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Low Growth Equational Complexity
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 62 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 25 September 2018, pp. 197-210
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- Article
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The 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.
bounds the size of equation required to determine the membership of 
. Known examples of finitely generated varieties 
with unbounded equational complexity have growth in Ω(
