On the isotopic characterizations of generalized Bol loops

Simple Bol Loops

Infinite Simple Bol Loops

If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.

BOL LOOPS AND BRUCK LOOPS OF ORDER pq

Right Bol loops are loops satisfying the identity ((zx)y)x = z((xy)x), and right Bruck loops are right Bol loops satisfying the identity (xy)−1 = x−1y−1. Let p and q be odd primes such that p > q. Advancing the research program of Niederreiter and Robinson from 1981, we classify right Bol loops of order pq. When q does not divide p−1, the only right Bol loop of order pq is the cyclic group of o...

Smarandache isotopy of second Smarandache Bol loops

The pair (GH , ·) is called a special loop if (G, ·) is a loop with an arbitrary subloop (H, ·) called its special subloop. A special loop (GH , ·) is called a second Smarandache Bol loop (S2ndBL) if and only if it obeys the second Smarandache Bol identity (xs · z)s = x(sz · s) for all x, z in G and s in H. The popularly known and well studied class of loops called Bol loops fall into this clas...

The Varieties of Loops of Bol-Moufang Type

A loop identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, and the order in which the variables appear on both sides is the same, viz. ((xy)x)z = x(y(xz)). Loop varieties defined by one identity of Bol-Moufang type include groups, Bol loops, Moufang loops and C-loops. We show that there are exactly 14 such varietie...