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Low Growth Equational Complexity

Part of: Representation theory of groups Varieties Semigroups

Published online by Cambridge University Press:  25 September 2018

Marcel Jackson
Marcel Jackson*
Affiliation:
Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia (m.g.jackson@latrobe.edu.au)
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Abstract

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.

Keywords

quasivarietygroupequational complexitygrowth

MSC classification

Primary: 08B05: Equational logic, Mal′cev (Mal′tsev) conditions
Secondary: 20M07: Varieties and pseudovarieties of semigroups 20D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 197 - 210
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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