scholarly journals Basarab loop and the generators of its total multiplication group

2021 ◽  
Author(s):  
T. G. Jaiyeola ◽  
Gideon Effiong
Keyword(s):  
Partial Answer ◽  
Multiplication Group ◽  
Regular Mapping ◽  
Inner Mapping Group ◽  
Group A ◽  
Left And Right ◽  
Mapping Group

A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop

scholarly journals Basarab loop and the generators of its total multiplication group

2021 ◽  
Author(s):  
Temitope Jaiyéolá ◽  
Gideon Effiong
Keyword(s):  
Partial Answer ◽  
Multiplication Group ◽  
Regular Mapping ◽  
Inner Mapping Group ◽  
Group A ◽  
Left And Right ◽  
Mapping Group

A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop.

scholarly journals On finite loops whose inner mapping groups are Abelian

2002 ◽  
Vol 65 (3) ◽  
pp. 477-484 ◽  
Author(s):  
Markku Niemenmaa
Keyword(s):  
Solvable Group ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Special Structure ◽  
Finite Abelian Groups ◽  
Multiplication Group ◽  
Nonassociative Algebras ◽  
Inner Mapping Group ◽  
Mapping Group

Loops are nonassociative algebras which can be investigated by using their multiplication groups and inner mapping groups If the inner mapping group of a loop is finite and Abelian, then the multiplication group is a solvable group. It is clear that not all finite Abelian groups can occur as inner mapping groups of loops. In this paper we show that certain finite Abelian groups with a special structure are not isomorphic to inner mapping groups of finite loops. We use our results and show how to construct solvable groups which are not isomorphic to multiplication groups of loops.

scholarly journals MULTIPLICATION GROUPS AND INNER MAPPING GROUPS OF CAYLEY–DICKSON LOOPS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350078
Author(s):  
JENYA KIRSHTEIN
Keyword(s):  
Semidirect Product ◽  
Complex Numbers ◽  
Real Numbers ◽  
Multiplication Group ◽  
Inner Mapping Group ◽  
Mapping Group

The Cayley–Dickson loop Qn is the multiplicative closure of basic elements of the algebra constructed by n applications of the Cayley–Dickson doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). We establish that the inner mapping group Inn (Qn) is an elementary abelian 2-group of order 22n-2 and describe the multiplication group Mlt (Qn) as a semidirect product of Inn (Qn) × ℤ2 and an elementary abelian 2-group of order 2n. We prove that one-sided inner mapping groups Inn l(Qn) and Inn r(Qn) are equal, elementary abelian 2-groups of order 22n-1-1. We establish that one-sided multiplication groups Mlt l(Qn) and Mlt r(Qn) are isomorphic, and show that Mlt l(Qn) is a semidirect product of Inn l(Qn) × ℤ2 and an elementary abelian 2-group of order 2n.

scholarly journals Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

2013 ◽  
Vol 1 ◽  
pp. 232-254 ◽  
Author(s):  
Zoltán M. Balogh ◽  
Jeremy T. Tyson ◽  
Kevin Wildrick
Keyword(s):  
Hausdorff Dimension ◽  
Heisenberg Group ◽  
Metric Space ◽  
Metric Spaces ◽  
Higher Dimension ◽  
Metric Measure Spaces ◽  
Regular Mapping ◽  
Sobolev Mappings ◽  
Measure Spaces ◽  
Left And Right

Abstract We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

scholarly journals Abelian groups as inner mapping groups of loops

2004 ◽  
Vol 70 (3) ◽  
pp. 481-488 ◽  
Author(s):  
Asif Ali ◽  
John Cossey
Keyword(s):  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Inner Mapping Group ◽  
Mapping Group

The question of which Abelian groups can be the inner mapping group of a loop has been considered by Niemenmaa, Kepka and others. We give a construction which shows that many finite Abelian groups can be the inner mapping group of a loop.

Finite classical groups and multiplication groups of loops

1995 ◽  
Vol 117 (3) ◽  
pp. 425-429 ◽  
Author(s):  
Ari Vesanen
Keyword(s):  
Permutation Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classical Groups ◽  
Alternating Groups ◽  
Multiplication Group ◽  
Finite Classical Groups ◽  
Left And Right

Let Q be a loop; then the left and right translations La(x) = ax and Ra(x) = xa are permutations of Q. The permutation group M(Q) = 〈La, Ra | a ε Q〉 is called the multiplication group of Q; it is well known that the structure of M(Q) reflects strongly the structure of Q (cf. [1] and [8], for example). It is thus an interesting question, which groups can be represented as multiplication groups of loops. In particular, it seems important to classify the finite simple groups that are multiplication groups of loops. In [3] it was proved that the alternating groups An are multiplication groups of loops, whenever n ≥ 6; in this paper we consider the finite classical groups and prove the following theorems

scholarly journals Can QRS morphology be used to differentiate between true septal vs. apparently septal lead placement? An analysis of ECG of real mid-septal, apparent mid-septal, and apical pacing

2020 ◽  
Vol 22 (Supplement_F) ◽  
pp. F14-F22
Author(s):  
Anna Mala ◽  
Pavel Osmancik ◽  
Dalibor Herman ◽  
Karol Curila ◽  
Petr Stros ◽  
...  
Keyword(s):  
Transition Zone ◽  
Anterior Wall ◽  
Lead Placement ◽  
Group A ◽  
Qrs Morphology ◽  
Exact Position ◽  
Left And Right ◽  
The Right ◽  
Lead Location ◽  
Ecg Criteria

Abstract The location of the pacemaker lead is based on the shape of the lead on fluoroscopy only, typically in the left and right anterior oblique positions. However, these fluoroscopy criteria are insufficient and many leads apparently considered to be in septum are in fact anchored in anterior wall. Periprocedural ECG could determine the correct lead location. The aim of the current analysis is to characterize ECG criteria associated with a correct position of the right ventricular (RV) lead in the mid-septum. Patients with indications for a pacemaker had the RV lead implanted in the apex (Group A) or mid-septum using the standard fluoroscopic criteria. The exact position of the RV lead was verified using computed tomography. Based on the findings, the mid-septal group was divided into two subgroups: (i) true septum, i.e. lead was found in the mid-septum, and (ii) false septum, i.e. lead was in the adjacent areas (anterior wall, anteroseptal groove). Paced ECGs were acquired from all patients and multiple criteria were analysed. Paced ECGs from 106 patients were analysed (27 in A, 36 in true septum, and 43 in false septum group). Group A had a significantly wider QRS, more left-deviated axis and later transition zone compared with the true septum and false septum groups. There were no differences in presence of q in lead I, or notching in inferior or lateral leads between the three groups. QRS patterns of true septum and false septum groups were similar with only one exception of the transition zone. In the multivariate model, the only ECG parameters associated with correct lead placement in the septum was an earlier transition zone (odds ratio (OR) 2.53, P = 0.001). ECGs can be easily used to differentiate apical pacing from septal or septum-close pacing. The only ECG characteristic that could help to identify true septum lead position was the transition zone in the precordial leads. ClinicalTrials.gov identifier: NCT02412176.

3D Assessment of Nasolabial Appearance in Patients With Complete Unilateral Cleft Lip and Palate

2017 ◽  
Vol 55 (2) ◽  
pp. 220-225 ◽  
Author(s):  
Ieva Bagante ◽  
Inta Zepa ◽  
Ilze Akota
Keyword(s):  
Cleft Lip ◽  
Cleft Lip And Palate ◽  
Cross Sectional Study ◽  
Control Group ◽  
Cross Sectional ◽  
3 Dimensional ◽  
Significant Difference ◽  
Group A ◽  
Left And Right ◽  
The Difference

Objective: Rhinoplasty in patients with complete unilateral cleft lip and palate (UCLP) is challenging, and the surgical outcome of the nose is complicated to evaluate. The aim of this study was to assess the nasolabial appearance of patients with UCLP compared with a control group. Design: Cross-sectional study. Setting: Riga Cleft Lip and Palate Centre, Latvia. Participants: All consecutive 35 patients born between 1994 and 2004 with nonsyndromic complete UCLP were included. Of 35 patients, 29 came for checkup; the mean age was 14.7 years (range 10-18). In the control group, 35 noncleft participants at 10 years of age were included. Interventions: Nasolabial appearance was evaluated from 3-dimensional images using a 3-dimensional stereo-photogrammetric camera setup (3dMDface System), the results being analysed statistically. Results: In UCLP group, a statistically significant difference between cleft and noncleft side was found only in alar wing length ( P < .05). The difference of nasolabial anthropometric distances in the control group between the left and right side was not significant. The difference between the UCLP group and the control group was significant in all anthropometric distances except the lateral lip length to cupid’s bow. Conclusions: The nasolabial appearance with acceptable symmetry after cleft lip and reconstructive surgery of the nose was achieved. Symmetry of the nasolabial appearance in patients with UCLP differed from those in the control group. The 3D photographs with a proposed set of anthropometric landmarks for evaluation of nasolabial appearance seems to be a convenient, accurate, and noninvasive way to follow and evaluate patients after surgery.

On the Structure of Half-Groups

1959 ◽  
Vol 11 ◽  
pp. 651-659
Author(s):  
James V. Whittaker
Keyword(s):  
Structure Theorem ◽  
Group A ◽  
Left And Right

Furstenberg (1) and the author (4), among others, have exhibited postulate systems for groups in terms of the operation x — y = x + ( — y).Furstenberg also investigated a system obtained by removing one of his postulates which defines what he called a half-group. A structure theorem for half-groups was given in (1). In the present paper, we prove another structure theorem for half-groups. A more restricted entity, called a pseudo-group, is introduced, and its structure, together with that of a half-group satisfying the left and right cancellation laws, is studied. Finally, some topological questions concerning their structure are also considered.

scholarly journals Buchsteiner loops: Associators and constructions

2015 ◽  
Vol 14 (04) ◽  
pp. 1550050
Author(s):  
Aleš Drápal ◽  
Michael Kinyon
Keyword(s):  
Nilpotency Class ◽  
General Construction ◽  
Inner Mapping Group ◽  
Mapping Group ◽  
Q 64

Let Q be a Buchsteiner loop. We describe the associator calculus in three variables, and show that |Q| ≥ 32 if Q is not conjugacy closed. We also show that |Q| ≥ 64 if there exists x ∈ Q such that x2 is not in the nucleus of Q. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that has both nilpotency class 3 and an abelian inner mapping group.

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