  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.
