A theorem of Mumford and Ramanujam for universal algebras (Q2165630)

A universal algebra \((A;\mathbb{F})\) is called an \textit{\(MR\)-algebra} if whenever a binary term function \(m(x, y)\) in the algebra admits a two-sided identity, then the reduct \((A; m(x,y))\) is associative. It is proved that every \(MR\)-quasigroup is isotopic to a group. \(MR\)-groups are the nilpotent groups of class 2. Commutative rings and complemented lattices are \(MR\)-algebras if and only if they are Boolean.