Every finite nilpotent loop has a supernilpotent loop as reduct

TL;DR

This paper proves that every finite nilpotent loop can be reduced to a supernilpotent loop with a specific binary operation, leading to finite bases for their equational theories in certain cases.

Contribution

It introduces a binary operation transforming finite nilpotent loops into supernilpotent loops, extending structural understanding beyond groups.

Findings

Existence of a binary operation making nilpotent loops supernilpotent.

Finite basis for equational theories of certain nilpotent loops.

Counterexample to direct product decomposition in loops.

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 $*$ such that $(A,*)$ is a direct product of nilpotent loops of prime power order, i.e., $(A,*)$ 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.