scholarly journals Stashing And Parallelization Pentagons

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Vasco Brattka
Keyword(s):  
Original Problem ◽  
Closure Operator ◽  
Algebraic Operation ◽  
Discontinuity Problem ◽  
Interior Operator ◽  
Natural Way

Parallelization is an algebraic operation that lifts problems to sequences in a natural way. Given a sequence as an instance of the parallelized problem, another sequence is a solution of this problem if every component is instance-wise a solution of the original problem. In the Weihrauch lattice parallelization is a closure operator. Here we introduce a dual operation that we call stashing and that also lifts problems to sequences, but such that only some component has to be an instance-wise solution. In this case the solution is stashed away in the sequence. This operation, if properly defined, induces an interior operator in the Weihrauch lattice. We also study the action of the monoid induced by stashing and parallelization on the Weihrauch lattice, and we prove that it leads to at most five distinct degrees, which (in the maximal case) are always organized in pentagons. We also introduce another closely related interior operator in the Weihrauch lattice that replaces solutions of problems by upper Turing cones that are strong enough to compute solutions. It turns out that on parallelizable degrees this interior operator corresponds to stashing. This implies that, somewhat surprisingly, all problems which are simultaneously parallelizable and stashable have computability-theoretic characterizations. Finally, we apply all these results in order to study the recently introduced discontinuity problem, which appears as the bottom of a number of natural stashing-parallelization pentagons. The discontinuity problem is not only the stashing of several variants of the lesser limited principle of omniscience, but it also parallelizes to the non-computability problem. This supports the slogan that "non-computability is the parallelization of discontinuity".

Fuzzy roughness via ideals

2020 ◽  
Vol 39 (5) ◽  
pp. 6869-6880
Author(s):  
S. H. Alsulami ◽  
Ismail Ibedou ◽  
S. E. Abbas
Keyword(s):  
Fuzzy Set ◽  
Closure Operator ◽  
Approximation Space ◽  
Approximation Spaces ◽  
Fuzzy Connectedness ◽  
Fuzzy Approximation ◽  
Separation Axioms ◽  
Fuzzy Ideal ◽  
Interior Operator

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

scholarly journals On induced map f# in fuzzy set theory and its applications to fuzzy continuity

2021 ◽  
Author(s):  
Sandeep Kaur ◽  
Nitakshi Goyal
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Closure Operator ◽  
Topological Spaces ◽  
Continuous Mappings ◽  
New Vision ◽  
Saturated Sets ◽  
Interior Operator ◽  
Fuzzy Topological Spaces

Abstract In this paper, we introduce # image of a fuzzy set which gives a induced map f # corresponding to any function f : X → Y , where X and Y are crisp sets. With this, we present a new vision of studying fuzzy continuous mappings in fuzzy topological spaces where fuzzy continuity explains the term of closeness in the mathematical models. We also define the concept of fuzzy saturated sets which helps us to prove some new characterizations of fuzzy continuous mappings in terms of interior operator rather than closure operator.

scholarly journals Closure System and Its Semantics

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 198
Author(s):  
Yinbin Lei ◽  
Jun Zhang
Keyword(s):  
Closure Operator ◽  
Spatial Relations ◽  
Accumulation Point ◽  
Boundary Operator ◽  
Topological Spaces ◽  
Closure System ◽  
Derived Set ◽  
Topological Closure ◽  
Interior Operator ◽  
Isolated Point

It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration.

Neighborhoods and convergence with respect to a closure operator

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
Josef Šlapal
Keyword(s):  
Closure Operator ◽  
Topological Spaces ◽  
Filter Convergence ◽  
Categorical Closure Operator ◽  
Natural Way

AbstractWe study neighborhoods with respect to a categorical closure operator. In particular, we discuss separation and compactness obtained from neighborhoods in a natural way and compare them with the usual closure separation and closure compactness. We also introduce a concept of convergence based on using centered systems of subobjects, which naturally generalizes the classical filter convergence in topological spaces. We investigate behavior of the convergence introduced and show, among others, that it relates to the separation and compactness in natural ways.

scholarly journals L-topological derived internal (resp. Enclosed) relation spaces

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2497-2516
Author(s):  
Xiu-Yun Wu ◽  
Qi Liu ◽  
Chun-Yan Liao ◽  
Yan-Hui Zhao
Keyword(s):  
Topological Space ◽  
Closure Operator ◽  
Operator Space ◽  
Internal Relation ◽  
Interior Operator

In this paper, notions of L-topological derived internal relation space, L-topological derived interior operator space, L-topological derived enclosed relation space and L-topological derived closure operator space are introduced. It is proved that all of these spaces are categorically isomorphic to L-topological space, L-topological internal relation space and L-topological enclosed relation space.

The lattices of families of regular sets in topological spaces

2020 ◽  
Vol 70 (2) ◽  
pp. 477-488
Author(s):  
Emilia Przemska
Keyword(s):  
Closure Operator ◽  
Boolean Algebras ◽  
Topological Spaces ◽  
Closed Sets ◽  
The Family ◽  
Interior Operator ◽  
Regular Sets ◽  
Regular Open ◽  
Complemented Lattice ◽  
Regular Closed

Abstract The question as to the number of sets obtainable from a given subset of a topological space using the operators derived by composing members of the set {b, i, ∨, ∧}, where b, i, ∨ and ∧ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski {b, i, ∨, ∧}-problem. This problem has been solved independently by Sherman [21] and, Gardner and Jackson [13], where the resulting 34 plus identity operators were depicted in the Hasse diagram. In this paper we investigate the sets of fixed points of these operators. We show that there are at most 23 such families of subsets. Twelve of them are the topology, the family of all closed subsets plus, well known generalizations of open sets, plus the families of their complements. Each of the other 11 families forms a complete complemented lattice under the operations of join, meet and negation defined according to a uniform procedure. Two of them are the well known Boolean algebras formed by the regular open sets and regular closed sets, any of the others in general need not be a Boolean algebras.

scholarly journals On gω-Open Sets in Grill Topological Spaces

2020 ◽  
pp. 132-143
Author(s):  
Amin Saif ◽  
Mohammed Al-Hawmi ◽  
Basheer Al-Refaei
Keyword(s):  
Closure Operator ◽  
Weak Form ◽  
Topological Spaces ◽  
Cluster Operator ◽  
Open Set ◽  
Interior Operator ◽  
Operator Closure

The propose of this paper is to introduce and investigate a weak form of ω-open set in grill topological spaces. We introduce the notion of -open set as a form stronger than βω-open set and weaker than ω-open set and -open set. By using this form, we study the generalization property, the interior operator, closure operator and θ-cluster operator.

scholarly journals Pointwise k-Pseudo Metric Space

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2505
Author(s):  
Yu Zhong ◽  
Alexander Šostak ◽  
Fu-Gui Shi
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
Topological Structures ◽  
Neighborhood System ◽  
Interior Operator

In this paper, the concept of a k-(quasi) pseudo metric is generalized to the L-fuzzy case, called a pointwise k-(quasi) pseudo metric, which is considered to be a map d:J(LX)×J(LX)⟶[0,∞) satisfying some conditions. What is more, it is proved that the category of pointwise k-pseudo metric spaces is isomorphic to the category of symmetric pointwise k-remote neighborhood ball spaces. Besides, some L-topological structures induced by a pointwise k-quasi-pseudo metric are obtained, including an L-quasi neighborhood system, an L-topology, an L-closure operator, an L-interior operator, and a pointwise quasi-uniformity.

General edge asymptotics of solutions of second-order elliptic boundary value problems II

1993 ◽  
Vol 123 (1) ◽  
pp. 157-184 ◽  
Author(s):  
Martin Costabel ◽  
Monique Dauge
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Elliptic Boundary Value Problems ◽  
Opening Angle ◽  
Elliptic Boundary Value ◽  
Singular Functions ◽  
Interior Operator ◽  
Natural Way

SynopsisThis is the second of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears trie phenomenon of “crossing” of the exponents of singularities. In Part I, we introduced for the Dirichlet problem appropriate combinations of the simple tensor product singularities.In this second part, we extend the results of Part I to general non-homogeneous boundary conditions. Moreover, we show how these combinations of singularities appear in a natural way as sections of an analytic vector bundle above the edge. In the case when the interior operator is the Laplacian, we give a simpler expression of the combined singular functions, involving divided differences of powers of a complex variable describing the coordinates in the normal plane to the edge.

scholarly journals On \(G^{\beta}\) -Proper ty of \(G\)-Metric Spaces

2021 ◽  
pp. 37-45
Author(s):  
Mubarak AL-Hubaishi ◽  
Amin Saif
Keyword(s):  
Metric Spaces ◽  
Closure Operator ◽  
Weak Form ◽  
The Other ◽  
Interior Operator

The purpose of this paper is to introduce and investigate weak form of G-open sets in G-metric spaces, namely \(G^{\beta}\)-open sets. The relationships among this form with the other known sets are introduced. We give the notions of the interior operator, the closure operator and frontier operator via \(G^{\beta}\)-open sets.

Sign in / Sign up

Export Citation Format

Share Document