====== Thomas Zaslavsky ======

====== Biased Graphs and the Associative Law ======

===== Abstract for the Combinatorics and Number Theory Seminar 2001 April 24 =====

A quasigroup is like a group but without the identity, inverses, or associativity; all that is left is the multiplication table, which is an arbitrary Latin square. This is worth something: one has unique solvability of equations xy=z. Also, an identity can always be found. The crucial missing property is the associative law.

A //biased graph// is a graph together with a distinguished class of circles (a.k.a. circuits, cycles, polygons) that satisfies a certain combinatorial property. Each quasigroup with //m// elements gives rise to a kind of biased graph called an //m//-fold biased expansion of //K//<sub>3</sub>, the complete graph of order 3. Conversely, every //m//-fold biased expansion of //K//<sub>3</sub> is obtained from a quasigroup. Trying to generalize this construction of quasigroups to //K<sub>n</sub>// fails to be interesting because //K//<sub>4</sub> implies the associative law, so that a biased expansion of //K<sub>n</sub>// must come from a group if //n// > 3.

We explore this fact and possible ways of getting around it. For instance, an //m//-fold biased expansion of //C<sub>n</sub>// , the circle of length //n//, encodes an //n//-dimensional Latin hypercube, which might be considered the multiplication table of an (//n//-1)-ary operation. Chords in the circle imply specific associative properties which get stronger and stronger until when all chords have been added (so the graph becomes //K<sub>n</sub>//) there is complete associativity, making the operation a group operation. What happens in between is completely unknown.