Low growth equational complexity

TL;DR

This paper demonstrates that certain algebraic varieties can have very slow, polylogarithmic growth in their equational complexity, contrasting with previously known examples of faster growth.

Contribution

It introduces examples of algebraic varieties with logarithmic and polylogarithmic equational complexity growth, expanding understanding of complexity bounds.

Findings

Varieties of semilattice ordered inverse semigroups have O(log^3 n) complexity.

Additive idempotent semirings also exhibit similar slow growth.

Finite groups have polylogarithmic quasi-equational complexity if and only if all Sylow subgroups are abelian.

Abstract

The equational complexity function $\beta_\mathscr{V}:\mathbb{N}\to\mathbb{N}$ of an equational class of algebras $\mathscr{V}$ bounds the size of equation required to determine membership of $n$-element algebras in $\mathscr{V}$. Known examples of finitely generated varieties $\mathscr{V}$ with unbounded equational complexity have growth in $\Omega(n^c)$, usually for $c\geq \frac{1}{2}$. We show that much slower growth is possible, exhibiting $O(\log_2^3(n))$ growth amongst varieties of semilattice ordered inverse semigroups and additive idempotent semirings. We also examine a quasivariety analogue of equational complexity, and show that a finite group has polylogarithmic quasi-equational complexity function, bounded if and only if all Sylow subgroups are abelian.