Bounding the free spectrum of nilpotent algebras of prime power order

TL;DR

This paper establishes a bound on the degree of the polynomial that limits the size of algebras generated by a finite nilpotent algebra of prime power order within a congruence modular variety.

Contribution

It provides a specific bound on the polynomial degree governing the size of generated algebras, extending known exponential bounds.

Findings

Bound on polynomial degree for algebra size growth

Polynomial degree depends on algebra's structure

Enhances understanding of algebraic growth constraints

Abstract

Let $\mathbf{A}$ be a finite nilpotent algebra in a congruence modular variety with finitely many fundamental operations. If $\mathbf{A}$ is of prime power order, then it is known that there is a polynomial $p$ such that for every $n \in \mathbb{N}$, every $n$-generated algebra in the variety generated by $\mathbf{A}$ has at most $2^{p(n)}$ elements. We present a bound on the degree of this polynomial.