Left orders in Garside groups

2016 ◽  
Vol 26 (07) ◽  
pp. 1349-1359 ◽  
Author(s):  
Fabienne Chouraqui
Keyword(s):  
Cantor Set ◽  
Total Order ◽  
Theoretical Solution ◽  
Structure Group ◽  
Baxter Equation ◽  
Unique Product ◽  
Bieberbach Group ◽  
Left Order ◽  
Left And Right

We consider the structure group of a non-degenerate symmetric (non-trivial) set-theoretical solution of the quantum Yang–Baxter equation. This is a Bieberbach group and also a Garside group. We show this group is not bi-orderable, that is it does not admit a total order which is invariant under left and right multiplications. Regarding the existence of a left invariant total ordering, there is a great diversity. There exist structure groups with a recurrent left order and with space of left orders homeomorphic to the Cantor set, while there exist others that are even not unique product groups.

scholarly journals Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces

2019 ◽  
Vol 30 (01) ◽  
pp. 91-115
Author(s):  
E. Acri ◽  
R. Lutowski ◽  
L. Vendramin
Keyword(s):  
Short Proof ◽  
Linear Representation ◽  
Structure Group ◽  
Baxter Equation ◽  
Product Property ◽  
Unique Product ◽  
Bieberbach Group

Using Bieberbach groups, we study multipermutation involutive solutions to the Yang–Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right [Formula: see text]-nilpotent skew braces. The theory of left [Formula: see text]-nilpotent skew braces is also developed and used to give a short proof of a theorem of Smoktunowicz in the context of skew braces.

scholarly journals On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation

2019 ◽  
Vol 62 (3) ◽  
pp. 683-717 ◽  
Author(s):  
Victoria Lebed ◽  
Leandro Vendramin
Keyword(s):  
Structure Group ◽  
Baxter Equation ◽  
Finite Quotient ◽  
Torsion Free

AbstractThis paper explores the structure groups G(X,r) of finite non-degenerate set-theoretic solutions (X,r) to the Yang–Baxter equation. Namely, we construct a finite quotient $\overline {G}_{(X,r)}$ of G(X,r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if X injects into G(X,r), then it also injects into $\overline {G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of G(X,r). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free.

scholarly journals COMPACTNESS OF THE SPACE OF LEFT ORDERS

2007 ◽  
Vol 16 (03) ◽  
pp. 257-266 ◽  
Author(s):  
M. A. DABKOWSKA ◽  
M. K. DABKOWSKI ◽  
V. S. HARIZANOV ◽  
J. H. PRZYTYCKI ◽  
M. A. VEVE
Keyword(s):  
Topological Space ◽  
Knot Theory ◽  
Total Order ◽  
Topological Properties ◽  
Natural Topology ◽  
Interesting Connection ◽  
Algebraic Result ◽  
Left Order

A left order on a magma (e.g. semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and Łoś [13] concerning the existence of left orders.

The three-dimensional structure of frozen-hydrated left- and right-handed forms of bacterial flagellar filaments from an identical serotype

1986 ◽  
Vol 44 ◽  
pp. 298-299
Author(s):  
S. Trachtenberg ◽  
D. J. DeRosier
Keyword(s):  
Ionic Strength ◽  
Basal Body ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Protein Species ◽  
Liquid Environment ◽  
Bacterial Flagellum ◽  
Left And Right ◽  
Rotary Motor ◽  
Helical Parameters

The bacterial cell is propelled through the liquid environment by means of one or more rotating flagella. The bacterial flagellum is composed of a basal body (rotary motor), hook (universal coupler), and filament (propellor). The filament is a rigid helical assembly of only one protein species — flagellin. The filament can adopt different morphologies and change, reversibly, its helical parameters (pitch and hand) as a function of mechanical stress and chemical changes (pH, ionic strength) in the environment.

Advantages of Complementary Stereo Images as Viewed with the Low-Temperature Field-Emission SEM

1992 ◽  
Vol 50 (2) ◽  
pp. 1314-1315
Author(s):  
William P. Wergin ◽  
Eric F. Erbe
Keyword(s):  
Fold Increase ◽  
Horizontal Displacement ◽  
Three Dimensions ◽  
Stereo Image ◽  
Stereo Pair ◽  
Sem Micrograph ◽  
Left And Right ◽  
The Common ◽  
The Brain ◽  
Pair Analysis

The eye-brain complex allows those of us with normal vision to perceive and evaluate our surroundings in three-dimensions (3-D). The principle factor that makes this possible is parallax - the horizontal displacement of objects that results from the independent views that the left and right eyes detect and simultaneously transmit to the brain for superimposition. The common SEM micrograph is a 2-D representation of a 3-D specimen. Depriving the brain of the 3-D view can lead to erroneous conclusions about the relative sizes, positions and convergence of structures within a specimen. In addition, Walter has suggested that the stereo image contains information equivalent to a two-fold increase in magnification over that found in a 2-D image. Because of these factors, stereo pair analysis should be routinely employed when studying specimens.Imaging complementary faces of a fractured specimen is a second method by which the topography of a specimen can be more accurately evaluated.

Histological evaluation of cochlear hair cell damage from noise-induced hearing loss in chinchillas

1990 ◽  
Vol 48 (3) ◽  
pp. 332-333
Author(s):  
R.V. Harrison ◽  
R.J. Mount ◽  
P. White ◽  
N. Fukushima
Keyword(s):  
Hair Cell ◽  
Evoked Potential ◽  
Cell Damage ◽  
Acoustic Trauma ◽  
Histological Evaluation ◽  
Cell Integrity ◽  
Functional Changes ◽  
Cochlear Hair Cell ◽  
Left And Right ◽  
Contralateral Ear

In studies which attempt to define the influence of various factors on recovery of hair cell integrity after acoustic trauma, an experimental and a control ear which initially have equal degrees of damage are required. With in a group of animals receiving an identical level of acoustic trauma there is more symmetry between the ears of each individual, in respect to function, than between animals. Figure 1 illustrates this, left and right cochlear evoked potential (CAP) audiograms are shown for two chinchillas receiving identical trauma. For this reason the contralateral ear is used as control.To compliment such functional evaluations we have devised a scoring system, based on the condition of hair cell stereocilia as revealed by scanning electron microscopy, which permits total stereociliar damage to be expressed numerically. This quantification permits correlation of the degree of structural pathology with functional changes. In this paper wereport experiments to verify the symmetry of stereociliar integrity between two ears, both for normal (non-exposed) animals and chinchillas in which each ear has received identical noise trauma.

The crystallography characteristic of (Mn,Fe)S single crystal

1990 ◽  
Vol 48 (4) ◽  
pp. 898-899
Author(s):  
Jiang Xishan
Keyword(s):  
Single Crystal ◽  
Cast Steel ◽  
Point Group ◽  
Central Area ◽  
Hexagonal Crystal ◽  
Growth Step ◽  
Left And Right ◽  
Relationship Of ◽  
Fcc Structure ◽  
New Discovery

This paper reports the growth step pattern and morphology at equilibrium and growth states of (Mn,Fe)S single crystal on the wall of micro-voids in ZG25 cast steel by using scanning electron microscope. Seldom report was presented on the growth morphology and steppattern of (Mn,Fe)S single crystal.Fig.1 shows the front half of the polyhedron of(Mn,Fe)S single crystal,its central area being the square crystal plane,the two pairs of hexagons symmetrically located in the high and low, the left and right with a certain, angle to the square crystal plane.According to the symmetrical relationship of crystal, it was defined that the (Mn,Fe)S single crystal at equilibrium state is tetrakaidecahedron consisted of eight hexagonal crystal planes and six square crystal planes. The macroscopic symmetry elements of the tetrakaidecahedron correpond to Oh—n3m symmetry class of fcc structure,in which the hexagonal crystal planes are the { 111 } crystal planes group,square crystal plaits are the { 100 } crystal planes group. This new discovery of the (Mn,Fe)S single crystal provides a typical example of the point group of Oh—n3m.

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