scholarly journals Closure operators

1975 ◽  
Vol 19 (3) ◽  
pp. 321-336
Author(s):  
P. J. Collyer ◽  
R. P. Sullivan
Keyword(s):  
Closure Operator ◽  
Closure Operators

A mapping κ: P(X) → P(X) is a quasi-closure operator (see Thron (1966) page 44) if (i) □κ = □, and for all A, B ∈ P(X) we have (ii) A ⊆ Aκ, and (iii) (A ⋓ B)κ = Aκ ∪ Bκ one easily deduces that such operators have the further property: (iv) if A ⊆ B ⊆ X, then Aκ if κ also satisfies: (v) Aκ2 ⊆ Aκ for all A ⊆ X, then κ is called a Kuratowski closure operator.

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

2021 ◽  
Vol 179 (1) ◽  
pp. 59-74
Author(s):  
Josef Šlapal
Keyword(s):  
Direct Product ◽  
Jordan Curve ◽  
Closure Operator ◽  
Jordan Curve Theorem ◽  
Closure Operators ◽  
Ternary Relation ◽  
Digital Space ◽  
Jordan Curves ◽  
Curves And Surfaces ◽  
Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

scholarly journals Path-set induced closure operators on graphs

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 863-871 ◽  
Author(s):  
Josef Slapal
Keyword(s):  
Image Analysis ◽  
Digital Image ◽  
Digital Image Analysis ◽  
Closure Operator ◽  
Simple Graph ◽  
Digital Topology ◽  
Closure Operators ◽  
Vertex Set ◽  
Positive Length ◽  
Path Connectedness

Given a simple graph, we associate with every set of paths of the same positive length a closure operator on the (vertex set of the) graph. These closure operators are then studied. In particular, it is shown that the connectedness with respect to them is a certain kind of path connectedness. Closure operators associated with sets of paths in some graphs with the vertex set Z2 are discussed which include the well known Marcus-Wyse and Khalimsky topologies used in digital topology. This demonstrates possible applications of the closure operators investigated in digital image analysis.

Generalized Fuzzy Closed Sets in Smooth Bitopological Spaces

2016 ◽  
pp. 419-456
Author(s):  
Osama A. El-Tantawy ◽  
Sobhy A. El-Sheikh ◽  
Rasha N. Majeed
Keyword(s):  
Closure Operator ◽  
Closure Operators ◽  
Closed Sets ◽  
Different Types ◽  
Definition Of ◽  
Bitopological Spaces

This chapter is devoted to the study of r-generalized fuzzy closed sets (briefly, gfc sets) in smooth bitopological spaces (briefly, smooth bts) in view definition of Šostak (1985). The chapter is divided into seven sections. The aim of Sections 1-2 is to introduce the fundamental concepts related to the work. In Section 3, the concept of r-(ti,tj)-gfc sets in the smooth bts's is introduce and investigate some notions of these sets, generalized fuzzy closure operator induced from these sets. In Section 4, (i,j)-GF-continuous (respectively, irresolute) mappings are introduced. In Section 5, the supra smooth topology which generated from a smooth bts is used to introduce and study the notion of r-t12-gfc sets for smooth bts's and supra generalized fuzzy closure operator . The present notion of gfc sets and the notion which introduced in section 3, are independent. In Section 6, two types of generalized supra fuzzy closure operators introduced by using two different approaches. Finally, Section 7 introduces and studies different types of fuzzy continuity which are related to closure.

The classification of small types of rank ω, Part I

2001 ◽  
Vol 66 (4) ◽  
pp. 1884-1898
Author(s):  
Steven Buechler ◽  
Colleen Hoover
Keyword(s):  
Stability Theory ◽  
Closure Operator ◽  
Algebraic Closure ◽  
Countable Model ◽  
Closure Operators ◽  
Basic Concepts ◽  
Countable Models

Abstract.Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we proveTheorem A. Let G be a superstate group of U-rank ω such that the generics of G are locally modular and Th(G) has few countable models. Let G− be the group of nongeneric elements of G. G+ = Go + G−. Let Π = {q ∈ S(∅): U(q) < ω}. For any countable model M of Th(G) there is a finite A ⊂ M such thai M is almost atomic over A ∪ (G+ ∩ M) ∪ ⋃p∈Πp(M).

The least extension of a closure operator on a sublattice

1967 ◽  
Vol 63 (1) ◽  
pp. 9-10 ◽  
Author(s):  
Roy O. Davies
Keyword(s):  
Complete Lattice ◽  
Closure Operator ◽  
Partial Ordering ◽  
Closure Operators ◽  
Image Position

Introduction. We recall (see Morgan Ward (3), Birkhoff (1), p. 49) that a closure operator on a complete lattice ℒ is a mapping f from ℒ into ℒ satisfying the conditionsfor all X, Y ∈ ℒ. The set of all closure operators on ℒ is itself a complete lattice with respect to the partial ordering in which f1 ≤ f2 if and only if f1(X) ≥ f2(X) for all X ∈ ℒ for every subset ℱ of and every element X ∈ ℒ we have.

scholarly journals Path-induced closure operators on graphs for defining digital Jordan surfaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1374-1380
Author(s):  
Josef Šlapal
Keyword(s):  
Positive Integer ◽  
Closure Operator ◽  
Simple Graph ◽  
Closure Operators ◽  
Adjacency Graph ◽  
Digital Space ◽  
Vertex Set

Abstract Given a simple graph with the vertex set X, we discuss a closure operator on X induced by a set of paths with identical lengths in the graph. We introduce a certain set of paths of the same length in the 2-adjacency graph on the digital line ℤ and consider the closure operators on ℤm (m a positive integer) that are induced by a special product of m copies of the introduced set of paths. We focus on the case m = 3 and show that the closure operator considered provides the digital space ℤ3 with a connectedness that may be used for defining digital surfaces satisfying a Jordan surface theorem.

scholarly journals The topology of θω-open sets

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5369-5377 ◽  
Author(s):  
Ghour Al ◽  
Bayan Irshedat
Keyword(s):  
Topological Space ◽  
Closure Operator ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Product Theorem ◽  
Closure Operators ◽  
Separation Axiom ◽  
New Class

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.

scholarly journals CLOSURE OPERATORS, FRAMES AND NEATEST REPRESENTATIONS

2017 ◽  
Vol 96 (3) ◽  
pp. 361-373
Author(s):  
ROB EGROT
Keyword(s):  
Closure Operator ◽  
Sufficient Condition ◽  
Recursive Construction ◽  
Necessary And Sufficient Condition ◽  
Closure Operators ◽  
Closed Sets ◽  
Necessary And Sufficient

Given a poset $P$ and a standard closure operator $\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$, we give a necessary and sufficient condition for the lattice of $\unicode[STIX]{x1D6E4}$-closed sets of ${\wp}(P)$ to be a frame in terms of the recursive construction of the $\unicode[STIX]{x1D6E4}$-closure of sets. We use this condition to show that, given a set ${\mathcal{U}}$ of distinguished joins from $P$, the lattice of ${\mathcal{U}}$-ideals of $P$ fails to be a frame if and only if it fails to be $\unicode[STIX]{x1D70E}$-distributive, with $\unicode[STIX]{x1D70E}$ depending on the cardinalities of sets in ${\mathcal{U}}$. From this we deduce that if a poset has the property that whenever $a\wedge (b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$, then it has an $(\unicode[STIX]{x1D714},3)$-representation.

A Condition for Equality of Cardinals of Minimal Generators Under Closure Operators

1971 ◽  
Vol 14 (4) ◽  
pp. 569-570 ◽  
Author(s):  
Japheth Hall
Keyword(s):  
Closure Operator ◽  
Closure Operators ◽  
Minimal Generators

Let C be an operator on the subsets of a set X with values among the subsets of X. We assume that C is a closure operator in X, i.e. a monotone, idempotent and extensive operator in X (cf., e.g., Birkhoff [3, p. 39], Schmidt [1], [2]). If A ⊆ X and B ⊆ X, we say that A and B are C-equivalent if C(A) = C(B) (Bleicher- Marczewski [4, p. 210]).

scholarly journals The orbit of closure-involution operations: the case of Boolean functions

2021 ◽  
Author(s):  
Jürgen Dassow
Keyword(s):  
Boolean Functions ◽  
Closure Operator ◽  
Closure Operators ◽  
Repeated Applications

AbstractFor a set A of Boolean functions, a closure operator c and an involution i, let $$\mathcal{N}_{c,i}(A)$$ N c , i ( A ) be the number of sets which can be obtained from A by repeated applications of c and i. The orbit $$\mathcal{O}(c,i)$$ O ( c , i ) is defined as the set of all these numbers. We determine the orbits $$\mathcal{O}(S,i)$$ O ( S , i ) where S is the closure defined by superposition and i is the complement or the duality. For the negation $${{\,\mathrm{non}\,}}$$ non , the orbit $$\mathcal{O}(S,{{\,\mathrm{non}\,}})$$ O ( S , non ) is almost determined. Especially, we show that the orbit in all these cases contains at most seven numbers. Moreover, we present some closure operators where the orbit with respect to duality and negation is arbitrarily large.

Sign in / Sign up

Export Citation Format

Share Document