL‐algebras is defined and several characterizations of the radical of a filter are given. Also we prove that A/F is an MV‐algebra if and only if D_s(A) ? F. After that we define the notion of semi maximal filter in BL‐algebras and we state and prove some theorems which determine the relationship between this notion and the other types of filters of a BL‐algebra. Moreover, we prove that A/F is a semi simple BL‐algebra if and only if F is a semi maximal filter of A. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Observability and nonlinear filtering

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability is explored. Finally, the results are partially extended to non-compact signal state spaces.     

Weak Equivalence of Internal Categories

Weak equivalence is defined as equivalence in the bicategory of modules between internal categories. It is known that two categories are weakly equivalent if and only if their Cauchy completions are equivalent. We prove that this condition can be generalized to a suitable notion of intermediate category, stable under composition with weak equivalences. Applications to categorical Morita theory are given.     

The Relationship Between Various Filter Notions on aGL-Monoid

The notion of a generalised filter is extended to the setting of aGL-monoid. It is shown that there exists a one-to-one correspondence between the collection of generalised filters on a setXand the collection of strongly stratifiedL-filters onX. Specialising to the case whereLis the closed unit interval [0, c] viewed as a Heyting algebra, we show that any strongly stratified [0, c]-filter onXcan be uniquely identified with a saturated filter onI^Xwith characteristic valuec. In this way, the notion of a generalised filter unifies various filter notions. In particular, necessity measures and finitely additive probability measures are specific examples of generalised filters.