An Effective Criterion for Nielsen–Schreier Varieties

DOI10.1093/IMRN/RNAD092arXiv2205.05364OpenAlexW4306828983MaRDI QIDQ6182667

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Publication date: 25 January 2024

Published in: Unnamed Author (Search for Journal in Brave)

Abstract: All algebras of a certain type are said to form a Nielsen-Schreier variety if every subalgebra of a free algebra is free. This property has been perceived as extremely rare; in particular, only six Nielsen-Schreier varieties of algebras with one binary operation have been discovered in prior work on this topic. We propose an effective combinatorial criterion for the Nielsen-Schreier property in the case of algebras over a field of zero characteristic; in our approach, operads play a crucial role. Using this criterion, we show that the well known varieties of all pre-Lie algebras and of all Lie-admissible algebras are Nielsen-Schreier, and, quite surprisingly, that there are already infinitely many non-equivalent Nielsen-Schreier varieties of algebras with one binary operation and identities of degree four.

Full work available at URL: https://arxiv.org/abs/2205.05364

Mathematics Subject Classification ID

Mathematical logic and foundations (03-XX) Algebraic geometry (14-XX)