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.

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 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.

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.

Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

2020 ◽  
Vol 63 (4) ◽  
pp. 1071-1091
Author(s):  
Luke Morgan ◽  
Cheryl E. Praeger ◽  
Kyle Rosa
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Symmetric Group ◽  
Minimal Normal Subgroup ◽  
Structural Information ◽  
Minimal Degree ◽  
Permutation Groups ◽  
Primitive Groups ◽  
Base Size ◽  
Abstract Structure

AbstractIn this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may be useful in place of precise structural information. We give bounds on the order, base size, minimal degree, fixed point ratio, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we study the structure of a finite semiprimitive group that induces the alternating or symmetric group on the set of orbits of an intransitive minimal normal subgroup.

scholarly journals Strongly Connectedness in Closure Space

2014 ◽  
Vol 9 ◽  
pp. 69-71
Author(s):  
U.D. Tapi ◽  
Bhagyashri A. Deole
Keyword(s):  
Topological Space ◽  
Disjoint Union ◽  
Closure Operator ◽  
Closure Space ◽  
Closed Set ◽  
Power Set ◽  
Strongly Connected

A Čech closure space (X, u) is a set X with Čech closure operator u: P(X) → P(X) where P(X) is a power set of X, which satisfies u𝝓=𝝓, A ⊆uA for every A⊆X, u (A⋃B) = uA⋃uB, for all A, B ⊆ X. Many properties which hold in topological space hold in closure space as well. A topological space X is strongly connected if and only if it is not a disjoint union of countably many but more than one closed set. If X is strongly connected, and Ei‟s are nonempty disjoint closed subsets of X, then X≠ E1∪E2∪. We further extend the concept of strongly connectedness in closure space. The aim of this paper is to introduce and study the concept of strongly connectedness in closure space.

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 Finitely presented algebras defined by permutation relations of dihedral type

2016 ◽  
Vol 26 (01) ◽  
pp. 171-202 ◽  
Author(s):  
Ferran Cedó ◽  
Eric Jespers ◽  
Georg Klein
Keyword(s):  
Normal Subgroup ◽  
Normal Form ◽  
Symmetric Group ◽  
Cyclic Group ◽  
Index Formula ◽  
Universal Group ◽  
Product Group ◽  
Unique Product ◽  
Dihedral Type ◽  
Finitely Presented

The class of finitely presented algebras over a field [Formula: see text] with a set of generators [Formula: see text] and defined by homogeneous relations of the form [Formula: see text], where [Formula: see text] runs through a subset [Formula: see text] of the symmetric group [Formula: see text] of degree [Formula: see text], is investigated. Groups [Formula: see text] in which the cyclic group [Formula: see text] is a normal subgroup of index [Formula: see text] are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid [Formula: see text] with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovskij. The universal group [Formula: see text] of [Formula: see text] is a unique product group, and it is the central localization of a cancellative subsemigroup of [Formula: see text]. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra [Formula: see text] is semiprimitive.

Unitary Representations of Generalized Symmetric Groups

1969 ◽  
Vol 21 ◽  
pp. 28-38 ◽  
Author(s):  
B. M. Puttaswamaiah
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Symmetric Groups ◽  
Unitary Representations ◽  
Complex Field ◽  
Permutation Matrices ◽  
Image Position

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute withObviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) andthen {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

scholarly journals Group closures of one-to-one transformations

2001 ◽  
Vol 64 (2) ◽  
pp. 177-188 ◽  
Author(s):  
Inessa Levi
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternating Group ◽  
One To One ◽  
Necessary And Sufficient ◽  
Infinite Set

For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

scholarly journals On the units of a modular group ring II

1973 ◽  
Vol 8 (3) ◽  
pp. 435-442 ◽  
Author(s):  
K.R. Pearson
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Symmetric Group ◽  
Group Ring ◽  
Modular Group ◽  
Group Of Units ◽  
Nonzero Characteristic ◽  
A Finite Group

Let R be a ring of nonzero characteristic and let G be a finite group with subgroup H. It is shown that H is a normal subgroup of the group of units of the group ring RG if and only if H is contained in the centre of G or R is the field with 2 elements, G is the symmetric group on 3 letters and H is normal in G.

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