bol loop
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2021 ◽  
Vol 40 (1) ◽  
pp. 85-106
Author(s):  
T. G. Jaiyéolá ◽  
S. P. David ◽  
O. O. Oyebola

A loop (Q, ·, \, /) is called a middle Bol loop (MBL) if it obeys the identity x(yz\x)=(x/z)(y\x). To every MBL corresponds a right Bol loop (RBL) and a left Bol loop (LBL). In this paper, some new algebraic properties of a middle Bol loop are established in a different style. Some new methods of constructing a MBL by using a non-abelian group, the holomorph of a right Bol loop and a ring are described. Some equivalent necessary and sufficient conditions for a right (left) Bol loop to be a middle Bol loop are established. A RBL (MBL, LBL, MBL) is shown to be a MBL (RBL, MBL, LBL) if and only if it is a Moufang loop.


Author(s):  
Gokila vani M ◽  
Vijay Amrtha S K ◽  
Hingis S
Keyword(s):  

2017 ◽  
Vol 9 (5) ◽  
pp. 37
Author(s):  
ALPER BULUT

A left Bol loop satisfying the automorphic inverse property is called a K-loop or a gyrocommutative gyrogroup. K-loops have been in the centre of attraction since its first discovery by A.A. Ungar in the context of Einstein's 1905 relativistic theory. In this paper some of the infinite dimensional K-loops are built from the direct limit of finite dimensional group transversals.


2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Jonathan D. H. Smith

Abstract.A classical construction associates a Bruck loop with a Moufang or Bol loop on which the squaring map is a permutation. The idea of that construction is now extended to a


2012 ◽  
Vol 19 (spec01) ◽  
pp. 927-946
Author(s):  
Orin Chein ◽  
Edgar G. Goodaire ◽  
Michael Kinyon
Keyword(s):  

There are a number of identities which, if satisfied by a Bol loop, imply that the loop is actually Moufang. In this paper we show that in a number of cases, the Moufang identity is also forced not by a single identity, but by giving elements a choice of equations to satisfy.


2010 ◽  
Vol 09 (04) ◽  
pp. 591-601 ◽  
Author(s):  
TUVAL FOGUEL ◽  
MICHAEL KINYON

Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.


2010 ◽  
Vol 117 (6) ◽  
pp. 790-798 ◽  
Author(s):  
K.W. Johnson ◽  
J.D.H. Smith
Keyword(s):  

2007 ◽  
Vol 360 (05) ◽  
pp. 2393-2409 ◽  
Author(s):  
Michael K. Kinyon ◽  
J. D. Phillips ◽  
Petr Vojtěchovský
Keyword(s):  

2007 ◽  
Vol 59 (2) ◽  
pp. 296-310 ◽  
Author(s):  
Orin Chein ◽  
Edgar G. Goodaire
Keyword(s):  

AbstractCall a non-Moufang Bol loop minimally non-Moufang if every proper subloop is Moufang and minimally nonassociative if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to 1. In the process, we derive many commutator and associator identities which hold in such loops.


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