Incidence properties of cosets in loops

Abstract

We study incidence properties among cosets of finite 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 finite 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 finite Bol loop $S$, can in some cases determine whether $|S|$ divides $|Q|$ for all finite Bol loops $Q$ with $S \le Q$, 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 $S\le Q$ are either disjoint or they intersect in a set whose cardinality equals that of some subloop of $S$.