Low Growth Equational Complexity

Abstract

AbstractThe equational complexity function $\beta \nu \,:\,{\open N} \to {\open N}$ of an equational class of algebras bounds the size of equation required to determine the membership of n-element algebras in . Known examples of finitely generated varieties with unbounded equational complexity have growth in Ω(nc), usually for c ≥ (1/2). We show that much slower growth is possible, exhibiting $O(\log_{2}^{3}(n))$ growth among 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.