Idempotent medial \(n\)-groupoids defined on fields
\textit{S. Fajtlowicz} and \textit{J. Mycielski} [ibid. 4, 244-249 (1974; Zbl 0307.12002)] showed that if we define an operation \(x\cdot y=px+(1-p)x\) on the real numbers \({\mathbb{R}}\) for some \(p\in {\mathbb{R}}\), then the idempotent and medial laws form a basis for the equational theory of the groupoid (\({\mathbb{R}},\cdot)\) if and only if \(p\) is a transcendental number. We generalize this to \(n\)-groupoids. Namely, if we define an \(n\)-ary operation \[ [x_ 1x_ 2...x_ n]=\alpha_ 1x_ 1+\alpha_ 2x_ 2+...+(1-\alpha_ 1-...-\alpha_{n-1}) \] on \({\mathbb{R}}\) for some \(\alpha_ 1,\alpha_ 2,...,\alpha_{n-1}\) in \({\mathbb{R}}\), then the idempotent and medial laws form a basis for the equational theory of the \(n\)-groupoid \(({\mathbb{R}},[.])\) if and only if \(\alpha_ 1,\alpha_ 2,...,\alpha_{n-1}\) are algebraically independent.
scientific article; zbMATH DE number 4087738