scholarly journals Equivalent conditions for digital covering maps

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4005-4014
Author(s):  
Ali Pakdaman ◽  
Mehdi Zakki
Keyword(s):  
Local Isomorphism ◽  
Covering Map ◽  
Unique Path ◽  
Continuous Surjection ◽  
Digital Covering ◽  
Equivalent Conditions ◽  
Digital Spaces

It is known that every digital covering map p:(E,k) ? (B,?) has the unique path lifting property. In this paper, we show that its inverse is true when the continuous surjective map p has no conciliator point. Also, we prove that a digital (k,?)-continuous surjection p:(E,k)? (B,?) is a digital covering map if and only if it is a local isomorphism, when all digital spaces are connected. Moreover, we find out a loop criterion for a digital covering map to be a radius n covering map.

scholarly journals Ultra regular covering space and its automorphism group

2010 ◽  
Vol 20 (4) ◽  
pp. 699-710 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Automorphism Group ◽  
Transformation Group ◽  
Covering Space ◽  
Fundamental Groups ◽  
Discrete Transformation ◽  
Local Isomorphism ◽  
Digital Space ◽  
Regular Covering ◽  
Digital Covering ◽  
Digital Spaces

Ultra regular covering space and its automorphism groupIn order to classify digital spaces in terms of digital-homotopic theoretical tools, a recent paper by Han (2006b) (see also the works of Boxer and Karaca (2008) as well as Han (2007b)) established the notion of regular covering space from the viewpoint of digital covering theory and studied an automorphism group (or Deck's discrete transformation group) of a digital covering. By using these tools, we can calculate digital fundamental groups of some digital spaces and classify digital covering spaces satisfying a radius 2 local isomorphism (Boxer and Karaca, 2008; Han, 2006b; 2008b; 2008d; 2009b). However, for a digital covering which does not satisfy a radius 2 local isomorphism, the study of a digital fundamental group of a digital space and its automorphism group remains open. In order to examine this problem, the present paper establishes the notion of an ultra regular covering space, studies its various properties and calculates an automorphism group of the ultra regular covering space. In particular, the paper develops the notion of compatible adjacency of a digital wedge. By comparing an ultra regular covering space with a regular covering space, we can propose strong merits of the former.

The Poset of Perfect Irreducible Images of a Space

1989 ◽  
Vol 41 (2) ◽  
pp. 213-233 ◽  
Author(s):  
Jack R. Porter ◽  
R. Grant Woods
Keyword(s):  
Topological Spaces ◽  
Covering Map ◽  
Continuous Surjection

We begin by briefly summarizing the contents of this paper; details, and some definitions of terminology, appear in subsequent sections. All hypothesized topological spaces are assumed to be Hausdorff. The reader is referred to [13] for undefined notation and terminology.A perfect irreducible continuous surjection is called a covering map. Let X be a space, let f and g be two such functions with domain X, and let Rf denote the range of (i.e., the set f [Z]). Then f and g are said to be equivalent (denoted f≈ g) if there is a homeomorphism h : Rf —” Rg such that h of = g. We identify equivalent covering maps with domain X, and then denote by IP(X) the set of such covering maps.

scholarly journals Unique pseudolifting property in digital topology

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 739-746
Author(s):  
Sang-Eon Han
Keyword(s):  
Homotopy Theory ◽  
Digital Topology ◽  
Fundamental Groups ◽  
Digital Version ◽  
Digital Covering ◽  
Digital Spaces

The generalized universal lifting property plays an important role in classical topology. In digital topology we also have its digital version [5, 6, 14]. More precisely, the paper [6] established the concept of a digital covering (see also [11, 17]). It has substantially contributed to the calculation of digital fundamental groups of some digital spaces, the classification of digital spaces and so forth. The paper [6] also established the unique lifting property of a digital covering which plays an important role in studying both digital covering and digital homotopy theory. Motivated by the unique lifting property, the paper develops a pseudocovering which is weaker than a digital covering and investigates its various properties. Furthermore, the paper proves that a pseudocovering with some hypothesis has the unique pseudolifting property which is weaker than the unique lifting property in digital covering theory.

scholarly journals The Most Refined Axiom for a Digital Covering Space and Its Utilities

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1868
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Covering Space ◽  
Fundamental Groups ◽  
Crucial Step ◽  
Covering Map ◽  
Five Elements ◽  
Digital Covering

This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local (k0,k1)-isomorphisms and use the most simplified local (k0,k1)-isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local (k0,k1)-isomorphism is proved to be a (k0,k1)-covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed k:=k(t,n)-curve with five elements in Zn, denoted by SCkn,5. After introducing the notion of digital topological imbedding, we investigate some properties of SCkn,5, where k:=k(t,n),3≤t≤n. Since SCkn,5 is the minimal and simple closed k-curve with odd elements in Zn which is not k-contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k-connected digital images.

scholarly journals Test map characterizations of local properties of fundamental groups

2018 ◽  
Vol 12 (01) ◽  
pp. 37-85 ◽  
Author(s):  
Jeremy Brazas ◽  
Hanspeter Fischer
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closure Operator ◽  
Fundamental Groups ◽  
Space Theory ◽  
Unified Approach ◽  
Covering Map ◽  
Unique Path ◽  
Local Properties ◽  
Path Connected

Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any path-connected metric space [Formula: see text], and a subgroup [Formula: see text], we characterize the unique path lifting property relative to [Formula: see text] in terms of a new closure operator on the [Formula: see text]-subgroup lattice that is induced by maps from a fixed “test” domain into [Formula: see text]. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.

scholarly journals Affine Subspaces of Curvature Functions from Closed Planar Curves

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Leonardo Alese
Keyword(s):  
Arc Length ◽  
Planar Curves ◽  
Affine Subspaces ◽  
Planar Curve ◽  
Real Functions ◽  
Discrete Counterpart ◽  
Equivalent Conditions

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

scholarly journals Quasi-ideal Ehresmann transversals: The spined product structure

2021 ◽  
Vol 19 (1) ◽  
pp. 77-86
Author(s):  
Xiangjun Kong ◽  
Pei Wang ◽  
Jian Tang
Keyword(s):  
Structure Theorem ◽  
Product Structure ◽  
Abundant Semigroup ◽  
Spined Product ◽  
Equivalent Conditions

Abstract In any U-abundant semigroup with an Ehresmann transversal, two significant components R and L are introduced in this paper and described by Green’s ∼ \sim -relations. Some interesting properties associated with R and L are explored and some equivalent conditions for the Ehresmann transversal to be a quasi-ideal are acquired. Finally, a spined product structure theorem is established for a U-abundant semigroup with a quasi-ideal Ehresmann transversal by means of R and L.

scholarly journals Some estimates for the commutators of multilinear maximal function on Morrey-type space

2021 ◽  
Vol 19 (1) ◽  
pp. 515-530
Author(s):  
Xiao Yu ◽  
Pu Zhang ◽  
Hongliang Li
Keyword(s):  
Maximal Function ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Bounded Mean Oscillation ◽  
Generalized Morrey Spaces ◽  
Type Space ◽  
Bmo Function ◽  
Mean Oscillation ◽  
Multilinear Maximal Function ◽  
Equivalent Conditions

Abstract In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

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