Low growth equational complexity

The equational complexity function ${\beta }_{\mathcal{V}}:\mathbb{N}\to \mathbb{N}$ of an equational class of algebras $\mathcal{V}$ bounds the size of equation required to determine membership of $n$-element algebras in $\mathcal{V}$. Known examples of finitely generated varieties $\mathcal{V}$ with unbounded equational complexity have growth in $\mathrm{\Omega }(n^c)$, usually for $c\ge \frac{1}{2}$. 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.