On loop identities that can be obtained by a nuclear identification

We start by describing all the varieties of loops Q that can be defined by autotopisms x, x Q, where x is a composition of two triples, each of which becomes an autotopism when the element x belongs to one of the nuclei. In this way we obtain a unifying approach to Bol, Moufang, extra, Buchsteiner and conjugacy closed loops. We reprove some classical facts in a new way and show how Buchsteiner loops fit into the traditional context. In Section 6 we describe a new class of loops with coincinding left and right nuclei. These loops have remarkable properties and do not belong to any of the classical classes. We start this paper by investigating interactions of loop nuclei and loop identities via loop autotopisms. We shall observe that this is a natural way how to obtain nearly all loop varieties that have been studied in the past. To be more exact, we shall get in this way Moufang loops, left Bol loops, right Bol loops, extra loops, left conjugacy closed (LCC) loops, RCC loops and Buchst...

• Type: Journal
• Year: 2010
• Where: EJC
• Authors: Ales Drápal, Premysl Jedlicka
• Keywords: EJC 2007, EJC 2010, Loop, Right Nucleus, Unifying Approach