**Problem of the Week**

**Math Club**

**BUGCAT 2020**

**Zassenhaus Conference**

**Hilton Memorial Lecture**

seminars:comb:abstract.200104zas

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*_{3}, the complete graph of order 3. Conversely, every *m*-fold biased expansion of *K*_{3} is obtained from a quasigroup. Trying to generalize this construction of quasigroups to *K _{n}* fails to be interesting because

We explore this fact and possible ways of getting around it. For instance, an *m*-fold biased expansion of *C _{n}* , the circle of length

seminars/comb/abstract.200104zas.txt · Last modified: 2020/01/29 14:03 (external edit)

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Noncommercial-Share Alike 3.0 Unported