Every finite nilpotent loop has a supernilpotent loop as reduct

TL;DR

The paper proves that every finite nilpotent loop has a supernilpotent loop reduct, using a constructive induction along a central series to refine quotients into coprime prime-power components. This yields a binary operation and compatible structure whose direct product is supernilpotent, enabling finite bases for equational theories in cases such as nilpotent loops of order pq. The work bridges loop theory with higher commutators and Mal'cev frameworks, and it clarifies limitations by showing that the main result does not extend to all loops or necessarily produce abelian reducts in the group setting. Overall, it provides a concrete route from finite nilpotence to supernilpotent reducts and offers concrete applications to equational bases and Mal'cev algebras.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm:loop']}
• proof : Proof of Corollary \ref{['cor:loop']}
• Lemma 3.1
• proof
• Theorem 3.2