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.

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.

On the extension of the Pflastersatz (Part II)

1937 ◽  
Vol 33 (1) ◽  
pp. 13-20 ◽  
Author(s):  
B. Kaufmann
Keyword(s):  
Euclidean Space ◽  
Closed Set ◽  
Closed Sets ◽  
Special Case

In the first part of this paper I announced some new Pflaster theorems for arbitrary r-dimensional closed sets lying in the Euclidean space Rn. In § III I proved them for the special case of an r-dimensional closed set F linked (rel a neighbourhood U) with an (n – r – 1)-dimensional spherical cycle. I shall now prove these theorems in the general case of a quite arbitrary closed set F.

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.

scholarly journals Some lattices of closure systems on a finite set

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Nathalie Caspard ◽  
Bernard Monjardet
Keyword(s):  
Systematic Study ◽  
Closed Set ◽  
Covering Relation ◽  
Closure Systems ◽  
Irreducible Elements ◽  
Finite Set ◽  
International Audience ◽  
Convex Geometries

International audience In this paper we study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set, we determine the covering relation \prec of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C' where C and C' are two such closure systems satisfying C \prec C'. We also do a systematic study of these lattices of closure systems, characterizing for instance their join-irreducible and their meet-irreducible elements.

scholarly journals Applications of operations on generalized topological spaces

2021 ◽  
Vol 103 (3) ◽  
pp. 96-104
Author(s):  
B. Roy ◽  
◽  
T. Noiri
Keyword(s):  
Closure Operator ◽  
Topological Spaces ◽  
Closure Operators ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
Separation Axioms ◽  
Weak Separation ◽  
General Collection ◽  
Generalized Type

In this paper γµ -open sets and γµ -closed sets in a GTS (X, µ) have been studied, where γµ is an operation from µ to P(X). In general, collection of γµ -open sets is smaller than the collection of µ-open sets. The condition under which both are same are also established here. Some properties of such sets have been discussed. Some closure like operators are also defined and their properties are discussed. The relation between similar types of closure operators on the GTS (X, µ) has been established. The condition under which the newly defined closure like operator is a Kuratowski closure operator is given. We have also defined a generalized type of closed sets termed as γµ -generalized closed set with the help of this newly defined closure operator and discussed some basic properties of such sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have shown some preservation theorems of such generalized concepts.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

A Classification and Closure Properties of Languages for Describing Concurrent System Behaviours

1981 ◽  
Vol 4 (3) ◽  
pp. 531-549 ◽  
Author(s):  
Miklós Szijártó
Keyword(s):  
Formal Languages ◽  
Parallel Program ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Concurrent System ◽  
Closure Properties ◽  
Sequential Program ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Independence Relations

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

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.

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

n-closure systems and n-closure operators

2006 ◽  
Vol 55 (2-3) ◽  
pp. 369-386 ◽  
Author(s):  
George Voutsadakis
Keyword(s):  
Closure Operators ◽  
Closure Systems
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