Every finite nilpotent loop has a supernilpotent loop as reduct

Abstract

A basic fact taught in undergraduate algebra courses is that every finite nilpotent group is a direct product of $p$-groups. Already Bruck observed that this does not generalize to loops. In particular, there exist nilpotent loops of size $6$ which are not direct products of loops of size $2$ and $3$. Still we show that every finite nilpotent loop $(A,\cdot )$ has a binary term operation $\ast$ such that $(A,\ast )$ is a direct product of nilpotent loops of prime power order, i.e., $(A,\ast )$ is supernilpotent. As an application we obtain that every nilpotent loop of order $pq$ for primes $p,q$ has a finite basis for its equational theory.