An Effective Criterion for Nielsen–Schreier Varieties

Abstract

Abstract To the memory of V. A. Artamonov (1946–2021) 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.