Quasigroup Associativity and Biased Expansion Graphs

We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We mention applications to transversal designs and generalized Dowling geometries. 1. Associativity in multary quasigroups A multary quasigroup is a set with an n-ary operation for some finite n ≥ 2, say f : Q → Q, such that the equation f(x1, x2, . . . , xn) = x0 is uniquely solvable for any one variable given the values of the other n variables. An (associative) factorization is an expression f(x1, . . . , xn) = g(x1, . . . , xi, h(xi+1, . . . , xj), . . . , xn) (1) where g and h are multary quasigroup operations. For instance, if f is constructed by iterating a group operation, f(x1, . . . , xn) = x1 · x2 · · · · · xn, then it has every possible factorization. We study the degree to which an arbitrary multary quasigroup with some known factorizations is an iterated group. We employ a new method, the structural analysis of biased expansion graphs. An operation may be disguised by isotopy, which means relabelling each variable separately; or by conjugation, which means permuting the variables. Precisely, we call operations f and f ′ isotopic if there exist bijections αi : Q→ Q such that f (x1, . . . , xn) α0 = f(x1 1 , . . . , x αn n ); we call them circularly conjugate if x0 = f (x1, . . . , xn) ⇐⇒ xi = f(xi+1, . . . , xn, x0, x1, . . . , xi−1) or x0 = f (x1, . . . , xn) ⇐⇒ xi = f(xi−1, . . . , x1, x0, xn, . . . , xi+1) for some i = 0, 1, . . . , n. Neither isotopy nor circular conjugation affects the existence of factorizations. The exact factorization formulas may change under circular conjugation, but the factorizations of f and f ′ correspond. If a ternary quasigroup factors in both possible ways, f(x1, x2, x3) = g1(h1(x1, x2), x3) = g2(x1, h2(x2, x3)) (2) Date: May, 2004. Version of February 18, 2006. 2000 Mathematics Subject Classification. Primary 05C22, 20N05, Secondary 05B15, 05B35.