Incidence properties of cosets in loops

Abstract

Abstract:We study incidence properties among cosets of infinite loops, with emphasis on well-structured varieties such as antiautomorphic loops and Bol loops. While cosets in groups are either disjoint or identical, we find that the incidence structure in general loops can be much richer. Every symmetric design, for example, can be realized as a canonical collection of cosets of a infinite loop. We show that in the variety of antiautomorphic loops the poset formed by set inclusion among intersections of left cosets is isomorphic to that formed by right cosets. We present an algorithm that, given a infinite Bol loop S, can in some cases determine whether |S| divides | Q| for all infinite Bol loops Q with S ≤ Q, and even whether there is a selection of left cosets of S that partitions Q. This method results in a positive confirmation of Lagrange's Theorem for Bol loops for a few new cases of subloops. Finally, we show that in a left automorphic Moufang loop Q (in particular, in a commutative Moufang loop Q), two left cosets of S ≤ Q are either disjoint or they intersect in a set whose cardinality equals that of some subloop of S. q

Sylow Theory for Quasigroups II: Sectional Action

“…for the colength of the pseudo-orbit generated by S. In this context, it is sometimes convenient to refer to the quantity |K| − 1 as the excess of a (ker c S )-class K, and to describe such a class as excessive if |K| > 1. From (13), one obtains the following. Corollary 6.3.…”

“…to the sum form (13) for co-l Q (S). Now consider progressively building up the sum (14) by taking the general elements G of M in the order that is determined by L. By Lemma 6.4, the elements of G make a total contribution of (|G|/|S|) (|S| − 1) to the sum (13). However, the only "new" contributions, that have not already arisen from elements of a subgroup H of G (which necessarily shows up earlier in L), are counted by the summand…”

“…The left multiplications are the permutations represented by the rows of the Latin square forming the body of the bordered multiplication table. The images under the left multiplications are the cosets that appeared in , but here they are described as left translates .…”