Low growth equational complexity

The equational complexity function βV : N → N of an equational class of algebras V bounds the size of equation required to determine membership of n-element algebras in V. Known examples of finitely generated varieties V 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 (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 sub-