Associativity in multiary quasigroups: the way of biased expansions

A multiary (polyadic, n-ary) quasigroup is an n-ary operation which is invertible with respect to each of its variables. A biased expansion of a graph is a kind of branched covering graph with an additional structure similar to the combinatorial homotopy of circles. A biased expansion of a circle with chords encodes a multiary quasigroup, the chords corresponding to factorizations, i.e., associative structure. Some but not all biased expansions are constructed from groups (group expansions); these include all biased expansions of complete graphs (with at least four nodes), which correspond to Dowling’s lattices of a group and encode an iterated group operation. We show that any biased expansion of a 3-connected graph (with at least four nodes) is a group expansion, and that all 2-connected biased expansions are constructed by the identification of edges from group expansions and irreducible multiary quasigroups. If a 2-connected biased expansion covers every base edge at most three times, or if every four-node minor that contains a fixed edge is a group expansion, then the whole biased expansion is a group expansion. We deduce that if a multiary quasigroup has a factorization graph that is 3-connected, or if every ternary principal retract is an iterated group isotope, it is isotopic to an iterated group. We mention applications of generalizing Dowling geometries and of transversal designs of high strength.