Mathematical Structures Defined by Identities II

TL;DR

This paper refines the asymptotic estimate for the number of algebras satisfying a specific irreducible identity, improving previous bounds by incorporating additional combinatorial tableaux series.

Contribution

It improves the asymptotic bound for the count of such algebras by utilizing an extended combinatorial approach with additional tableaux series.

Findings

Refined asymptotic bound for algebra count

Utilized additional tableaux series for sharper estimates

Enhanced understanding of algebra structures defined by identities

Abstract

In our paper arXiv: math.RA/0110333 v1 Oct 2001 we showed that the number of algebras defined by a binary operation satisfying a formally irreducible identity between two n-iterates is O( e^{-n/16}S_{n}^{2} for n --> infinity, S_{n} being the nth-Catalan number. This was proved by using exclusively the series of tableaux A_{n}. By using also the series of tableaux B_{n}, we now sharpen this result to O{(n+2)/n|e^{-n/16}-2/n)|S_{n}^{2}}