Associativity in multary quasigroups: The way of biased expansions
  A "biased expansion" of a graph is a kind of branched covering graph with additional structure related to combinatorial homotopy of circles. Some but not all biased expansions are constructed from groups ("group expansions"); these include all biased expansions of complete graphs (assuming order at least four), which correspond to Dowling's lattices of a group and encode an iterated group operation. A biased expansion of a circle with chords encodes a multary (polyadic, n-ary) quasigroup, the chords corresponding to factorizations, i.e., associative structure. We show that any biased expansion of a 3-connected graph (of order at least four) is a group expansion, and that all 2-connected biased expansions are constructed by expanded edge amalgamation from group expansions and irreducible multary quasigroups. If a 2-connected biased expansion covers every base edge at most three times, or if every four-node minor is a group expansion, then the whole biased expansion is a group expansion. In particular, if a multary quasigroup has a factorization graph that is 3-connected, if it has order 3, or if every residual ternary quasigroup is an iterated group isotope, it is isotopic to an iterated group. We mention applications to generalizing Dowling geometries and to transversal designs of high strength. 
Thomas Zaslavsky