Idempotent medialn-groupoids defined on fields

Fajtlowicz and Mycielski ([2]) showed that if we define an operationx·y=px+(1-p)x on the real numbersR for somep R, then the idempotent and medial laws form a basis for the equational theory of the groupoid (R,) if and only ifp is a transcendental number. In this paper, we generalize this ton-groupoids. Namely, if we define ann-ary operation [x₁x₂h.x_n]=₁x₁ + ₂x₂+h.+(1-₁-h._n–1) onR for some ₁, ₂, h., _n–1 in R, then the idempotent and medial laws form a basis for the equational theory of then-groupoid (R, [ ]) if and only if ₁, ₂,..., _n–1 are algebraically independent.This is a part of Ph.D. dissertation developed under the direction of professor Trevor Evans while the author was at Emory University.