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Quasigroup actions: Markov chains, pseudoinverses, and linear representations. (English) Zbl 0944.20059

This paper initiates the extension to quasigroups of a further aspect of group representation theory: transitive permutation representations. Using pseudoinverses of incidence matrices of quasigroups in partitions induced by left multiplications of subquasigroups, a transitive permutation action of a quasigroup is defined as a set of Markov chain actions indexed by the quasigroup. Finally, Burnside’s lemma for transitive quasigroup actions is derived.

MSC:

20N05 Loops, quasigroups
20C15 Ordinary representations and characters
20B05 General theory for finite permutation groups
20M30 Representation of semigroups; actions of semigroups on sets
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