Self-Complementary Generalized Orbits of a Permutation Group

1974 ◽  
Vol 17 (2) ◽  
pp. 203-208 ◽  
Author(s):  
Roberto Frucht ◽  
Frank Harary
Keyword(s):  
Permutation Group ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Cycle Index ◽  
Arbitrary Permutation ◽  
Group A ◽  
Special Classes

AbstractA permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k) the set of all k subsets of X, and the resulting equivalence classes are called k orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X—S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits.

SINGULAR SWITCHED LINEAR SYSTEMS: SOME GEOMETRIC ASPECTS

2012 ◽  
Vol 26 (25) ◽  
pp. 1246006
Author(s):  
H. DIEZ-MACHÍO ◽  
J. CLOTET ◽  
M. I. GARCÍA-PLANAS ◽  
M. D. MAGRET ◽  
M. E. MONTORO
Keyword(s):  
Linear Systems ◽  
Group Action ◽  
Lie Group ◽  
Equivalence Relation ◽  
Geometric Approach ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Switched Linear Systems ◽  
Lie Group Action

We present a geometric approach to the study of singular switched linear systems, defining a Lie group action on the differentiable manifold consisting of the matrices defining their subsystems with orbits coinciding with equivalence classes under an equivalence relation which preserves reachability and derive miniversal (orthogonal) deformations of the system. We relate this with some new results on reachability of such systems.

scholarly journals Invariant connections and invariant holomorphic bundles on homogeneous manifolds

2014 ◽  
Vol 12 (1) ◽  
pp. 1-13
Author(s):  
Indranil Biswas ◽  
Andrei Teleman
Keyword(s):  
Lie Group ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Transitive Action ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Group A ◽  
Holomorphic Bundles ◽  
Invariant Gauge

AbstractLet X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

Maximal Order of an NG-group

2021 ◽  
pp. 33-38
Author(s):  
Faraj. A. Abdunabi
Keyword(s):  
Permutation Group ◽  
Equivalence Relation ◽  
Maximal Order ◽  
Quotient Set

This study was aimed to consider the NG-group that consisting of transformations on a nonempty set A has no bijection as its element. In addition, it tried to find the maximal order of these groups. It found the order of NG-group not greater than n. Our results proved by showing that any kind of NG-group in the theorem be isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A. Keywords: NG-group; Permutation group; Equivalence relation; -subgroup

scholarly journals A General Theory of Wilf-Equivalence for Catalan Structures

10.37236/5629 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
General Theory ◽  
Equivalence Relation ◽  
Generating Functions ◽  
Asymptotic Estimate ◽  
Equivalence Classes ◽  
Catalan Number ◽  
Combinatorial Structures ◽  
Dyck Paths ◽  
Significant Interest ◽  
Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

Cauchy Points of Metric Locales

1989 ◽  
Vol 41 (5) ◽  
pp. 830-854 ◽  
Author(s):  
B. Banaschewski ◽  
A. Pultr
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Equivalence Classes ◽  
Classical Description ◽  
Natural Approach ◽  
Precise Meaning ◽  
Natural Equivalence ◽  
Cauchy Sequences

A natural approach to topology which emphasizes its geometric essence independent of the notion of points is given by the concept of frame (for instance [4], [8]). We consider this a good formalization of the intuitive perception of a space as given by the “places” of non-trivial extent with appropriate geometric relations between them. Viewed from this position, points are artefacts determined by collections of places which may in some sense by considered as collapsing or contracting; the precise meaning of the latter as well as possible notions of equivalence being largely arbitrary, one may indeed have different notions of point on the same “space”. Of course, the well-known notion of a point as a homomorphism into 2 evidently fits into this pattern by the familiar correspondence between these and the completely prime filters. For frames equipped with a diameter as considered in this paper, we introduce a natural alternative, the Cauchy points. These are the obvious counterparts, for metric locales, of equivalence classes of Cauchy sequences familiar from the classical description of completion of metric spaces: indeed they are decreasing sequences for which the diameters tend to zero, identified by a natural equivalence relation.

scholarly journals On the Groups of Cobordism Ωk

1958 ◽  
Vol 13 ◽  
pp. 135-156 ◽  
Author(s):  
Masahisa Adachi
Keyword(s):  
Equivalence Relation ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Graded Algebra ◽  
Differentiable Manifolds ◽  
Free Part ◽  
Natural Way

In the papers [11] and [18] Rohlin and Thom have introduced an equivalence relation into the set of compact orientable (not necessarily connected) differentiable manifolds, which, roughly speaking, is described in the following manner: two differentiable manifolds are equivalent (cobordantes), when they together form the boundary of a bounded differentiable manifold. The equivalence classes can be added and multiplied in a natural way and form a graded algebra Ω relative to the addition, the multiplication and the dimension of manifolds. The precise structures of the groups of cobordism Ωk of dimension k are not known thoroughly. Thom [18] has determined the free part of Ω and also calculated explicitly Ωk for 0 ≦ k ≦ 7.

scholarly journals Bipolar Fuzzy Relations

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1044 ◽  
Author(s):  
Jeong-Gon Lee ◽  
Kul Hur
Keyword(s):  
Equivalence Class ◽  
Level Set ◽  
Equivalence Relation ◽  
Level Sets ◽  
Equivalence Classes ◽  
Fuzzy Relation ◽  
Fuzzy Relations ◽  
Transitive Relation ◽  
Fuzzy Partition

We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an ( a , b ) -level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their ( a , b ) -level sets.

scholarly journals Borel structures and Borel theories

2011 ◽  
Vol 76 (2) ◽  
pp. 461-476 ◽  
Author(s):  
Greg Hjorth ◽  
André Nies
Keyword(s):  
Equivalence Relation ◽  
Equivalence Classes ◽  
Strong Sense ◽  
Borel Equivalence Relation ◽  
And Function

AbstractWe show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.We also investigate Borel isomorphisms between Borel structures.

scholarly journals When are Two Algorithms the Same?

2009 ◽  
Vol 15 (2) ◽  
pp. 145-168 ◽  
Author(s):  
Andreas Blass ◽  
Nachum Dershowitz ◽  
Yuri Gurevich
Keyword(s):  
Equivalence Relation ◽  
Equivalence Classes ◽  
Natural Way

AbstractPeople usually regard algorithms as more abstract than the programs that implement them. The natural way to formalize this idea is that algorithms are equivalence classes of programs with respect to a suitable equivalence relation. We argue that no such equivalence relation exists.

scholarly journals On an equivalence of fuzzy subgroups III

2003 ◽  
Vol 2003 (36) ◽  
pp. 2303-2313 ◽  
Author(s):  
V. Murali ◽  
B. B. Makamba
Keyword(s):  
Equivalence Relation ◽  
Equivalence Classes ◽  
The Third ◽  
Fuzzy Subgroups

This paper is the third in a series of papers studying equivalence classes of fuzzy subgroups of a given group under a suitable equivalence relation. We introduce the notion of a pinned flag in order to study the operations sum, intersection and union, and their behavior with respect to the equivalence. Further, we investigate the extent to which a homomorphism preserves the equivalence. Whenever the equivalences are not preserved, we have provided suitable counterexamples.

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