Vaughan--Lee's nilpotent loop of size $12$ is finitely based

TL;DR

This paper proves that Vaughan--Lee's specific nilpotent loop of size 12 has a finite basis for its equational theory, expanding understanding of finite basis problems in nilpotent loops.

Contribution

The paper provides a finite basis for Vaughan--Lee's non-prime-power order nilpotent loop and characterizes its term functions, addressing an open question.

Findings

Finite basis established for the loop of order 12.

Subpower membership problem solved in polynomial time.

Explicit characterization of the loop's term functions.

Abstract

In 1983 Vaughan--Lee showed that if a finite nilpotent loop splits into a direct product of factors of prime power order, then its equational theory has a finite basis. Whether the condition on the direct decomposition is necessary has remained open since. In the same paper, Vaughan--Lee gives an explicit example of a nilpotent loop of order $12$ that does not factor into loops of prime power order and asks whether it is finitely based. We give a finite basis for his example by explicitly characterizing its term functions. This also allows us to show that the subpower membership problem for this loop can be solved in polynomial time.