scholarly journals Some Variants of Normal Čech Closure Spaces via Canonically Closed Sets

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1225
Author(s):  
Ria Gupta ◽  
Ananga Kumar Das
Keyword(s):  
Closure Space ◽  
Closed Sets ◽  
Closure Spaces

New generalizations of normality in Čech closure space such as π-normal, weakly π-normal and κ-normal are introduced and studied using canonically closed sets. It is observed that the class of κ-normal spaces contains both the classes of weakly π-normal and almost normal Čech closure spaces.

Representation of bifinite domains by BF-closure spaces

2021 ◽  
Vol 71 (3) ◽  
pp. 565-572
Author(s):  
Lingjuan Yao ◽  
Qingguo Li
Keyword(s):  
Continuous Functions ◽  
Closure Space ◽  
Closed Sets ◽  
Set Inclusion ◽  
Closure Spaces ◽  
Scott Continuous

Abstract In this paper, we propose the notion of BF-closure spaces as concrete representation of bifinite domains. We prove that every bifinite domain can be obtained as the set of F-closed sets of some BF-closure space under set inclusion. Furthermore, we obtain that the category of bifinite domains and Scott-continuous functions is equivalent to that of BF-closure spaces and F-morphisms.

scholarly journals The Prime Stems of Rooted Circuits of Closure Spaces and Minimum Implicational Bases

10.37236/3068 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Masataka Nakamura ◽  
Kenji Kashiwabara
Keyword(s):  
Convex Geometry ◽  
Closure Operators ◽  
Optimal Basis ◽  
Closed Set ◽  
Closed Sets ◽  
Closure Systems ◽  
Closure Spaces ◽  
Convex Geometries ◽  
Necessary And Sufficient ◽  
Special Case

A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for closure systems or equivalently for closure operators. A rooted circuit is a specific type of a pair $(X,e)$ of a subset $X$, called a stem, and an element $e\not\in X$, called a root. We introduce a notion called a 'prime stem', which plays the key role in this article. Every prime stem is shown to be a pseudo-closed set of an implicational system. If the sizes of stems are all the same, the stems are all pseudo-closed sets, and they give rise to a canonical minimum implicational basis. For an affine convex geometry, the prime stems determine a canonical minimum basis, and furthermore  gives rise to an optimal basis. A 'critical rooted circuit' is a special case of a rooted circuit defined for an antimatroid. As a precedence structure, 'critical rooted circuits' are necessary and sufficient to fix an antimatroid whereas critical rooted circuits are not necessarily sufficient to restore the original antimatroid as an implicational system. It is shown through an example.

On normality of a closure space generated from a tree by relation

2021 ◽  
pp. 2250110
Author(s):  
A. K. Das ◽  
Ria Gupta
Keyword(s):  
Comparative Study ◽  
Binary Relation ◽  
Applied Mathematics ◽  
Closure Space ◽  
Topological Properties ◽  
Closure Spaces

Binary relation plays a prominent role in the study of mathematics in particular applied mathematics. Recently, some authors generated closure spaces through relation and made a comparative study of topological properties in the space by varying the property on the relation. In this paper, we have studied closure spaces generated from a tree through binary relation and observed that under certain situation the space generated from a tree is normal.

scholarly journals Extensions of closure spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 223
Author(s):  
D. Deses ◽  
A. De Groot-Van der Voorde ◽  
E. Lowen-Colebunders
Keyword(s):  
Closure Operator ◽  
Finite Additivity ◽  
Closure Space ◽  
Closure Spaces ◽  
The One ◽  
Separation Conditions

<p>A closure space X is a set endowed with a closure operator P(X) → P(X), satisfying the usual topological axioms, except finite additivity. A T<sub>1</sub> closure extension Y of a closure space X induces a structure ϒ on X satisfying the smallness axioms introduced by H. Herrlich [?], except the one on finite unions of collections. We'll use the word seminearness for a smallness structure of this type, i.e. satisfying the conditions (S1),(S2),(S3) and (S5) from [?]. In this paper we show that every T<sub>1</sub> seminearness structure ϒ on X can in fact be induced by a T<sub>1</sub> closure extension. This result is quite different from its topological counterpart which was treated by S.A. Naimpally and J.H.M. Whitfield in [?]. Also in the topological setting the existence of (strict) extensions satisfying higher separation conditions such as T<sub>2</sub> and T<sub>3</sub> has been completely characterized by means of concreteness, separatedness and regularity [?]. In the closure setting these conditions will appear to be too weak to ensure the existence of suitable (strict) extensions. In this paper we introduce stronger alternatives in order to present internal characterizations of the existence of (strict) T<sub>2</sub> or strict regular closure extensions.</p>

scholarly journals Lattices of regular closed subsets of closure spaces

2014 ◽  
Vol 24 (07) ◽  
pp. 969-1030 ◽  
Author(s):  
Luigi Santocanale ◽  
Friedrich Wehrung
Keyword(s):  
Hyperplane Arrangement ◽  
Convex Geometry ◽  
Macneille Completion ◽  
Closed Set ◽  
Closed Sets ◽  
Closure Spaces ◽  
Infinite Collection ◽  
Convex Geometries ◽  
Poset Of Regions ◽  
Regular Closed

For a closure space (P, φ) with φ(ø) = ø, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg (P, φ), extending the poset Clop (P, φ) of all clopen subsets. If (P, φ) is a finite convex geometry, then Reg (P, φ) is pseudocomplemented. The Dedekind–MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg (P, φ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces,• Reg (P, φ) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity.• If Reg (P, φ) is semidistributive, then it is a bounded homomorphic image of a free lattice.• Clop (P, φ) is a lattice if and only if every regular closed set is clopen.The extended permutohedron R (G) on a graph G and the extended permutohedron Reg S on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of all regular closed sets is, in the semilattice context, always the Dedekind–MacNeille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R (G) and Reg S are bounded homomorphic images of free lattices.

C ech ψ∞g-Closed sets in Closure Spaces

2016 ◽  
Vol 33 (2) ◽  
pp. 130-134
Author(s):  
V Kokilavani ◽  
◽  
P.R Kavitha
Keyword(s):  
Closed Sets ◽  
Closure Spaces

Group of closure isomorphisms of Cech closure spaces

2016 ◽  
Vol 7 (3) ◽  
pp. 132
Author(s):  
Kavitha T. ◽  
Sini P. ◽  
Ramachandran P. T.
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Closure Operator ◽  
Closure Space ◽  
Closure Spaces

Here we discuss some results on the group of all closure isomorphisms of a \u{C}ech closure space. A subgroup \(H\) of the symmetric group \(S(X)\) is \(c\) representable on $X$ if there exists a closure operator \(V\) on $X$ such that the group of closure isomorphisms of the closure space \((X,\ V)\) is \(H\). In this paper, we prove a non trivial normal subgroup of the symmetric group \(S(X)\) is \(c\)-representable on \(X\) if and only if the cardinality of \(X\) is three.

C̆ech Fuzzy Soft Closure Spaces

2018 ◽  
Vol 7 (2) ◽  
pp. 62-74 ◽  
Author(s):  
Rasha Naser Majeed
Keyword(s):  
Topological Space ◽  
Subject Classification ◽  
Closure Space ◽  
Soft Sets ◽  
Basic Properties ◽  
Closed Mapping ◽  
Closure Spaces ◽  
Fuzzy Soft Sets ◽  
Soft Topological Space

In this paper, the C̆ech fuzzy soft closure spaces are defined and their basic properties are studied. Closed (respectively, open) fuzzy soft sets is defined in C̆ech fuzzy-soft closure spaces. It has been shown that for each C̆ech fuzzy soft closure space there is an associated fuzzy soft topological space. In addition, the concepts of a subspace and a sum are defined in C̆ech fuzzy soft closure space. Finally, fuzzy soft continuous (respectively, open and closed) mapping between C̆ech fuzzy soft closure spaces are introduced. Mathematics Subject Classification: 54A40, 54B05, 54C05.

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