The Burnside algebra of a quasigroup

Abstract

The paper extends the concept of a Burnside algebra from finite groups to finite quasigroups, based on an earlier specification of quasigroup permutation actions as members of a certain covariety of coalgebras. For quasigroups, the Burnside algebra carries two distinct product operations, denoted respectively as “(direct) product” and “reduced tensor product.” The direct product arises from categorical products in the covariety, while the reduced tensor product arises from bisimulations. In the associative case, the direct product and reduced tensor product agree. With respect to the direct product, the Burnside algebra of a quasigroup is semisimple, and its structure may be described in terms of marks, exactly generalizing the associative case. With respect to the reduced tensor product, nilpotent elements may exist. The problem of finding a primitive set of idempotents for the semisimple part of the reduced tensor product reduct of the Burnside algebra is raised. As a first step towards the solution of this problem, the primitive reduced tensor idempotent multiple of the regular representation is identified in terms of the orbitals of the action of the right multiplication group. There is a characterization of those quasigroups for which this multiple is the same as in the group case.