Permutation representations of left quasigroups

The concept of a permutation representation has recently been extended from groups to quasigroups. Following a suggestion of Walter Taylor, the concept is now further extended to left quasigroups. The paper surveys the current state of the theory, giving new proofs where necessary to cover the general case of left quasigroups. Both the Burnside Lemma and the Burnside algebra appear in this new context. This paper is dedicated to Walter Taylor. Received August 9, 2005; accepted in final form March 7, 2006.