Association schemes and S-rings coming from loops and quasigroups

Kenneth Johnson
Penn State University



Content: Given any finite quasigroup $Q$ an association scheme can be automatically constructed from the "class algebra". Moreover if $Q$ is a loop (i.e there is an identity element) the association scheme may be regarded as an S-ring over $Q$ (with suitable modifications to the definition). Recently Humphries and I have been examining commutative fissions of S-rings over groups and it is a natural extension of the work to look at S-rings over loops. The family of Moufang loops retain some of the properties of groups and provide interesting examples. There is a connection between this work and the theory of random walks on the corresponding objects.

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